On an Inverse Problem of Reconstructing a Heat Conduction ...AdvancesinMathematicalPhysics 5....
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Research Article On an Inverse Problem of Reconstructing a Heat Conduction Process from Nonlocal Data Makhmud A. Sadybekov , 1 Gulnar Dildabek, 1,2 and Marina B. Ivanova 1,3 1 Institute of Mathematics and Mathematical Modeling, 125 Pushkin Str., 050010 Almaty, Kazakhstan 2 Al-Farabi Kazakh National University, 71 Al-Farabi Ave., 050040 Almaty, Kazakhstan 3 South-Kazakhstan State Pharmaceutical Academy, 1 al-Farabi Sq., 160019 Shymkent, Kazakhstan Correspondence should be addressed to Makhmud A. Sadybekov; [email protected] Received 28 February 2018; Accepted 21 May 2018; Published 12 June 2018 Academic Editor: Pavel Kurasov Copyright © 2018 Makhmud A. Sadybekov et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We consider an inverse problem for a one-dimensional heat equation with involution and with periodic boundary conditions with respect to a space variable. is problem simulates the process of heat propagation in a thin closed wire wrapped around a weakly permeable insulation. e inverse problem consists in the restoration (simultaneously with the solution) of an unknown right-hand side of the equation, which depends only on the spatial variable. e conditions for redefinition are initial and final states. Existence and uniqueness results for the given problem are obtained via the method of separation of variables. 1. Introduction and Statement of the Problem e problems that imply the determination of coefficients or the right-hand side of a differential equation (together with its solution) are commonly referred to as inverse problems of mathematical physics. In this paper we consider one family of problems implying the determination of the density distribution and of heat sources from given values of initial and final distributions. is problem simulates the process of heat propagation in a thin closed wire wrapped around a weakly permeable insulation. e mathematical statement of such problems leads to an inverse problem for the heat equation, where it is required to find not only a solution of the problem, but also its right-hand side that depends only on a spatial variable. In this paper, we will consider the inverse problem close to that investigated in [1, 2]. Together with the solution it is necessary to find an unknown right-hand side of the equation. e equation contains the usual time derivative and an involution with respect to the spatial variable. In contrast to [1], we investigate the problem under nonlocal boundary conditions with respect to the spatial variable. e conditions for overdetermination are initial and final states. e second of the main differences in the investigated inverse problem being studied is that the unknown function enters, both in the right-hand side of the equation and in the conditions of the initial and final overdetermination. Let us consider a problem of modeling the thermal diffusion process which is close to that described in the report of Cabada and Tojo [2], where the example that describes a concrete situation in physics is given. Consider a closed metal wire (length 2) wrapped around a thin sheet of insulation material in the manner shown in Figure 1. Assume that the position =0 is the lowest of the wire, and the insulation goes up to the leſt at − and to the right up to . Since the wire is closed, points − and coincide. e layer of insulation is assumed to be slightly perme- able. erefore, the temperature value from one side affects the diffusion process on the other side. For this reason, the standard heat equation is modified and to its right-hand side ( 2 Φ/ 2 )(, ) a third term ( 2 Φ/ 2 )(−, ) (where || < 1) is added. Here Φ(, ) is the temperature at point of the wire at time . us, this process is described by the following: Φ (, ) − Φ (, ) + Φ (−, ) = () (1) in the domain Ω = {(,) : − < < , 0 < < }. Here () is the influence of an internal source that does not Hindawi Advances in Mathematical Physics Volume 2018, Article ID 8301656, 8 pages https://doi.org/10.1155/2018/8301656
Transcript of On an Inverse Problem of Reconstructing a Heat Conduction ...AdvancesinMathematicalPhysics 5....
Research ArticleOn an Inverse Problem of Reconstructing a Heat ConductionProcess from Nonlocal Data
Makhmud A Sadybekov 1 Gulnar Dildabek12 andMarina B Ivanova13
1 Institute of Mathematics and Mathematical Modeling 125 Pushkin Str 050010 Almaty Kazakhstan2Al-Farabi Kazakh National University 71 Al-Farabi Ave 050040 Almaty Kazakhstan3South-Kazakhstan State Pharmaceutical Academy 1 al-Farabi Sq 160019 Shymkent Kazakhstan
Correspondence should be addressed to Makhmud A Sadybekov sadybekovmathkz
Received 28 February 2018 Accepted 21 May 2018 Published 12 June 2018
Academic Editor Pavel Kurasov
Copyright copy 2018 MakhmudA Sadybekov et alThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited
We consider an inverse problem for a one-dimensional heat equation with involution and with periodic boundary conditions withrespect to a space variable This problem simulates the process of heat propagation in a thin closed wire wrapped around a weaklypermeable insulationThe inverse problem consists in the restoration (simultaneously with the solution) of an unknown right-handside of the equation which depends only on the spatial variableThe conditions for redefinition are initial and final states Existenceand uniqueness results for the given problem are obtained via the method of separation of variables
1 Introduction and Statement of the Problem
The problems that imply the determination of coefficients orthe right-hand side of a differential equation (together withits solution) are commonly referred to as inverse problemsof mathematical physics In this paper we consider onefamily of problems implying the determination of the densitydistribution and of heat sources from given values of initialand final distributions This problem simulates the processof heat propagation in a thin closed wire wrapped arounda weakly permeable insulation The mathematical statementof such problems leads to an inverse problem for the heatequation where it is required to find not only a solution ofthe problem but also its right-hand side that depends onlyon a spatial variable
In this paper we will consider the inverse problem closeto that investigated in [1 2] Together with the solutionit is necessary to find an unknown right-hand side of theequationThe equation contains the usual time derivative andan involution with respect to the spatial variable In contrastto [1] we investigate the problem under nonlocal boundaryconditions with respect to the spatial variableThe conditionsfor overdetermination are initial and final states
The second of the main differences in the investigatedinverse problem being studied is that the unknown function
enters both in the right-hand side of the equation and in theconditions of the initial and final overdetermination
Let us consider a problem of modeling the thermaldiffusion process which is close to that described in the reportof Cabada and Tojo [2] where the example that describes aconcrete situation in physics is given Consider a closedmetalwire (length 2120587) wrapped around a thin sheet of insulationmaterial in the manner shown in Figure 1
Assume that the position 119909 = 0 is the lowest of the wireand the insulation goes up to the left at minus120587 and to the right upto 120587 Since the wire is closed points minus120587 and 120587 coincide
The layer of insulation is assumed to be slightly perme-able Therefore the temperature value from one side affectsthe diffusion process on the other side For this reason thestandard heat equation is modified and to its right-hand side(1205972Φ1205971199092)(119909 119905) a third term 120576(1205972Φ1205971199092)(minus119909 119905) (where |120576| lt1) is added Here Φ(119909 119905) is the temperature at point 119909 of thewire at time 119905
Thus this process is described by the following
Φ119905 (119909 119905) minus Φ119909119909 (119909 119905) + 120576Φ119909119909 (minus119909 119905) = 119891 (119909) (1)
in the domain Ω = (119909 119905) minus120587 lt 119909 lt 120587 0 lt 119905 lt 119879Here 119891(119909) is the influence of an internal source that does not
HindawiAdvances in Mathematical PhysicsVolume 2018 Article ID 8301656 8 pageshttpsdoiorg10115520188301656
2 Advances in Mathematical Physics
Φ(minusx)
Φ(minus) = Φ()
Φ(x)
Φ(0)
Figure 1 The closed metal wire wrapped around a thin sheet ofinsulation material
change with time 119905 = 0 is an initial time point and 119905 = 119879 is afinal one
As the additional information we take values of the initialand final conditions of the temperature
Φ (119909 0) = 120601 (119909) Φ (119909 119879) = 120595 (119909)
119909 isin [minus120587 120587] (2)
Since the wire is closed it is natural to assume that thetemperature at the ends of the wire is the same at all times
Φ (minus120587 119905) = Φ (120587 119905) 119905 isin [0 119879] (3)
Since Φ(119909 119905) is the temperature at point 119909 of the wire attime 119905 average temperature along the entire length of the wireat time 119905 can be calculated from formula Φ(119905) = int120587
minus120587Φ(120585 119905)119889120585
If additional heating (cooling) is applied through the bound-ary point 119909 = minus120587 then the average temperature of the wireΦ(119905) increases (decreases)The greater the importance of thisheating (cooling) is the faster the average temperature of thewire Φ(119905) changes
Consider a process in which the temperature at one end atevery time point 119905 is proportional to the change speed of theaverage value of the temperature throughout the wire Then
Φ (minus120587 119905) = 120574 119889119889119905 int120587
minus120587Φ (120585 119905) 119889120585 119905 isin [0 119879] (4)
Here 120574 is a proportionality coefficientThus the investigated process is reduced to the following
mathematical inverse problem Find the right-hand side 119891(119909)of the heat equation (1) and its solution Φ(119909 119905) subject to theinitial and final conditions (2) the boundary condition (3) andcondition (4)
2 Reduction to a Mathematical Problem
Condition (4) is significantly nonlocal (ie values of theunknown function over the entire domain are used) The
integral along inner lines of the domain is present in thisconditionUsing the idea ofAA Samarskii we transform thiscondition Taking into account (1) from (4) we get
Φ (minus120587 119905) = 120574int120587minus120587Φ120585120585 (120585 119905) minus 120576Φ120585120585 (minus120585 119905) + 119891 (120585) 119889120585
119905 isin [0 119879] (5)
Hence
Φ (minus120587 119905) = 120574 (1 minus 120576) [Φ119909 (120587 119905) minus Φ119909 (minus120587 119905)]+ 120574int120587minus120587119891 (120585) 119889120585 119905 isin [0 119879] (6)
Let us introduce the following notations
119906 (119909 119905) = Φ (119909 119905) minus 120574int120587minus120587119891 (120585) 119889120585 (7)
Then in terms of a new function 119906(119909 119905) we get thefollowing inverse problem In the domain Ω = (119909 119905) minus120587 lt119909 lt 120587 0 lt 119905 lt 119879 find the right-hand side 119891(119909) of the heatequation with involution
119906119905 (119909 119905) minus 119906119909119909 (119909 119905) + 120576119906119909119909 (minus119909 119905) = 119891 (119909) (8)
and its solution 119906(119909 119905) satisfying the initial conditions119906 (119909 0) = 120601 (119909) minus 120574int120587
minus120587119891 (120585) 119889120585 119909 isin [minus120587 120587] (9)
and the final conditions
119906 (119909 119879) = 120595 (119909) minus 120574int120587minus120587119891 (120585) 119889120585 119909 isin [minus120587 120587] (10)
and the boundary ones
119906119909 (minus120587 119905) minus 119906119909 (120587 119905) minus 119886119906 (120587 119905) = 0119906 (minus120587 119905) minus 119906 (120587 119905) = 0
119905 isin [0 119879] (11)
where 120601(119909) and 120595(119909) are given sufficiently smooth functions120576 is a nonzero real number such that |120576| lt 1 and 119886 = 1120574(120576 minus1)In the physical sense the second of conditions (11)
means the equality of the distribution densities at the endsof the interval And the first of conditions (11) means theproportionality of the difference of fluxes across oppositeboundaries to the density value at the boundaryWe note thatin [1] the Dirichlet boundary conditions 119906(minus120587 119905) = 119906(120587 119905) =0 were used instead of condition (11)
The well-posedness of direct and inverse problems forparabolic equations with involution was considered in [3ndash5]
The solvability of various inverse problems for parabolicequations was studied in papers of Anikonov YuE andBelov YuYa Bubnov BA Prilepko AI and Kostin ABMonakhov VN Kozhanov AI Kaliev I A Sabitov KBand many others These citations can be seen in our papers
Advances in Mathematical Physics 3
[6 7] In [1] there are good references to publications onrelated issues We note [3 8ndash28] from recent papers close tothe theme of our article In these papers different variants ofdirect and inverse initial-boundary value problems for evo-lutionary equations are considered including problems withnonlocal boundary conditions and problems for equationswith fractional derivatives
In this paper we shall use a spectral problem for anordinary differential operators with involution Such andsimilar spectral problems were considered in [29ndash40]
Definition 1 By a regular solution of the inverse prob-lem (8)-(11) we mean a pair of functions (119906(119909 119905) 119891(119909)) of theclass 119906(119909 119905) isin 11986221119909119905(Ω) 119891(119909) isin 119862[minus120587 120587] that inverts (8) andconditions (9)-(11) into an identity
Definition 2 By a generalized solution of the inverseproblem (8)-(11) we mean a pair of functions (119906(119909 119905) 119891(119909))of the class 119906(119909 119905) isin 119882212 (Ω) cap 119862(Ω) 119891(119909) isin 1198712(minus120587 120587) thatsatisfies (8) and conditions (9)-(11) almost everywhere
When one uses the method of separation of variablesto solve the problem a spectral problem appears which ismentioned in the next section
3 Spectral Problem
The use of the Fourier method for solving problem (8)ndash(11)leads to a spectral problem for an operator L given by thedifferential expression
L119883 (119909) equiv minus11988310158401015840 (119909) + 12057611988310158401015840 (minus119909) = 120582119883 (119909) minus120587 lt 119909 lt 120587 (12)
and the boundary conditions
1198831015840 (minus120587) minus 1198831015840 (120587) minus 119886119883 (120587) = 0119883 (minus120587) minus 119883 (120587) = 0 (13)
where 120582 is a spectral parameterThe spectral problems for (12) were first considered
apparently in [37] There was considered a case of Dirichletand Neumann boundary conditions and cases of conditionsin the form (13) for 119886 = 0 Here we consider the case 119886 = 0We assume that 119886 gt 0
We represent a general solution of (12) in the followingform
119883 (119909) = 119860 sin (1205831119909) + 119861 cos (1205832119909) 1205831 = radic 1205821 + 120576
1205832 = radic 1205821 minus 120576 (14)
where119860 and 119861 are arbitrary complex numbers Satisfying theboundary conditions (13) for finding eigenvalues we obtainthe following
sin (1205831120587) = 0tan (1205832120587) = 11988621205832
(15)
Therefore the spectral problem (12)-(13) has two series ofeigenvalues
1205821198961 = (1 + 120576) 1198962 119896 isin N (16)
1205821198962 = (1 minus 120576)(119896 + 120575119896)2 119896 isin N0 equiv N cup 0 where 120575119896 =119874(119886(119896 + 1)) gt 0 as 119896 rarr infin with corresponding normalizedeigenfunctions given by
1198831198961 (119909) = 1radic120587 sin (119896119909) 119896 isin N1198831198962 (119909) = ]119896 cos ((119896 + 120575119896) 119909) 119896 isin N0
(17)
Here ]119896 is a normalization coefficient
]minus2119896 = 1003817100381710038171003817cos ((119896 + 120575119896) 119909)100381710038171003817100381721198712(minus120587120587)= 120587 + 1198862
(119896 + 120575119896) [1198862 + (119896 + 120575119896)2 1205872] (18)
Lemma 3 The system of functions (17) is complete andorthonormal in 1198712(minus120587 120587)Proof We note that the system of eigenfunctions (17) doesnot depend on the parameter 120576 Only the eigenvalues ofproblem (12)-(13) depend on 120576
In the case when 120576 = 0 system (17) is a system ofeigenfunctions of the classical Strum-Liouville problem
minus11988310158401015840 (119909) = 120582119883 (119909) minus 120587 lt 119909 lt 120587 (19)
with the self-adjoint boundary conditions (13)Consequently system (17) forms the complete orthonor-
mal system in 1198712(minus120587 120587) that is it is the orthonormal basis ofthe space
4 Uniqueness of the Solution of the Problem
Let the pair of functions (119906(119909 119905) 119891(119909)) be a solution of theinverse problem (8)-(11) Let us introduce notations
119906119896119894 (119905) = int120587minus120587119906 (119909 119905)119883119896119894 (119909) 119889119909
119891119896119894 = int120587minus120587119891 (119909)119883119896119894 (119909) 119889119909
119894 = 1 2(20)
4 Advances in Mathematical Physics
We apply the operator 119889119889119905 to 119906119896119894(119905) Then using (8) byintegrating by parts we obtain the following problem
1199061015840119896119894 (119905) + 120582119896119894119906119896119894 (119905) = 119891119896119894 0 lt 119905 lt 119879 119894 = 1 2 (21)
119906119896119894 (0) = 120601119896119894 minus 120590119896119894 119894 = 1 2 (22)
119906119896119894 (119879) = 120595119896119894 minus 120590119896119894 119894 = 1 2 (23)
Here we use the following notations
120601119896119894 = int120587minus120587120601 (119909)119883119896119894 (119909) 119889119909
120595119896119894 = int120587minus120587120595 (119909)119883119896119894 (119909) 119889119909
120590119896119894 = 120574int120587minus120587119891 (120585) 119889120585int120587
minus120587119883119896119894 (119909) 119889119909
(24)
It is easy to see that the function 1198961(119905) = (120582119896119894)minus1119891119896119894is a partial solution of the inhomogeneous equation (21)Therefore we obtain a unique solution of theCauchy problem(21)-(22) in the following form
119906119896119894 (119905) = (120601119896119894 minus 120590119896119894 minus 119891119896119894120582119896119894) 119890minus120582119896119894119905 + 119891119896119894120582119896119894 (25)
Since |120576| lt 1 119886 gt 0 we have 120582119896119894 gt 0 for all 119896 119894Consequently
1 minus 119890minus120582119896119894119879 ge 1 minus 119890minus12058202119879 gt 0for all the values of the indexes 119896 119894 (26)
Therefore using the condition of rdquofinal overdetermina-tionrdquo (23) we get
119891119896119894 = 120582119896119894120595119896119894 minus 120601119896119894119890minus120582119896119894119879
1 minus 119890minus120582119896119894119879 minus 120582119896119894120590119896119894 (27)
Lemma 4 If 119886 gt 0 then the generalized solution(119906(119909 119905) 119891(119909)) of the inverse problem (8)-(11) is unique
Proof Suppose that there are two generalized solutionsof the inverse problem (8)-(11) (1199061(119909 119905) 1198911(119909)) and(1199062(119909 119905) 1198912(119909)) Denote
119906 (119909 119905) = 1199061 (119909 119905) minus 1199062 (119909 119905) 119891 (119909) = 1198911 (119909) minus 1198912 (119909) (28)
Then the functions 119906(119909 119905) and 119891(119909) satisfy (8) the boundaryconditions (11) and the homogeneous conditions (9) and (10)
119906 (119909 0) = minus120574int120587minus120587119891 (120585) 119889120585 119909 isin [minus120587 120587]
119906 (119909 119879) = minus120574int120587minus120587119891 (120585) 119889120585 119909 isin [minus120587 120587]
(29)
Therefore by using the notations (20) from (27) we find
119891119896119894 = minus120582119896119894120590119896119894 equiv minus120582119896119894120574int120587minus120587119891 (120585) 119889120585int120587
minus120587119883119896119894 (119909) 119889119909 (30)
Since
120582119896119894 int120587minus120587119883119896119894 (119909) 119889119909 = int120587
minus120587minus11988310158401015840119896119894 (119909) + 12057611988310158401015840119896119894 (minus119909) 119889119909
= (1 minus 120576) 1198831015840119896119894 (minus120587) minus 1198831015840119896119894 (120587)= 119886 (1 minus 120576)119883119896119894 (120587)
(31)
this gives
119891119896119894 = minus119886 (1 minus 120576) (120574int120587minus120587119891 (120585) 119889120585)119883119896119894 (120587) (32)
Since1198831198961(120587) = 0 then for 119894 = 1 from (32) we have 1198911198961 =0For the generalized solution of the problem (8)-(11) it
is necessary that 119891 isin 1198712(minus120587 120587) From this by virtue ofParsevalrsquos equality for orthogonal bases we have that
infinsum119896=1
1003816100381610038161003816119891119896210038161003816100381610038162 le 100381710038171003817100381711989110038171003817100381710038172 lt infin (33)
is necessarySince 119886(1 minus 120576) = 0
1198831198962 (120587) = ]119896 cos ((119896 + 120575119896) 120587) lim119896rarrinfin
]119896 = 1radic120587lim119896rarrinfin
120575119896 = 0(34)
then (32) and inequality (33) are possible only at(120574 int120587minus120587119891(120585)119889120585) = 0 Hence we obtain 1198911198962 = 0
Therefore using this result from (25) and (27) we find
119906119896119894 (119905) equiv int120587minus120587119906 (119909 119905) 119883119896119894 (119909) 119889119909 = 0
119891119896119894 equiv int120587minus120587119891 (119909)119883119896119894 (119909) 119889119909 = 0
(35)
for all values of the indexes 119896 isin N for 119894 = 1 and 119896 isin N0 for 119894 =2 Further by the completeness of system (17) in 1198712(minus120587 120587)we obtain
119906 (119909 119905) equiv 0119891 (119909) equiv 0
forall (119909 119905) isin Ω(36)
The uniqueness of the generalized solution of the inverseproblem (8)-(11) is proved
Corollary 5 If 119886 gt 0 then the regular solution (119906(119909 119905) 119891(119909))of the inverse problem (8)-(11) is unique
Advances in Mathematical Physics 5
5 Construction of a Formal Solution ofthe Problem
As the eigenfunctions system (17) forms the orthonormalbasis in 1198712(minus120587 120587) then the unknown functions 119906(119909 119905) and119891(119909) can be represented as
119906 (119909 119905) = infinsum119896=1
1199061198961 (119905) 1198831198961 (119909) + infinsum119896=0
1199061198962 (119905) 1198831198962 (119909) (37)
119891 (119909) = infinsum119896=1
11989111989611198831198961 (119909) + infinsum119896=0
11989111989621198831198962 (119909) (38)
where 1199061198961(119905) and 1199061198962(119905) are unknown functions 1198911198961 and 1198911198962are unknown constants
Substituting (37) and (38) into (8) we obtain the inverseproblems (21)-(23) If the constants 120590119896119894 are assumed to begiven then the solutions of these inverse problems existare unique and are represented by formulas (25) and (27)Substituting (25) and (27) into series (37) and (38) we obtaina formal solution of the inverse problem (8)-(11)
Let 120601(119909) 120595(119909) isin 11988222 (minus120587 120587) and satisfy the boundaryconditions (13) Since 120582119896119894 gt 0 then
infinsum119896=1
1003816100381610038161003816100381610038161003816100381610038161003816120582119896119894120595119896119894 minus 120601119896119894119890minus1205821198961198941198791 minus 119890minus120582119896119894119879
10038161003816100381610038161003816100381610038161003816100381610038162
le 11986210038171003817100381710038171206011003817100381710038171003817211988222(minus120587120587)
+ 10038171003817100381710038171205951003817100381710038171003817211988222(minus120587120587)
119894 = 1 2(39)
Therefore from the analysis of formula (27) it is easy tosee that in order for the formal solution (37) of the problem(8)-(11) to be a generalized solution it is necessary that
lim119896rarrinfin
120582119896119894120590119896119894 = 0 119894 = 1 2 (40)
holdsAs above we calculate
120582119896119894120590119896119894 equiv 119886 (1 minus 120576) (120574int120587minus120587119891 (120585) 119889120585)119883119896119894 (120587) (41)
where1198831198961(120587) = 0 and1198831198962 (120587) = ]119896 cos ((119896 + 120575119896) 120587) lim119896rarrinfin
]119896 = 1radic120587lim119896rarrinfin
120575119896 = 0(42)
Thus (40) holds if and only if 120590119896119894 = 0 for all values of theindexes 119896 119894 This is possible only in the case
int120587minus120587119891 (120585) 119889120585 = 0 (43)
Thus condition (43) is a necessary condition for theexistence of the generalized solution of the inverse problem
(8)-(11) In this case problems (8)-(11) and (1)-(4) coincideIndeed from (43) and (1) we have
0 = int120587minus120587119891 (120585) 119889120585
= int120587minus120587Φ119905 (120585 119905) 119889120585
minus int120587minus120587Φ120585120585 (120585 119905) + 120576Φ120585120585 (minus120585 119905) 119889120585
(44)
For the first integral we apply condition (4) and calculatethe second integral Then we obtain
0 = (1 minus 120576)sdot [Φ119909 (minus120587 119905) minus Φ119909 (120587 119905) + 1120574 (1 minus 120576)Φ (minus120587 119905)] (45)
This means that the boundary conditions (4) and (11) coin-cide Hence the problems (8)-(11) and (1)-(4) also coincide
Thus in what follows we shall consider problem (1)-(3)with the boundary condition
Φ119909 (minus120587 119905) minus Φ119909 (120587 119905) minus 119886Φ (minus120587 119905) = 0 (46)Thus in what follows we will consider the inverse
problem (1)-(3) (46)Similarly as before the formal solution of this problem
can be constructed in the form of series
Φ (119909 119905) = infinsum119896=1
Φ1198961 (119905) 1198831198961 (119909) + infinsum119896=0
Φ1198962 (119905) 1198831198962 (119909) (47)
119891 (119909) = infinsum119896=1
11989111989611198831198961 (119909) + infinsum119896=0
11989111989621198831198962 (119909) (48)
where
Φ119896119894 (119905) = (120601119896119894 minus 119891119896119894120582119896119894) 119890minus120582119896119894119905 + 119891119896119894120582119896119894 (49)
119891119896119894 = 120582119896119894120595119896119894 minus 120601119896119894119890minus120582119896119894119879
1 minus 119890minus120582119896119894119879 (50)
In order to complete our study it is necessary as in theFourier method to justify the smoothness of the resultingformal solutions and the convergence of all appearing series
6 Main Results
Here we present the existence and uniqueness results for ourinverse problem
Theorem 6 (let 119886 gt 0) (A) Let 120601(119909) 120595(119909) isin 11988222 (minus120587 120587)and let the functions120601(119909) 120595(119909) satisfy the boundary conditions(13) Then for a real number 120576 such that |120576| lt 1 the inverseproblem (1)-(3) (46) has a unique generalized solution whichis stable in norm1003817100381710038171003817Φ119905100381710038171003817100381721198712(Ω) + 1003817100381710038171003817Φ119909119909100381710038171003817100381721198712(Ω) + 1003817100381710038171003817119891100381710038171003817100381721198712(minus120587120587)
le 11986210038171003817100381710038171206011003817100381710038171003817211988222(minus120587120587)
+ 10038171003817100381710038171205951003817100381710038171003817211988222(minus120587120587)
(51)
where the constant 119862 does not depend on 120601(119909) and 120595(119909)
6 Advances in Mathematical Physics
(B) Let 120601(119909) 120595(119909) isin 1198624[minus120587 120587] and the functions120601(119909) 120595(119909) and 12060110158401015840(119909) 12059510158401015840(119909) satisfy the boundary conditions(13) then for a real number 120576 such that |120576| lt 1 the inverseproblem (1)-(3) (46) has a unique regular solution
Proof Since |120576| lt 1 119886 gt 0 we have 120582119896119894 gt 0 for all 119896 119894Consequently there exists a constant 120573 gt 0 such that for allthe values of the indexes 119896 119894 we have
1 minus 119890minus120582119896119894119879 ge 1 minus 119890minus12058202119879 ge 120573 gt 0 (52)
Since 120582119896119894 gt 0 then 119890minus120582119896119894119879 le 119890minus120582119896119894119905 lt 1 Therefore fromrepresentations (49) and (50) we get estimates
10038161003816100381610038161198911198961198941003816100381610038161003816 le10038161003816100381610038161205821198961198941003816100381610038161003816120573 10038161003816100381610038161206011198961198941003816100381610038161003816 + 10038161003816100381610038161205951198961198941003816100381610038161003816 (53)
1003816100381610038161003816Φ119896119894 (119905)1003816100381610038161003816 le (1 + 2120573) 10038161003816100381610038161206011198961198941003816100381610038161003816 + 10038161003816100381610038161205951198961198941003816100381610038161003816 (54)
where constant 120573 does not depend on the indexes 119896 119894 and onthe functions 120601(119909) and 120595(119909)
As proved in Lemma 3 the eigenfunctions system (17)forms the orthonormal basis in 1198712(minus120587 120587)Therefore we havean expansion
120601 (119909) = infinsum119896=1
12060111989611198831198961 (119909) + infinsum119896=0
12060111989621198831198962 (119909) (55)
Taking into account (12) we obtain
minus12060110158401015840 (119909) + 12057612060110158401015840 (minus119909) = infinsum119896=1
120582119896112060111989611198831198961 (119909)
+ infinsum119896=0
120582119896212060111989621198831198962 (119909) (56)
From this by Parsevalrsquos equality it is easy to obtain anestimate
infinsum119896=1
1003816100381610038161003816120582119896110038161003816100381610038162 1003816100381610038161003816120601119896110038161003816100381610038162 +infinsum119896=1
1003816100381610038161003816120582119896210038161003816100381610038162 1003816100381610038161003816120601119896210038161003816100381610038162
le 2 (1 + 1205762) 10038171003817100381710038171003817120601101584010158401003817100381710038171003817100381721198712(minus120587120587) (57)
Similarly for the coefficients of the expansion
120595 (119909) = infinsum119896=1
12059511989611198831198961 (119909) + infinsum119896=0
12059511989621198831198962 (119909) (58)
we have the inequality
infinsum119896=1
1003816100381610038161003816120582119896110038161003816100381610038162 1003816100381610038161003816120595119896110038161003816100381610038162 +infinsum119896=1
1003816100381610038161003816120582119896210038161003816100381610038162 1003816100381610038161003816120595119896210038161003816100381610038162
le 2 (1 + 1205762) 10038171003817100381710038171003817120595101584010158401003817100381710038171003817100381721198712(minus120587120587) (59)
Now from (53) (57) and (59) we see that the series (48)converges and the estimate
1003817100381710038171003817119891100381710038171003817100381721198712(minus120587120587) le 21 + 12057621205732 10038171003817100381710038171003817120601101584010158401003817100381710038171003817100381721198712(minus120587120587) + 10038171003817100381710038171003817120595101584010158401003817100381710038171003817100381721198712(minus120587120587) (60)
holdsTaking into account (12) from (47) we obtain
minus Φ119909119909 (119909 119905) + 120576Φ119909119909 (minus119909 119905)= infinsum119896=1
1205821198961Φ1198961 (119905) 1198831198961 (119909)
+ infinsum119896=0
1205821198962Φ1198962 (119905) 1198831198962 (119909) (61)
Substituting the value (minus119909) instead of 119909 we findminus Φ119909119909 (minus119909 119905) + 120576Φ119909119909 (119909 119905)= minusinfinsum119896=1
1205821198961Φ1198961 (119905) 1198831198961 (119909)
+ infinsum119896=0
1205821198962Φ1198962 (119905) 1198831198962 (119909) (62)
Here we use the properties 1198831198961(minus119909) = minus1198831198961(119909) 1198831198962(minus119909) =1198831198962(119909) obtained from the explicit form (12) of the eigenfunc-tions
Multiplying (62) by 120576 and summing with (61) we have
(minus1 + 1205762)Φ119909119909 (119909 119905)= (1 minus 120576) infinsum
119896=1
1205821198961Φ1198961 (119905) 1198831198961 (119909)
+ (1 + 120576) infinsum119896=0
1205821198962Φ1198962 (119905) 1198831198962 (119909) (63)
Since |120576| lt 1 from this by Parsevalrsquos equality we find
1003817100381710038171003817Φ119909119909 (sdot 119905)100381710038171003817100381721198712(minus120587120587) = 1(1 + 120576)2
infinsum119896=1
1003816100381610038161003816120582119896110038161003816100381610038162 1003816100381610038161003816Φ1198961 (119905)10038161003816100381610038162
+ 1(1 minus 120576)2
infinsum119896=0
1003816100381610038161003816120582119896210038161003816100381610038162 1003816100381610038161003816Φ1198962 (119905)10038161003816100381610038162(64)
for each 119905 isin [0 119879] Integrating this equation with respect to119905 isin [0 119879] taking (54) into account we have1003817100381710038171003817Φ119909119909100381710038171003817100381721198712(Ω) le 8119879
(1 minus 1205762)2 (1 +2120573)2
sdot 2sum119894=1
infinsum119896=1
100381610038161003816100381612058211989611989410038161003816100381610038162 100381610038161003816100381612060111989611989410038161003816100381610038162 + 100381610038161003816100381612059511989611989410038161003816100381610038162 (65)
Advances in Mathematical Physics 7
Hence using (57) and (59) we obtain
1003817100381710038171003817Φ119909119909100381710038171003817100381721198712(Ω) le 16119879 1 + 1205762(1 minus 1205762)2 (1 +
2120573)2
sdot 10038171003817100381710038171003817120601101584010158401003817100381710038171003817100381721198712(minus120587120587) + 10038171003817100381710038171003817120595101584010158401003817100381710038171003817100381721198712(minus120587120587) (66)
Now we can easily obtain an estimate for Ψ119905(119909 119905) from(1) This fact together with (60) and (66) gives the necessaryestimate (51) for the solution
From the obtained estimates it also follows that in theconstructed by us formal solution of the inverse problem allthe series converge they can be term-by-term differentiatedand the series obtained during differentiation also convergein sense of the metrics 1198712
From (47) and (54) by using the Holderrsquos inequality it iseasy to justify the inequality
max(119909119905)isinΩ
|Φ (119909 119905)|2 le 11986210038171003817100381710038171003817120601101584010158401003817100381710038171003817100381721198712(minus120587120587) + 10038171003817100381710038171003817120595101584010158401003817100381710038171003817100381721198712(minus120587120587) (67)
which justifies the continuity ofΦ(119909 119905) in the closed domainΩFrom the representation of the solution in the form of
series (37) (38) and inequalities (53) (54) it is easy to justifythe estimates 1003816100381610038161003816Φ119909119909 (119909 119905)1003816100381610038161003816 + 1003816100381610038161003816Φ119905 (119909 119905)1003816100381610038161003816 + 1003816100381610038161003816119891 (119909)1003816100381610038161003816
le 119862infinsum119896=1
100381610038161003816100381612058211989611989410038161003816100381610038162 10038161003816100381610038161206011198961198941003816100381610038161003816 + 10038161003816100381610038161205951198961198941003816100381610038161003816 (68)
Let 120601(119909) 120595(119909) isin 1198624[minus120587 120587] and let the functions 120601(119909) 120595(119909)and 12060110158401015840(119909) 12059510158401015840(119909) satisfy the boundary conditions (13) thenthe number series in the right-hand side of (68) convergesTherefore in such case the constructed by us formal solutiongives the regular solution of the inverse problem (1)-(3) (46)
The uniqueness of this constructed solution follows fromLemma 4
The theorem is completely proved
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This paper was published with the support of the ScienceCommittee of the Ministry of Education and Science of theRepublic of Kazakhstan under Project AP05133271 and TargetProgram BR05236656
References
[1] B Ahmad A Alsaedi M Kirane and R G TapdigogluldquoAn inverse problem for space and time fractional evolution
equations with an involution perturbationrdquo Quaestiones Math-ematicae vol 40 no 2 pp 151ndash160 2017
[2] A Cabada and A F Tojo ldquoEquations with involutionsrdquo inProceedings of the Workshop on Differential Equations p 240MallaMoravka CzechRepublicMarch 2014 httpusersmathcascz
[3] A Ashyralyev and A Sarsenbi ldquoWell-posedness of a parabolicequation with nonlocal boundary conditionrdquo Boundary ValueProblems vol 2015 no 1 2015
[4] M Kirane and N Al-Salti ldquoInverse problems for a nonlocalwave equation with an involution perturbationrdquo Journal ofNonlinear Science and Applications vol 9 no 3 pp 1243ndash12512016
[5] A Ashyralyev and A Sarsenbi ldquoWell-Posedness of a ParabolicEquation with Involutionrdquo Numerical Functional Analysis andOptimization vol 38 no 10 pp 1295ndash1304 2017
[6] I Orazov and M A Sadybekov ldquoOne nonlocal problem ofdetermination of the temperature and density of heat sourcesrdquoRussian Mathematics vol 56 no 2 pp 60ndash64 2012
[7] I Orazov and M A Sadybekov ldquoOn a class of problems ofdetermining the temperature and density of heat sources giveninitial and final temperaturerdquo Siberian Mathematical Journalvol 53 no 1 pp 146ndash151 2012
[8] N I Ivanchov ldquoInverse problems for the heat-conductionequation with nonlocal boundary conditionsrdquoUkrainianMath-ematical Journal vol 45 no 8 pp 1186ndash1192 1993
[9] I A Kaliev and M M Sabitova ldquoProblems of determiningthe temperature and density of heat sources from the initialand final temperaturesrdquo Journal of Applied and IndustrialMathematics vol 4 no 3 pp 332ndash339 2010
[10] I A KalievM FMugafarov andOV Fattahova ldquoInverse prob-lem for forwardbackward parabolic equation with generalizedconjugation conditionsrdquo Ufa Mathematical Journal vol 3 no2 pp 33ndash41 2011
[11] M Kirane and S A Malik ldquoDetermination of an unknownsource term and the temperature distribution for the linearheat equation involving fractional derivative in timerdquo AppliedMathematics and Computation vol 218 no 1 pp 163ndash170 2011
[12] F Kanca andM I Ismailov ldquoThe inverse problem of finding thetime-dependent diffusion coefficient of the heat equation fromintegral overdetermination datardquo Inverse Problems in Scienceand Engineering vol 20 no 4 pp 463ndash476 2012
[13] M Kirane S A Malik and M Al-Gwaiz ldquoAn inverse sourceproblem for a two dimensional time fractional diffusionequation with nonlocal boundary conditionsrdquo MathematicalMethods in the Applied Sciences vol 36 no 9 pp 1056ndash10692013
[14] A Ashyralyev and Y A Sharifov ldquoExistence and uniquenessof solutions for nonlinear impulsive differential equations withtwo-point and integral boundary conditionsrdquo Advances inDifference Equations vol 2013 no 1 p 173 2013
[15] F Kanca ldquoInverse coefficient problem of the parabolic equa-tion with periodic boundary and integral overdeterminationconditionsrdquo Abstract and Applied Analysis vol 2013 Article ID659804 7 pages 2013
[16] D Lesnic S A Yousefi and M Ivanchov ldquoDetermination of atime-dependent diffusivity from nonlocal conditionsrdquo AppliedMathematics and Computation vol 41 no 1-2 pp 301ndash3202013
[17] L Miller and M Yamamoto ldquoCoefficient inverse problem for afractional diffusion equationrdquo Inverse Problems vol 29 no 7 p075013 2013
8 Advances in Mathematical Physics
[18] G Li D Zhang X Jia and M Yamamoto ldquoSimultaneousinversion for the space-dependent diffusion coefficient andthe fractional order in the time-fractional diffusion equationrdquoInverse Problems vol 29 no 6 Article ID 065014 2013
[19] A B Kostin ldquoCounterexamples in inverse problems forparabolic elliptic and hyperbolic equationsrdquo ComputationalMathematics and Mathematical Physics vol 54 no 5 pp 797ndash810 2014
[20] A Ashyralyev and A Hanalyev ldquoWell-posedness of nonlocalparabolic differential problems with dependent operatorsrdquoTheScientific World Journal vol 2014 Article ID 519814 2014
[21] I Orazov and M A Sadybekov ldquoOn an inverse problemof mathematical modeling of the extraction process of poly-disperse porous materialsrdquo in Proceedings of the ADVANCE-MENTS IN MATHEMATICAL SCIENCES Proceedings of theInternational Conference on Advancements in MathematicalSciences p 020005 Antalya Turkey
[22] I Orazov and M A Sadybekov ldquoOne-dimensional diffusionproblem with not strengthened regular boundary conditionsrdquoin Proceedings of the 41ST International Conference ldquoApplicationsof Mathematics in Engineering And EconomicSrdquo AMEE rsquo15 p040007 Sozopol Bulgaria
[23] N H Tuan D N D Hai L D Long V T Nguyen and MKirane ldquoOn a RieszndashFeller space fractional backward diffusionproblemwith a nonlinear sourcerdquo Journal of Computational andApplied Mathematics vol 312 pp 103ndash126 2017
[24] M Sadybekov G Oralsyn and M Ismailov ldquoAn inverseproblem of finding the time-dependent heat transfer coefficientfrom an integral conditionrdquo International Journal of Pure andApplied Mathematics vol 113 no 4 pp 139ndash149 2017
[25] M Kirane andMNH Tuan ldquoIdentification and regularizationfor unknown source for a time-fractional diffusion equationrdquoComputersMathematics withApplications vol 73 no 6 pp 931ndash950 2017
[26] B T Torebek and R Tapdigoglu ldquoSome inverse problems forthe nonlocal heat equation with Caputo fractional derivativerdquoMathematical Methods in the Applied Sciences vol 40 no 18pp 6468ndash6479 2017
[27] M Kirane B Samet and B T Torebek ldquoDetermination ofan unknown source term temperature distribution for thesub-diffusion equation at the initial and final datardquo ElectronicJournal of Differential Equations vol 2017 2017
[28] O Taki-Eddine and B Abdelfatah ldquoOn determining the coef-ficient in a parabolic equation with nonlocal boundary andintegral conditionrdquo Electronic Journal of Mathematical Analysisand Applications vol 6 no 1 pp 94ndash102 2018
[29] A M Sarsenbi ldquoUnconditional bases related to a nonclassicalsecond-order differential operatorrdquo Journal of Differential Equa-tions vol 46 no 4 pp 509ndash514 2010
[30] V PKurdyumov andA PKhromov ldquoTheRiesz bases consistingof eigen and associated functions for a functional differentialoperator with variable structurerdquo Russian Mathematics vol 54no 2 pp 33ndash45 2010
[31] A M Sarsenbi and A A Tengaeva ldquoOn the basis propertiesof the root functions of two generalized spectral problemsrdquoJournal of Differential Equations vol 48 no 2 pp 306ndash3082012
[32] M A Sadybekov and A M Sarsenbi ldquoCriterion for the basisproperty of the eigenfunction system of a multiple differen-tiation operator with an involutionrdquo Journal of DifferentialEquations vol 48 no 8 pp 1112ndash1118 2012
[33] A Kopzhassarova and A Sarsenbi ldquoBasis properties of eigen-functions of second-order differential operators with invo-lutionrdquo Abstract and Applied Analysis vol 2012 Article ID576843 6 pages 2012
[34] AAKopzhassarova A L Lukashov andAM Sarsenbi ldquoSpec-tral properties of non-self-adjoint perturbations for a spectralproblem with involutionrdquo Abstract and Applied Analysis vol2012 Article ID 590781 5 pages 2012
[35] A Sarsenbi and M Sadybekov ldquoEigenfunctions of a fourthorder operator pencilrdquo in Proceedings of the 2nd InternationalConference on Analysis and Applied Mathematics ICAAM 2014pp 241ndash245 kaz September 2014
[36] L V Kritskov and A M Sarsenbi ldquoSpectral properties of anonlocal problem for a second-order differential equation withan involutionrdquo Journal of Differential Equations vol 51 no 8pp 984ndash990 2015
[37] L V Kritskov and A M Sarsenbi ldquoBasicity in Lp of rootfunctions for differential equations with involutionrdquo ElectronicJournal of Differential Equations vol 2015 2015
[38] M A Sadybekov A Sarsenbi and A Tengayeva ldquoDescriptionof spectral properties of a generalized spectral problem withinvolution for differentiation operator of the second orderrdquo inProceedings of the 3rd International Conference on Analysis andApplied Mathematics ICAAM 2016 kaz September 2016
[39] A G Baskakov I A Krishtal and E Y Romanova ldquoSpectralanalysis of a differential operator with an involutionrdquo Journal ofEvolution Equations (JEE) vol 17 no 2 pp 669ndash684 2017
[40] L V Kritskov and A M Sarsenbi ldquoRiesz basis property ofsystem of root functions of second-order differential operatorwith involutionrdquo Journal of Differential Equations vol 53 no 1pp 33ndash46 2017
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2 Advances in Mathematical Physics
Φ(minusx)
Φ(minus) = Φ()
Φ(x)
Φ(0)
Figure 1 The closed metal wire wrapped around a thin sheet ofinsulation material
change with time 119905 = 0 is an initial time point and 119905 = 119879 is afinal one
As the additional information we take values of the initialand final conditions of the temperature
Φ (119909 0) = 120601 (119909) Φ (119909 119879) = 120595 (119909)
119909 isin [minus120587 120587] (2)
Since the wire is closed it is natural to assume that thetemperature at the ends of the wire is the same at all times
Φ (minus120587 119905) = Φ (120587 119905) 119905 isin [0 119879] (3)
Since Φ(119909 119905) is the temperature at point 119909 of the wire attime 119905 average temperature along the entire length of the wireat time 119905 can be calculated from formula Φ(119905) = int120587
minus120587Φ(120585 119905)119889120585
If additional heating (cooling) is applied through the bound-ary point 119909 = minus120587 then the average temperature of the wireΦ(119905) increases (decreases)The greater the importance of thisheating (cooling) is the faster the average temperature of thewire Φ(119905) changes
Consider a process in which the temperature at one end atevery time point 119905 is proportional to the change speed of theaverage value of the temperature throughout the wire Then
Φ (minus120587 119905) = 120574 119889119889119905 int120587
minus120587Φ (120585 119905) 119889120585 119905 isin [0 119879] (4)
Here 120574 is a proportionality coefficientThus the investigated process is reduced to the following
mathematical inverse problem Find the right-hand side 119891(119909)of the heat equation (1) and its solution Φ(119909 119905) subject to theinitial and final conditions (2) the boundary condition (3) andcondition (4)
2 Reduction to a Mathematical Problem
Condition (4) is significantly nonlocal (ie values of theunknown function over the entire domain are used) The
integral along inner lines of the domain is present in thisconditionUsing the idea ofAA Samarskii we transform thiscondition Taking into account (1) from (4) we get
Φ (minus120587 119905) = 120574int120587minus120587Φ120585120585 (120585 119905) minus 120576Φ120585120585 (minus120585 119905) + 119891 (120585) 119889120585
119905 isin [0 119879] (5)
Hence
Φ (minus120587 119905) = 120574 (1 minus 120576) [Φ119909 (120587 119905) minus Φ119909 (minus120587 119905)]+ 120574int120587minus120587119891 (120585) 119889120585 119905 isin [0 119879] (6)
Let us introduce the following notations
119906 (119909 119905) = Φ (119909 119905) minus 120574int120587minus120587119891 (120585) 119889120585 (7)
Then in terms of a new function 119906(119909 119905) we get thefollowing inverse problem In the domain Ω = (119909 119905) minus120587 lt119909 lt 120587 0 lt 119905 lt 119879 find the right-hand side 119891(119909) of the heatequation with involution
119906119905 (119909 119905) minus 119906119909119909 (119909 119905) + 120576119906119909119909 (minus119909 119905) = 119891 (119909) (8)
and its solution 119906(119909 119905) satisfying the initial conditions119906 (119909 0) = 120601 (119909) minus 120574int120587
minus120587119891 (120585) 119889120585 119909 isin [minus120587 120587] (9)
and the final conditions
119906 (119909 119879) = 120595 (119909) minus 120574int120587minus120587119891 (120585) 119889120585 119909 isin [minus120587 120587] (10)
and the boundary ones
119906119909 (minus120587 119905) minus 119906119909 (120587 119905) minus 119886119906 (120587 119905) = 0119906 (minus120587 119905) minus 119906 (120587 119905) = 0
119905 isin [0 119879] (11)
where 120601(119909) and 120595(119909) are given sufficiently smooth functions120576 is a nonzero real number such that |120576| lt 1 and 119886 = 1120574(120576 minus1)In the physical sense the second of conditions (11)
means the equality of the distribution densities at the endsof the interval And the first of conditions (11) means theproportionality of the difference of fluxes across oppositeboundaries to the density value at the boundaryWe note thatin [1] the Dirichlet boundary conditions 119906(minus120587 119905) = 119906(120587 119905) =0 were used instead of condition (11)
The well-posedness of direct and inverse problems forparabolic equations with involution was considered in [3ndash5]
The solvability of various inverse problems for parabolicequations was studied in papers of Anikonov YuE andBelov YuYa Bubnov BA Prilepko AI and Kostin ABMonakhov VN Kozhanov AI Kaliev I A Sabitov KBand many others These citations can be seen in our papers
Advances in Mathematical Physics 3
[6 7] In [1] there are good references to publications onrelated issues We note [3 8ndash28] from recent papers close tothe theme of our article In these papers different variants ofdirect and inverse initial-boundary value problems for evo-lutionary equations are considered including problems withnonlocal boundary conditions and problems for equationswith fractional derivatives
In this paper we shall use a spectral problem for anordinary differential operators with involution Such andsimilar spectral problems were considered in [29ndash40]
Definition 1 By a regular solution of the inverse prob-lem (8)-(11) we mean a pair of functions (119906(119909 119905) 119891(119909)) of theclass 119906(119909 119905) isin 11986221119909119905(Ω) 119891(119909) isin 119862[minus120587 120587] that inverts (8) andconditions (9)-(11) into an identity
Definition 2 By a generalized solution of the inverseproblem (8)-(11) we mean a pair of functions (119906(119909 119905) 119891(119909))of the class 119906(119909 119905) isin 119882212 (Ω) cap 119862(Ω) 119891(119909) isin 1198712(minus120587 120587) thatsatisfies (8) and conditions (9)-(11) almost everywhere
When one uses the method of separation of variablesto solve the problem a spectral problem appears which ismentioned in the next section
3 Spectral Problem
The use of the Fourier method for solving problem (8)ndash(11)leads to a spectral problem for an operator L given by thedifferential expression
L119883 (119909) equiv minus11988310158401015840 (119909) + 12057611988310158401015840 (minus119909) = 120582119883 (119909) minus120587 lt 119909 lt 120587 (12)
and the boundary conditions
1198831015840 (minus120587) minus 1198831015840 (120587) minus 119886119883 (120587) = 0119883 (minus120587) minus 119883 (120587) = 0 (13)
where 120582 is a spectral parameterThe spectral problems for (12) were first considered
apparently in [37] There was considered a case of Dirichletand Neumann boundary conditions and cases of conditionsin the form (13) for 119886 = 0 Here we consider the case 119886 = 0We assume that 119886 gt 0
We represent a general solution of (12) in the followingform
119883 (119909) = 119860 sin (1205831119909) + 119861 cos (1205832119909) 1205831 = radic 1205821 + 120576
1205832 = radic 1205821 minus 120576 (14)
where119860 and 119861 are arbitrary complex numbers Satisfying theboundary conditions (13) for finding eigenvalues we obtainthe following
sin (1205831120587) = 0tan (1205832120587) = 11988621205832
(15)
Therefore the spectral problem (12)-(13) has two series ofeigenvalues
1205821198961 = (1 + 120576) 1198962 119896 isin N (16)
1205821198962 = (1 minus 120576)(119896 + 120575119896)2 119896 isin N0 equiv N cup 0 where 120575119896 =119874(119886(119896 + 1)) gt 0 as 119896 rarr infin with corresponding normalizedeigenfunctions given by
1198831198961 (119909) = 1radic120587 sin (119896119909) 119896 isin N1198831198962 (119909) = ]119896 cos ((119896 + 120575119896) 119909) 119896 isin N0
(17)
Here ]119896 is a normalization coefficient
]minus2119896 = 1003817100381710038171003817cos ((119896 + 120575119896) 119909)100381710038171003817100381721198712(minus120587120587)= 120587 + 1198862
(119896 + 120575119896) [1198862 + (119896 + 120575119896)2 1205872] (18)
Lemma 3 The system of functions (17) is complete andorthonormal in 1198712(minus120587 120587)Proof We note that the system of eigenfunctions (17) doesnot depend on the parameter 120576 Only the eigenvalues ofproblem (12)-(13) depend on 120576
In the case when 120576 = 0 system (17) is a system ofeigenfunctions of the classical Strum-Liouville problem
minus11988310158401015840 (119909) = 120582119883 (119909) minus 120587 lt 119909 lt 120587 (19)
with the self-adjoint boundary conditions (13)Consequently system (17) forms the complete orthonor-
mal system in 1198712(minus120587 120587) that is it is the orthonormal basis ofthe space
4 Uniqueness of the Solution of the Problem
Let the pair of functions (119906(119909 119905) 119891(119909)) be a solution of theinverse problem (8)-(11) Let us introduce notations
119906119896119894 (119905) = int120587minus120587119906 (119909 119905)119883119896119894 (119909) 119889119909
119891119896119894 = int120587minus120587119891 (119909)119883119896119894 (119909) 119889119909
119894 = 1 2(20)
4 Advances in Mathematical Physics
We apply the operator 119889119889119905 to 119906119896119894(119905) Then using (8) byintegrating by parts we obtain the following problem
1199061015840119896119894 (119905) + 120582119896119894119906119896119894 (119905) = 119891119896119894 0 lt 119905 lt 119879 119894 = 1 2 (21)
119906119896119894 (0) = 120601119896119894 minus 120590119896119894 119894 = 1 2 (22)
119906119896119894 (119879) = 120595119896119894 minus 120590119896119894 119894 = 1 2 (23)
Here we use the following notations
120601119896119894 = int120587minus120587120601 (119909)119883119896119894 (119909) 119889119909
120595119896119894 = int120587minus120587120595 (119909)119883119896119894 (119909) 119889119909
120590119896119894 = 120574int120587minus120587119891 (120585) 119889120585int120587
minus120587119883119896119894 (119909) 119889119909
(24)
It is easy to see that the function 1198961(119905) = (120582119896119894)minus1119891119896119894is a partial solution of the inhomogeneous equation (21)Therefore we obtain a unique solution of theCauchy problem(21)-(22) in the following form
119906119896119894 (119905) = (120601119896119894 minus 120590119896119894 minus 119891119896119894120582119896119894) 119890minus120582119896119894119905 + 119891119896119894120582119896119894 (25)
Since |120576| lt 1 119886 gt 0 we have 120582119896119894 gt 0 for all 119896 119894Consequently
1 minus 119890minus120582119896119894119879 ge 1 minus 119890minus12058202119879 gt 0for all the values of the indexes 119896 119894 (26)
Therefore using the condition of rdquofinal overdetermina-tionrdquo (23) we get
119891119896119894 = 120582119896119894120595119896119894 minus 120601119896119894119890minus120582119896119894119879
1 minus 119890minus120582119896119894119879 minus 120582119896119894120590119896119894 (27)
Lemma 4 If 119886 gt 0 then the generalized solution(119906(119909 119905) 119891(119909)) of the inverse problem (8)-(11) is unique
Proof Suppose that there are two generalized solutionsof the inverse problem (8)-(11) (1199061(119909 119905) 1198911(119909)) and(1199062(119909 119905) 1198912(119909)) Denote
119906 (119909 119905) = 1199061 (119909 119905) minus 1199062 (119909 119905) 119891 (119909) = 1198911 (119909) minus 1198912 (119909) (28)
Then the functions 119906(119909 119905) and 119891(119909) satisfy (8) the boundaryconditions (11) and the homogeneous conditions (9) and (10)
119906 (119909 0) = minus120574int120587minus120587119891 (120585) 119889120585 119909 isin [minus120587 120587]
119906 (119909 119879) = minus120574int120587minus120587119891 (120585) 119889120585 119909 isin [minus120587 120587]
(29)
Therefore by using the notations (20) from (27) we find
119891119896119894 = minus120582119896119894120590119896119894 equiv minus120582119896119894120574int120587minus120587119891 (120585) 119889120585int120587
minus120587119883119896119894 (119909) 119889119909 (30)
Since
120582119896119894 int120587minus120587119883119896119894 (119909) 119889119909 = int120587
minus120587minus11988310158401015840119896119894 (119909) + 12057611988310158401015840119896119894 (minus119909) 119889119909
= (1 minus 120576) 1198831015840119896119894 (minus120587) minus 1198831015840119896119894 (120587)= 119886 (1 minus 120576)119883119896119894 (120587)
(31)
this gives
119891119896119894 = minus119886 (1 minus 120576) (120574int120587minus120587119891 (120585) 119889120585)119883119896119894 (120587) (32)
Since1198831198961(120587) = 0 then for 119894 = 1 from (32) we have 1198911198961 =0For the generalized solution of the problem (8)-(11) it
is necessary that 119891 isin 1198712(minus120587 120587) From this by virtue ofParsevalrsquos equality for orthogonal bases we have that
infinsum119896=1
1003816100381610038161003816119891119896210038161003816100381610038162 le 100381710038171003817100381711989110038171003817100381710038172 lt infin (33)
is necessarySince 119886(1 minus 120576) = 0
1198831198962 (120587) = ]119896 cos ((119896 + 120575119896) 120587) lim119896rarrinfin
]119896 = 1radic120587lim119896rarrinfin
120575119896 = 0(34)
then (32) and inequality (33) are possible only at(120574 int120587minus120587119891(120585)119889120585) = 0 Hence we obtain 1198911198962 = 0
Therefore using this result from (25) and (27) we find
119906119896119894 (119905) equiv int120587minus120587119906 (119909 119905) 119883119896119894 (119909) 119889119909 = 0
119891119896119894 equiv int120587minus120587119891 (119909)119883119896119894 (119909) 119889119909 = 0
(35)
for all values of the indexes 119896 isin N for 119894 = 1 and 119896 isin N0 for 119894 =2 Further by the completeness of system (17) in 1198712(minus120587 120587)we obtain
119906 (119909 119905) equiv 0119891 (119909) equiv 0
forall (119909 119905) isin Ω(36)
The uniqueness of the generalized solution of the inverseproblem (8)-(11) is proved
Corollary 5 If 119886 gt 0 then the regular solution (119906(119909 119905) 119891(119909))of the inverse problem (8)-(11) is unique
Advances in Mathematical Physics 5
5 Construction of a Formal Solution ofthe Problem
As the eigenfunctions system (17) forms the orthonormalbasis in 1198712(minus120587 120587) then the unknown functions 119906(119909 119905) and119891(119909) can be represented as
119906 (119909 119905) = infinsum119896=1
1199061198961 (119905) 1198831198961 (119909) + infinsum119896=0
1199061198962 (119905) 1198831198962 (119909) (37)
119891 (119909) = infinsum119896=1
11989111989611198831198961 (119909) + infinsum119896=0
11989111989621198831198962 (119909) (38)
where 1199061198961(119905) and 1199061198962(119905) are unknown functions 1198911198961 and 1198911198962are unknown constants
Substituting (37) and (38) into (8) we obtain the inverseproblems (21)-(23) If the constants 120590119896119894 are assumed to begiven then the solutions of these inverse problems existare unique and are represented by formulas (25) and (27)Substituting (25) and (27) into series (37) and (38) we obtaina formal solution of the inverse problem (8)-(11)
Let 120601(119909) 120595(119909) isin 11988222 (minus120587 120587) and satisfy the boundaryconditions (13) Since 120582119896119894 gt 0 then
infinsum119896=1
1003816100381610038161003816100381610038161003816100381610038161003816120582119896119894120595119896119894 minus 120601119896119894119890minus1205821198961198941198791 minus 119890minus120582119896119894119879
10038161003816100381610038161003816100381610038161003816100381610038162
le 11986210038171003817100381710038171206011003817100381710038171003817211988222(minus120587120587)
+ 10038171003817100381710038171205951003817100381710038171003817211988222(minus120587120587)
119894 = 1 2(39)
Therefore from the analysis of formula (27) it is easy tosee that in order for the formal solution (37) of the problem(8)-(11) to be a generalized solution it is necessary that
lim119896rarrinfin
120582119896119894120590119896119894 = 0 119894 = 1 2 (40)
holdsAs above we calculate
120582119896119894120590119896119894 equiv 119886 (1 minus 120576) (120574int120587minus120587119891 (120585) 119889120585)119883119896119894 (120587) (41)
where1198831198961(120587) = 0 and1198831198962 (120587) = ]119896 cos ((119896 + 120575119896) 120587) lim119896rarrinfin
]119896 = 1radic120587lim119896rarrinfin
120575119896 = 0(42)
Thus (40) holds if and only if 120590119896119894 = 0 for all values of theindexes 119896 119894 This is possible only in the case
int120587minus120587119891 (120585) 119889120585 = 0 (43)
Thus condition (43) is a necessary condition for theexistence of the generalized solution of the inverse problem
(8)-(11) In this case problems (8)-(11) and (1)-(4) coincideIndeed from (43) and (1) we have
0 = int120587minus120587119891 (120585) 119889120585
= int120587minus120587Φ119905 (120585 119905) 119889120585
minus int120587minus120587Φ120585120585 (120585 119905) + 120576Φ120585120585 (minus120585 119905) 119889120585
(44)
For the first integral we apply condition (4) and calculatethe second integral Then we obtain
0 = (1 minus 120576)sdot [Φ119909 (minus120587 119905) minus Φ119909 (120587 119905) + 1120574 (1 minus 120576)Φ (minus120587 119905)] (45)
This means that the boundary conditions (4) and (11) coin-cide Hence the problems (8)-(11) and (1)-(4) also coincide
Thus in what follows we shall consider problem (1)-(3)with the boundary condition
Φ119909 (minus120587 119905) minus Φ119909 (120587 119905) minus 119886Φ (minus120587 119905) = 0 (46)Thus in what follows we will consider the inverse
problem (1)-(3) (46)Similarly as before the formal solution of this problem
can be constructed in the form of series
Φ (119909 119905) = infinsum119896=1
Φ1198961 (119905) 1198831198961 (119909) + infinsum119896=0
Φ1198962 (119905) 1198831198962 (119909) (47)
119891 (119909) = infinsum119896=1
11989111989611198831198961 (119909) + infinsum119896=0
11989111989621198831198962 (119909) (48)
where
Φ119896119894 (119905) = (120601119896119894 minus 119891119896119894120582119896119894) 119890minus120582119896119894119905 + 119891119896119894120582119896119894 (49)
119891119896119894 = 120582119896119894120595119896119894 minus 120601119896119894119890minus120582119896119894119879
1 minus 119890minus120582119896119894119879 (50)
In order to complete our study it is necessary as in theFourier method to justify the smoothness of the resultingformal solutions and the convergence of all appearing series
6 Main Results
Here we present the existence and uniqueness results for ourinverse problem
Theorem 6 (let 119886 gt 0) (A) Let 120601(119909) 120595(119909) isin 11988222 (minus120587 120587)and let the functions120601(119909) 120595(119909) satisfy the boundary conditions(13) Then for a real number 120576 such that |120576| lt 1 the inverseproblem (1)-(3) (46) has a unique generalized solution whichis stable in norm1003817100381710038171003817Φ119905100381710038171003817100381721198712(Ω) + 1003817100381710038171003817Φ119909119909100381710038171003817100381721198712(Ω) + 1003817100381710038171003817119891100381710038171003817100381721198712(minus120587120587)
le 11986210038171003817100381710038171206011003817100381710038171003817211988222(minus120587120587)
+ 10038171003817100381710038171205951003817100381710038171003817211988222(minus120587120587)
(51)
where the constant 119862 does not depend on 120601(119909) and 120595(119909)
6 Advances in Mathematical Physics
(B) Let 120601(119909) 120595(119909) isin 1198624[minus120587 120587] and the functions120601(119909) 120595(119909) and 12060110158401015840(119909) 12059510158401015840(119909) satisfy the boundary conditions(13) then for a real number 120576 such that |120576| lt 1 the inverseproblem (1)-(3) (46) has a unique regular solution
Proof Since |120576| lt 1 119886 gt 0 we have 120582119896119894 gt 0 for all 119896 119894Consequently there exists a constant 120573 gt 0 such that for allthe values of the indexes 119896 119894 we have
1 minus 119890minus120582119896119894119879 ge 1 minus 119890minus12058202119879 ge 120573 gt 0 (52)
Since 120582119896119894 gt 0 then 119890minus120582119896119894119879 le 119890minus120582119896119894119905 lt 1 Therefore fromrepresentations (49) and (50) we get estimates
10038161003816100381610038161198911198961198941003816100381610038161003816 le10038161003816100381610038161205821198961198941003816100381610038161003816120573 10038161003816100381610038161206011198961198941003816100381610038161003816 + 10038161003816100381610038161205951198961198941003816100381610038161003816 (53)
1003816100381610038161003816Φ119896119894 (119905)1003816100381610038161003816 le (1 + 2120573) 10038161003816100381610038161206011198961198941003816100381610038161003816 + 10038161003816100381610038161205951198961198941003816100381610038161003816 (54)
where constant 120573 does not depend on the indexes 119896 119894 and onthe functions 120601(119909) and 120595(119909)
As proved in Lemma 3 the eigenfunctions system (17)forms the orthonormal basis in 1198712(minus120587 120587)Therefore we havean expansion
120601 (119909) = infinsum119896=1
12060111989611198831198961 (119909) + infinsum119896=0
12060111989621198831198962 (119909) (55)
Taking into account (12) we obtain
minus12060110158401015840 (119909) + 12057612060110158401015840 (minus119909) = infinsum119896=1
120582119896112060111989611198831198961 (119909)
+ infinsum119896=0
120582119896212060111989621198831198962 (119909) (56)
From this by Parsevalrsquos equality it is easy to obtain anestimate
infinsum119896=1
1003816100381610038161003816120582119896110038161003816100381610038162 1003816100381610038161003816120601119896110038161003816100381610038162 +infinsum119896=1
1003816100381610038161003816120582119896210038161003816100381610038162 1003816100381610038161003816120601119896210038161003816100381610038162
le 2 (1 + 1205762) 10038171003817100381710038171003817120601101584010158401003817100381710038171003817100381721198712(minus120587120587) (57)
Similarly for the coefficients of the expansion
120595 (119909) = infinsum119896=1
12059511989611198831198961 (119909) + infinsum119896=0
12059511989621198831198962 (119909) (58)
we have the inequality
infinsum119896=1
1003816100381610038161003816120582119896110038161003816100381610038162 1003816100381610038161003816120595119896110038161003816100381610038162 +infinsum119896=1
1003816100381610038161003816120582119896210038161003816100381610038162 1003816100381610038161003816120595119896210038161003816100381610038162
le 2 (1 + 1205762) 10038171003817100381710038171003817120595101584010158401003817100381710038171003817100381721198712(minus120587120587) (59)
Now from (53) (57) and (59) we see that the series (48)converges and the estimate
1003817100381710038171003817119891100381710038171003817100381721198712(minus120587120587) le 21 + 12057621205732 10038171003817100381710038171003817120601101584010158401003817100381710038171003817100381721198712(minus120587120587) + 10038171003817100381710038171003817120595101584010158401003817100381710038171003817100381721198712(minus120587120587) (60)
holdsTaking into account (12) from (47) we obtain
minus Φ119909119909 (119909 119905) + 120576Φ119909119909 (minus119909 119905)= infinsum119896=1
1205821198961Φ1198961 (119905) 1198831198961 (119909)
+ infinsum119896=0
1205821198962Φ1198962 (119905) 1198831198962 (119909) (61)
Substituting the value (minus119909) instead of 119909 we findminus Φ119909119909 (minus119909 119905) + 120576Φ119909119909 (119909 119905)= minusinfinsum119896=1
1205821198961Φ1198961 (119905) 1198831198961 (119909)
+ infinsum119896=0
1205821198962Φ1198962 (119905) 1198831198962 (119909) (62)
Here we use the properties 1198831198961(minus119909) = minus1198831198961(119909) 1198831198962(minus119909) =1198831198962(119909) obtained from the explicit form (12) of the eigenfunc-tions
Multiplying (62) by 120576 and summing with (61) we have
(minus1 + 1205762)Φ119909119909 (119909 119905)= (1 minus 120576) infinsum
119896=1
1205821198961Φ1198961 (119905) 1198831198961 (119909)
+ (1 + 120576) infinsum119896=0
1205821198962Φ1198962 (119905) 1198831198962 (119909) (63)
Since |120576| lt 1 from this by Parsevalrsquos equality we find
1003817100381710038171003817Φ119909119909 (sdot 119905)100381710038171003817100381721198712(minus120587120587) = 1(1 + 120576)2
infinsum119896=1
1003816100381610038161003816120582119896110038161003816100381610038162 1003816100381610038161003816Φ1198961 (119905)10038161003816100381610038162
+ 1(1 minus 120576)2
infinsum119896=0
1003816100381610038161003816120582119896210038161003816100381610038162 1003816100381610038161003816Φ1198962 (119905)10038161003816100381610038162(64)
for each 119905 isin [0 119879] Integrating this equation with respect to119905 isin [0 119879] taking (54) into account we have1003817100381710038171003817Φ119909119909100381710038171003817100381721198712(Ω) le 8119879
(1 minus 1205762)2 (1 +2120573)2
sdot 2sum119894=1
infinsum119896=1
100381610038161003816100381612058211989611989410038161003816100381610038162 100381610038161003816100381612060111989611989410038161003816100381610038162 + 100381610038161003816100381612059511989611989410038161003816100381610038162 (65)
Advances in Mathematical Physics 7
Hence using (57) and (59) we obtain
1003817100381710038171003817Φ119909119909100381710038171003817100381721198712(Ω) le 16119879 1 + 1205762(1 minus 1205762)2 (1 +
2120573)2
sdot 10038171003817100381710038171003817120601101584010158401003817100381710038171003817100381721198712(minus120587120587) + 10038171003817100381710038171003817120595101584010158401003817100381710038171003817100381721198712(minus120587120587) (66)
Now we can easily obtain an estimate for Ψ119905(119909 119905) from(1) This fact together with (60) and (66) gives the necessaryestimate (51) for the solution
From the obtained estimates it also follows that in theconstructed by us formal solution of the inverse problem allthe series converge they can be term-by-term differentiatedand the series obtained during differentiation also convergein sense of the metrics 1198712
From (47) and (54) by using the Holderrsquos inequality it iseasy to justify the inequality
max(119909119905)isinΩ
|Φ (119909 119905)|2 le 11986210038171003817100381710038171003817120601101584010158401003817100381710038171003817100381721198712(minus120587120587) + 10038171003817100381710038171003817120595101584010158401003817100381710038171003817100381721198712(minus120587120587) (67)
which justifies the continuity ofΦ(119909 119905) in the closed domainΩFrom the representation of the solution in the form of
series (37) (38) and inequalities (53) (54) it is easy to justifythe estimates 1003816100381610038161003816Φ119909119909 (119909 119905)1003816100381610038161003816 + 1003816100381610038161003816Φ119905 (119909 119905)1003816100381610038161003816 + 1003816100381610038161003816119891 (119909)1003816100381610038161003816
le 119862infinsum119896=1
100381610038161003816100381612058211989611989410038161003816100381610038162 10038161003816100381610038161206011198961198941003816100381610038161003816 + 10038161003816100381610038161205951198961198941003816100381610038161003816 (68)
Let 120601(119909) 120595(119909) isin 1198624[minus120587 120587] and let the functions 120601(119909) 120595(119909)and 12060110158401015840(119909) 12059510158401015840(119909) satisfy the boundary conditions (13) thenthe number series in the right-hand side of (68) convergesTherefore in such case the constructed by us formal solutiongives the regular solution of the inverse problem (1)-(3) (46)
The uniqueness of this constructed solution follows fromLemma 4
The theorem is completely proved
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This paper was published with the support of the ScienceCommittee of the Ministry of Education and Science of theRepublic of Kazakhstan under Project AP05133271 and TargetProgram BR05236656
References
[1] B Ahmad A Alsaedi M Kirane and R G TapdigogluldquoAn inverse problem for space and time fractional evolution
equations with an involution perturbationrdquo Quaestiones Math-ematicae vol 40 no 2 pp 151ndash160 2017
[2] A Cabada and A F Tojo ldquoEquations with involutionsrdquo inProceedings of the Workshop on Differential Equations p 240MallaMoravka CzechRepublicMarch 2014 httpusersmathcascz
[3] A Ashyralyev and A Sarsenbi ldquoWell-posedness of a parabolicequation with nonlocal boundary conditionrdquo Boundary ValueProblems vol 2015 no 1 2015
[4] M Kirane and N Al-Salti ldquoInverse problems for a nonlocalwave equation with an involution perturbationrdquo Journal ofNonlinear Science and Applications vol 9 no 3 pp 1243ndash12512016
[5] A Ashyralyev and A Sarsenbi ldquoWell-Posedness of a ParabolicEquation with Involutionrdquo Numerical Functional Analysis andOptimization vol 38 no 10 pp 1295ndash1304 2017
[6] I Orazov and M A Sadybekov ldquoOne nonlocal problem ofdetermination of the temperature and density of heat sourcesrdquoRussian Mathematics vol 56 no 2 pp 60ndash64 2012
[7] I Orazov and M A Sadybekov ldquoOn a class of problems ofdetermining the temperature and density of heat sources giveninitial and final temperaturerdquo Siberian Mathematical Journalvol 53 no 1 pp 146ndash151 2012
[8] N I Ivanchov ldquoInverse problems for the heat-conductionequation with nonlocal boundary conditionsrdquoUkrainianMath-ematical Journal vol 45 no 8 pp 1186ndash1192 1993
[9] I A Kaliev and M M Sabitova ldquoProblems of determiningthe temperature and density of heat sources from the initialand final temperaturesrdquo Journal of Applied and IndustrialMathematics vol 4 no 3 pp 332ndash339 2010
[10] I A KalievM FMugafarov andOV Fattahova ldquoInverse prob-lem for forwardbackward parabolic equation with generalizedconjugation conditionsrdquo Ufa Mathematical Journal vol 3 no2 pp 33ndash41 2011
[11] M Kirane and S A Malik ldquoDetermination of an unknownsource term and the temperature distribution for the linearheat equation involving fractional derivative in timerdquo AppliedMathematics and Computation vol 218 no 1 pp 163ndash170 2011
[12] F Kanca andM I Ismailov ldquoThe inverse problem of finding thetime-dependent diffusion coefficient of the heat equation fromintegral overdetermination datardquo Inverse Problems in Scienceand Engineering vol 20 no 4 pp 463ndash476 2012
[13] M Kirane S A Malik and M Al-Gwaiz ldquoAn inverse sourceproblem for a two dimensional time fractional diffusionequation with nonlocal boundary conditionsrdquo MathematicalMethods in the Applied Sciences vol 36 no 9 pp 1056ndash10692013
[14] A Ashyralyev and Y A Sharifov ldquoExistence and uniquenessof solutions for nonlinear impulsive differential equations withtwo-point and integral boundary conditionsrdquo Advances inDifference Equations vol 2013 no 1 p 173 2013
[15] F Kanca ldquoInverse coefficient problem of the parabolic equa-tion with periodic boundary and integral overdeterminationconditionsrdquo Abstract and Applied Analysis vol 2013 Article ID659804 7 pages 2013
[16] D Lesnic S A Yousefi and M Ivanchov ldquoDetermination of atime-dependent diffusivity from nonlocal conditionsrdquo AppliedMathematics and Computation vol 41 no 1-2 pp 301ndash3202013
[17] L Miller and M Yamamoto ldquoCoefficient inverse problem for afractional diffusion equationrdquo Inverse Problems vol 29 no 7 p075013 2013
8 Advances in Mathematical Physics
[18] G Li D Zhang X Jia and M Yamamoto ldquoSimultaneousinversion for the space-dependent diffusion coefficient andthe fractional order in the time-fractional diffusion equationrdquoInverse Problems vol 29 no 6 Article ID 065014 2013
[19] A B Kostin ldquoCounterexamples in inverse problems forparabolic elliptic and hyperbolic equationsrdquo ComputationalMathematics and Mathematical Physics vol 54 no 5 pp 797ndash810 2014
[20] A Ashyralyev and A Hanalyev ldquoWell-posedness of nonlocalparabolic differential problems with dependent operatorsrdquoTheScientific World Journal vol 2014 Article ID 519814 2014
[21] I Orazov and M A Sadybekov ldquoOn an inverse problemof mathematical modeling of the extraction process of poly-disperse porous materialsrdquo in Proceedings of the ADVANCE-MENTS IN MATHEMATICAL SCIENCES Proceedings of theInternational Conference on Advancements in MathematicalSciences p 020005 Antalya Turkey
[22] I Orazov and M A Sadybekov ldquoOne-dimensional diffusionproblem with not strengthened regular boundary conditionsrdquoin Proceedings of the 41ST International Conference ldquoApplicationsof Mathematics in Engineering And EconomicSrdquo AMEE rsquo15 p040007 Sozopol Bulgaria
[23] N H Tuan D N D Hai L D Long V T Nguyen and MKirane ldquoOn a RieszndashFeller space fractional backward diffusionproblemwith a nonlinear sourcerdquo Journal of Computational andApplied Mathematics vol 312 pp 103ndash126 2017
[24] M Sadybekov G Oralsyn and M Ismailov ldquoAn inverseproblem of finding the time-dependent heat transfer coefficientfrom an integral conditionrdquo International Journal of Pure andApplied Mathematics vol 113 no 4 pp 139ndash149 2017
[25] M Kirane andMNH Tuan ldquoIdentification and regularizationfor unknown source for a time-fractional diffusion equationrdquoComputersMathematics withApplications vol 73 no 6 pp 931ndash950 2017
[26] B T Torebek and R Tapdigoglu ldquoSome inverse problems forthe nonlocal heat equation with Caputo fractional derivativerdquoMathematical Methods in the Applied Sciences vol 40 no 18pp 6468ndash6479 2017
[27] M Kirane B Samet and B T Torebek ldquoDetermination ofan unknown source term temperature distribution for thesub-diffusion equation at the initial and final datardquo ElectronicJournal of Differential Equations vol 2017 2017
[28] O Taki-Eddine and B Abdelfatah ldquoOn determining the coef-ficient in a parabolic equation with nonlocal boundary andintegral conditionrdquo Electronic Journal of Mathematical Analysisand Applications vol 6 no 1 pp 94ndash102 2018
[29] A M Sarsenbi ldquoUnconditional bases related to a nonclassicalsecond-order differential operatorrdquo Journal of Differential Equa-tions vol 46 no 4 pp 509ndash514 2010
[30] V PKurdyumov andA PKhromov ldquoTheRiesz bases consistingof eigen and associated functions for a functional differentialoperator with variable structurerdquo Russian Mathematics vol 54no 2 pp 33ndash45 2010
[31] A M Sarsenbi and A A Tengaeva ldquoOn the basis propertiesof the root functions of two generalized spectral problemsrdquoJournal of Differential Equations vol 48 no 2 pp 306ndash3082012
[32] M A Sadybekov and A M Sarsenbi ldquoCriterion for the basisproperty of the eigenfunction system of a multiple differen-tiation operator with an involutionrdquo Journal of DifferentialEquations vol 48 no 8 pp 1112ndash1118 2012
[33] A Kopzhassarova and A Sarsenbi ldquoBasis properties of eigen-functions of second-order differential operators with invo-lutionrdquo Abstract and Applied Analysis vol 2012 Article ID576843 6 pages 2012
[34] AAKopzhassarova A L Lukashov andAM Sarsenbi ldquoSpec-tral properties of non-self-adjoint perturbations for a spectralproblem with involutionrdquo Abstract and Applied Analysis vol2012 Article ID 590781 5 pages 2012
[35] A Sarsenbi and M Sadybekov ldquoEigenfunctions of a fourthorder operator pencilrdquo in Proceedings of the 2nd InternationalConference on Analysis and Applied Mathematics ICAAM 2014pp 241ndash245 kaz September 2014
[36] L V Kritskov and A M Sarsenbi ldquoSpectral properties of anonlocal problem for a second-order differential equation withan involutionrdquo Journal of Differential Equations vol 51 no 8pp 984ndash990 2015
[37] L V Kritskov and A M Sarsenbi ldquoBasicity in Lp of rootfunctions for differential equations with involutionrdquo ElectronicJournal of Differential Equations vol 2015 2015
[38] M A Sadybekov A Sarsenbi and A Tengayeva ldquoDescriptionof spectral properties of a generalized spectral problem withinvolution for differentiation operator of the second orderrdquo inProceedings of the 3rd International Conference on Analysis andApplied Mathematics ICAAM 2016 kaz September 2016
[39] A G Baskakov I A Krishtal and E Y Romanova ldquoSpectralanalysis of a differential operator with an involutionrdquo Journal ofEvolution Equations (JEE) vol 17 no 2 pp 669ndash684 2017
[40] L V Kritskov and A M Sarsenbi ldquoRiesz basis property ofsystem of root functions of second-order differential operatorwith involutionrdquo Journal of Differential Equations vol 53 no 1pp 33ndash46 2017
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Advances in Mathematical Physics 3
[6 7] In [1] there are good references to publications onrelated issues We note [3 8ndash28] from recent papers close tothe theme of our article In these papers different variants ofdirect and inverse initial-boundary value problems for evo-lutionary equations are considered including problems withnonlocal boundary conditions and problems for equationswith fractional derivatives
In this paper we shall use a spectral problem for anordinary differential operators with involution Such andsimilar spectral problems were considered in [29ndash40]
Definition 1 By a regular solution of the inverse prob-lem (8)-(11) we mean a pair of functions (119906(119909 119905) 119891(119909)) of theclass 119906(119909 119905) isin 11986221119909119905(Ω) 119891(119909) isin 119862[minus120587 120587] that inverts (8) andconditions (9)-(11) into an identity
Definition 2 By a generalized solution of the inverseproblem (8)-(11) we mean a pair of functions (119906(119909 119905) 119891(119909))of the class 119906(119909 119905) isin 119882212 (Ω) cap 119862(Ω) 119891(119909) isin 1198712(minus120587 120587) thatsatisfies (8) and conditions (9)-(11) almost everywhere
When one uses the method of separation of variablesto solve the problem a spectral problem appears which ismentioned in the next section
3 Spectral Problem
The use of the Fourier method for solving problem (8)ndash(11)leads to a spectral problem for an operator L given by thedifferential expression
L119883 (119909) equiv minus11988310158401015840 (119909) + 12057611988310158401015840 (minus119909) = 120582119883 (119909) minus120587 lt 119909 lt 120587 (12)
and the boundary conditions
1198831015840 (minus120587) minus 1198831015840 (120587) minus 119886119883 (120587) = 0119883 (minus120587) minus 119883 (120587) = 0 (13)
where 120582 is a spectral parameterThe spectral problems for (12) were first considered
apparently in [37] There was considered a case of Dirichletand Neumann boundary conditions and cases of conditionsin the form (13) for 119886 = 0 Here we consider the case 119886 = 0We assume that 119886 gt 0
We represent a general solution of (12) in the followingform
119883 (119909) = 119860 sin (1205831119909) + 119861 cos (1205832119909) 1205831 = radic 1205821 + 120576
1205832 = radic 1205821 minus 120576 (14)
where119860 and 119861 are arbitrary complex numbers Satisfying theboundary conditions (13) for finding eigenvalues we obtainthe following
sin (1205831120587) = 0tan (1205832120587) = 11988621205832
(15)
Therefore the spectral problem (12)-(13) has two series ofeigenvalues
1205821198961 = (1 + 120576) 1198962 119896 isin N (16)
1205821198962 = (1 minus 120576)(119896 + 120575119896)2 119896 isin N0 equiv N cup 0 where 120575119896 =119874(119886(119896 + 1)) gt 0 as 119896 rarr infin with corresponding normalizedeigenfunctions given by
1198831198961 (119909) = 1radic120587 sin (119896119909) 119896 isin N1198831198962 (119909) = ]119896 cos ((119896 + 120575119896) 119909) 119896 isin N0
(17)
Here ]119896 is a normalization coefficient
]minus2119896 = 1003817100381710038171003817cos ((119896 + 120575119896) 119909)100381710038171003817100381721198712(minus120587120587)= 120587 + 1198862
(119896 + 120575119896) [1198862 + (119896 + 120575119896)2 1205872] (18)
Lemma 3 The system of functions (17) is complete andorthonormal in 1198712(minus120587 120587)Proof We note that the system of eigenfunctions (17) doesnot depend on the parameter 120576 Only the eigenvalues ofproblem (12)-(13) depend on 120576
In the case when 120576 = 0 system (17) is a system ofeigenfunctions of the classical Strum-Liouville problem
minus11988310158401015840 (119909) = 120582119883 (119909) minus 120587 lt 119909 lt 120587 (19)
with the self-adjoint boundary conditions (13)Consequently system (17) forms the complete orthonor-
mal system in 1198712(minus120587 120587) that is it is the orthonormal basis ofthe space
4 Uniqueness of the Solution of the Problem
Let the pair of functions (119906(119909 119905) 119891(119909)) be a solution of theinverse problem (8)-(11) Let us introduce notations
119906119896119894 (119905) = int120587minus120587119906 (119909 119905)119883119896119894 (119909) 119889119909
119891119896119894 = int120587minus120587119891 (119909)119883119896119894 (119909) 119889119909
119894 = 1 2(20)
4 Advances in Mathematical Physics
We apply the operator 119889119889119905 to 119906119896119894(119905) Then using (8) byintegrating by parts we obtain the following problem
1199061015840119896119894 (119905) + 120582119896119894119906119896119894 (119905) = 119891119896119894 0 lt 119905 lt 119879 119894 = 1 2 (21)
119906119896119894 (0) = 120601119896119894 minus 120590119896119894 119894 = 1 2 (22)
119906119896119894 (119879) = 120595119896119894 minus 120590119896119894 119894 = 1 2 (23)
Here we use the following notations
120601119896119894 = int120587minus120587120601 (119909)119883119896119894 (119909) 119889119909
120595119896119894 = int120587minus120587120595 (119909)119883119896119894 (119909) 119889119909
120590119896119894 = 120574int120587minus120587119891 (120585) 119889120585int120587
minus120587119883119896119894 (119909) 119889119909
(24)
It is easy to see that the function 1198961(119905) = (120582119896119894)minus1119891119896119894is a partial solution of the inhomogeneous equation (21)Therefore we obtain a unique solution of theCauchy problem(21)-(22) in the following form
119906119896119894 (119905) = (120601119896119894 minus 120590119896119894 minus 119891119896119894120582119896119894) 119890minus120582119896119894119905 + 119891119896119894120582119896119894 (25)
Since |120576| lt 1 119886 gt 0 we have 120582119896119894 gt 0 for all 119896 119894Consequently
1 minus 119890minus120582119896119894119879 ge 1 minus 119890minus12058202119879 gt 0for all the values of the indexes 119896 119894 (26)
Therefore using the condition of rdquofinal overdetermina-tionrdquo (23) we get
119891119896119894 = 120582119896119894120595119896119894 minus 120601119896119894119890minus120582119896119894119879
1 minus 119890minus120582119896119894119879 minus 120582119896119894120590119896119894 (27)
Lemma 4 If 119886 gt 0 then the generalized solution(119906(119909 119905) 119891(119909)) of the inverse problem (8)-(11) is unique
Proof Suppose that there are two generalized solutionsof the inverse problem (8)-(11) (1199061(119909 119905) 1198911(119909)) and(1199062(119909 119905) 1198912(119909)) Denote
119906 (119909 119905) = 1199061 (119909 119905) minus 1199062 (119909 119905) 119891 (119909) = 1198911 (119909) minus 1198912 (119909) (28)
Then the functions 119906(119909 119905) and 119891(119909) satisfy (8) the boundaryconditions (11) and the homogeneous conditions (9) and (10)
119906 (119909 0) = minus120574int120587minus120587119891 (120585) 119889120585 119909 isin [minus120587 120587]
119906 (119909 119879) = minus120574int120587minus120587119891 (120585) 119889120585 119909 isin [minus120587 120587]
(29)
Therefore by using the notations (20) from (27) we find
119891119896119894 = minus120582119896119894120590119896119894 equiv minus120582119896119894120574int120587minus120587119891 (120585) 119889120585int120587
minus120587119883119896119894 (119909) 119889119909 (30)
Since
120582119896119894 int120587minus120587119883119896119894 (119909) 119889119909 = int120587
minus120587minus11988310158401015840119896119894 (119909) + 12057611988310158401015840119896119894 (minus119909) 119889119909
= (1 minus 120576) 1198831015840119896119894 (minus120587) minus 1198831015840119896119894 (120587)= 119886 (1 minus 120576)119883119896119894 (120587)
(31)
this gives
119891119896119894 = minus119886 (1 minus 120576) (120574int120587minus120587119891 (120585) 119889120585)119883119896119894 (120587) (32)
Since1198831198961(120587) = 0 then for 119894 = 1 from (32) we have 1198911198961 =0For the generalized solution of the problem (8)-(11) it
is necessary that 119891 isin 1198712(minus120587 120587) From this by virtue ofParsevalrsquos equality for orthogonal bases we have that
infinsum119896=1
1003816100381610038161003816119891119896210038161003816100381610038162 le 100381710038171003817100381711989110038171003817100381710038172 lt infin (33)
is necessarySince 119886(1 minus 120576) = 0
1198831198962 (120587) = ]119896 cos ((119896 + 120575119896) 120587) lim119896rarrinfin
]119896 = 1radic120587lim119896rarrinfin
120575119896 = 0(34)
then (32) and inequality (33) are possible only at(120574 int120587minus120587119891(120585)119889120585) = 0 Hence we obtain 1198911198962 = 0
Therefore using this result from (25) and (27) we find
119906119896119894 (119905) equiv int120587minus120587119906 (119909 119905) 119883119896119894 (119909) 119889119909 = 0
119891119896119894 equiv int120587minus120587119891 (119909)119883119896119894 (119909) 119889119909 = 0
(35)
for all values of the indexes 119896 isin N for 119894 = 1 and 119896 isin N0 for 119894 =2 Further by the completeness of system (17) in 1198712(minus120587 120587)we obtain
119906 (119909 119905) equiv 0119891 (119909) equiv 0
forall (119909 119905) isin Ω(36)
The uniqueness of the generalized solution of the inverseproblem (8)-(11) is proved
Corollary 5 If 119886 gt 0 then the regular solution (119906(119909 119905) 119891(119909))of the inverse problem (8)-(11) is unique
Advances in Mathematical Physics 5
5 Construction of a Formal Solution ofthe Problem
As the eigenfunctions system (17) forms the orthonormalbasis in 1198712(minus120587 120587) then the unknown functions 119906(119909 119905) and119891(119909) can be represented as
119906 (119909 119905) = infinsum119896=1
1199061198961 (119905) 1198831198961 (119909) + infinsum119896=0
1199061198962 (119905) 1198831198962 (119909) (37)
119891 (119909) = infinsum119896=1
11989111989611198831198961 (119909) + infinsum119896=0
11989111989621198831198962 (119909) (38)
where 1199061198961(119905) and 1199061198962(119905) are unknown functions 1198911198961 and 1198911198962are unknown constants
Substituting (37) and (38) into (8) we obtain the inverseproblems (21)-(23) If the constants 120590119896119894 are assumed to begiven then the solutions of these inverse problems existare unique and are represented by formulas (25) and (27)Substituting (25) and (27) into series (37) and (38) we obtaina formal solution of the inverse problem (8)-(11)
Let 120601(119909) 120595(119909) isin 11988222 (minus120587 120587) and satisfy the boundaryconditions (13) Since 120582119896119894 gt 0 then
infinsum119896=1
1003816100381610038161003816100381610038161003816100381610038161003816120582119896119894120595119896119894 minus 120601119896119894119890minus1205821198961198941198791 minus 119890minus120582119896119894119879
10038161003816100381610038161003816100381610038161003816100381610038162
le 11986210038171003817100381710038171206011003817100381710038171003817211988222(minus120587120587)
+ 10038171003817100381710038171205951003817100381710038171003817211988222(minus120587120587)
119894 = 1 2(39)
Therefore from the analysis of formula (27) it is easy tosee that in order for the formal solution (37) of the problem(8)-(11) to be a generalized solution it is necessary that
lim119896rarrinfin
120582119896119894120590119896119894 = 0 119894 = 1 2 (40)
holdsAs above we calculate
120582119896119894120590119896119894 equiv 119886 (1 minus 120576) (120574int120587minus120587119891 (120585) 119889120585)119883119896119894 (120587) (41)
where1198831198961(120587) = 0 and1198831198962 (120587) = ]119896 cos ((119896 + 120575119896) 120587) lim119896rarrinfin
]119896 = 1radic120587lim119896rarrinfin
120575119896 = 0(42)
Thus (40) holds if and only if 120590119896119894 = 0 for all values of theindexes 119896 119894 This is possible only in the case
int120587minus120587119891 (120585) 119889120585 = 0 (43)
Thus condition (43) is a necessary condition for theexistence of the generalized solution of the inverse problem
(8)-(11) In this case problems (8)-(11) and (1)-(4) coincideIndeed from (43) and (1) we have
0 = int120587minus120587119891 (120585) 119889120585
= int120587minus120587Φ119905 (120585 119905) 119889120585
minus int120587minus120587Φ120585120585 (120585 119905) + 120576Φ120585120585 (minus120585 119905) 119889120585
(44)
For the first integral we apply condition (4) and calculatethe second integral Then we obtain
0 = (1 minus 120576)sdot [Φ119909 (minus120587 119905) minus Φ119909 (120587 119905) + 1120574 (1 minus 120576)Φ (minus120587 119905)] (45)
This means that the boundary conditions (4) and (11) coin-cide Hence the problems (8)-(11) and (1)-(4) also coincide
Thus in what follows we shall consider problem (1)-(3)with the boundary condition
Φ119909 (minus120587 119905) minus Φ119909 (120587 119905) minus 119886Φ (minus120587 119905) = 0 (46)Thus in what follows we will consider the inverse
problem (1)-(3) (46)Similarly as before the formal solution of this problem
can be constructed in the form of series
Φ (119909 119905) = infinsum119896=1
Φ1198961 (119905) 1198831198961 (119909) + infinsum119896=0
Φ1198962 (119905) 1198831198962 (119909) (47)
119891 (119909) = infinsum119896=1
11989111989611198831198961 (119909) + infinsum119896=0
11989111989621198831198962 (119909) (48)
where
Φ119896119894 (119905) = (120601119896119894 minus 119891119896119894120582119896119894) 119890minus120582119896119894119905 + 119891119896119894120582119896119894 (49)
119891119896119894 = 120582119896119894120595119896119894 minus 120601119896119894119890minus120582119896119894119879
1 minus 119890minus120582119896119894119879 (50)
In order to complete our study it is necessary as in theFourier method to justify the smoothness of the resultingformal solutions and the convergence of all appearing series
6 Main Results
Here we present the existence and uniqueness results for ourinverse problem
Theorem 6 (let 119886 gt 0) (A) Let 120601(119909) 120595(119909) isin 11988222 (minus120587 120587)and let the functions120601(119909) 120595(119909) satisfy the boundary conditions(13) Then for a real number 120576 such that |120576| lt 1 the inverseproblem (1)-(3) (46) has a unique generalized solution whichis stable in norm1003817100381710038171003817Φ119905100381710038171003817100381721198712(Ω) + 1003817100381710038171003817Φ119909119909100381710038171003817100381721198712(Ω) + 1003817100381710038171003817119891100381710038171003817100381721198712(minus120587120587)
le 11986210038171003817100381710038171206011003817100381710038171003817211988222(minus120587120587)
+ 10038171003817100381710038171205951003817100381710038171003817211988222(minus120587120587)
(51)
where the constant 119862 does not depend on 120601(119909) and 120595(119909)
6 Advances in Mathematical Physics
(B) Let 120601(119909) 120595(119909) isin 1198624[minus120587 120587] and the functions120601(119909) 120595(119909) and 12060110158401015840(119909) 12059510158401015840(119909) satisfy the boundary conditions(13) then for a real number 120576 such that |120576| lt 1 the inverseproblem (1)-(3) (46) has a unique regular solution
Proof Since |120576| lt 1 119886 gt 0 we have 120582119896119894 gt 0 for all 119896 119894Consequently there exists a constant 120573 gt 0 such that for allthe values of the indexes 119896 119894 we have
1 minus 119890minus120582119896119894119879 ge 1 minus 119890minus12058202119879 ge 120573 gt 0 (52)
Since 120582119896119894 gt 0 then 119890minus120582119896119894119879 le 119890minus120582119896119894119905 lt 1 Therefore fromrepresentations (49) and (50) we get estimates
10038161003816100381610038161198911198961198941003816100381610038161003816 le10038161003816100381610038161205821198961198941003816100381610038161003816120573 10038161003816100381610038161206011198961198941003816100381610038161003816 + 10038161003816100381610038161205951198961198941003816100381610038161003816 (53)
1003816100381610038161003816Φ119896119894 (119905)1003816100381610038161003816 le (1 + 2120573) 10038161003816100381610038161206011198961198941003816100381610038161003816 + 10038161003816100381610038161205951198961198941003816100381610038161003816 (54)
where constant 120573 does not depend on the indexes 119896 119894 and onthe functions 120601(119909) and 120595(119909)
As proved in Lemma 3 the eigenfunctions system (17)forms the orthonormal basis in 1198712(minus120587 120587)Therefore we havean expansion
120601 (119909) = infinsum119896=1
12060111989611198831198961 (119909) + infinsum119896=0
12060111989621198831198962 (119909) (55)
Taking into account (12) we obtain
minus12060110158401015840 (119909) + 12057612060110158401015840 (minus119909) = infinsum119896=1
120582119896112060111989611198831198961 (119909)
+ infinsum119896=0
120582119896212060111989621198831198962 (119909) (56)
From this by Parsevalrsquos equality it is easy to obtain anestimate
infinsum119896=1
1003816100381610038161003816120582119896110038161003816100381610038162 1003816100381610038161003816120601119896110038161003816100381610038162 +infinsum119896=1
1003816100381610038161003816120582119896210038161003816100381610038162 1003816100381610038161003816120601119896210038161003816100381610038162
le 2 (1 + 1205762) 10038171003817100381710038171003817120601101584010158401003817100381710038171003817100381721198712(minus120587120587) (57)
Similarly for the coefficients of the expansion
120595 (119909) = infinsum119896=1
12059511989611198831198961 (119909) + infinsum119896=0
12059511989621198831198962 (119909) (58)
we have the inequality
infinsum119896=1
1003816100381610038161003816120582119896110038161003816100381610038162 1003816100381610038161003816120595119896110038161003816100381610038162 +infinsum119896=1
1003816100381610038161003816120582119896210038161003816100381610038162 1003816100381610038161003816120595119896210038161003816100381610038162
le 2 (1 + 1205762) 10038171003817100381710038171003817120595101584010158401003817100381710038171003817100381721198712(minus120587120587) (59)
Now from (53) (57) and (59) we see that the series (48)converges and the estimate
1003817100381710038171003817119891100381710038171003817100381721198712(minus120587120587) le 21 + 12057621205732 10038171003817100381710038171003817120601101584010158401003817100381710038171003817100381721198712(minus120587120587) + 10038171003817100381710038171003817120595101584010158401003817100381710038171003817100381721198712(minus120587120587) (60)
holdsTaking into account (12) from (47) we obtain
minus Φ119909119909 (119909 119905) + 120576Φ119909119909 (minus119909 119905)= infinsum119896=1
1205821198961Φ1198961 (119905) 1198831198961 (119909)
+ infinsum119896=0
1205821198962Φ1198962 (119905) 1198831198962 (119909) (61)
Substituting the value (minus119909) instead of 119909 we findminus Φ119909119909 (minus119909 119905) + 120576Φ119909119909 (119909 119905)= minusinfinsum119896=1
1205821198961Φ1198961 (119905) 1198831198961 (119909)
+ infinsum119896=0
1205821198962Φ1198962 (119905) 1198831198962 (119909) (62)
Here we use the properties 1198831198961(minus119909) = minus1198831198961(119909) 1198831198962(minus119909) =1198831198962(119909) obtained from the explicit form (12) of the eigenfunc-tions
Multiplying (62) by 120576 and summing with (61) we have
(minus1 + 1205762)Φ119909119909 (119909 119905)= (1 minus 120576) infinsum
119896=1
1205821198961Φ1198961 (119905) 1198831198961 (119909)
+ (1 + 120576) infinsum119896=0
1205821198962Φ1198962 (119905) 1198831198962 (119909) (63)
Since |120576| lt 1 from this by Parsevalrsquos equality we find
1003817100381710038171003817Φ119909119909 (sdot 119905)100381710038171003817100381721198712(minus120587120587) = 1(1 + 120576)2
infinsum119896=1
1003816100381610038161003816120582119896110038161003816100381610038162 1003816100381610038161003816Φ1198961 (119905)10038161003816100381610038162
+ 1(1 minus 120576)2
infinsum119896=0
1003816100381610038161003816120582119896210038161003816100381610038162 1003816100381610038161003816Φ1198962 (119905)10038161003816100381610038162(64)
for each 119905 isin [0 119879] Integrating this equation with respect to119905 isin [0 119879] taking (54) into account we have1003817100381710038171003817Φ119909119909100381710038171003817100381721198712(Ω) le 8119879
(1 minus 1205762)2 (1 +2120573)2
sdot 2sum119894=1
infinsum119896=1
100381610038161003816100381612058211989611989410038161003816100381610038162 100381610038161003816100381612060111989611989410038161003816100381610038162 + 100381610038161003816100381612059511989611989410038161003816100381610038162 (65)
Advances in Mathematical Physics 7
Hence using (57) and (59) we obtain
1003817100381710038171003817Φ119909119909100381710038171003817100381721198712(Ω) le 16119879 1 + 1205762(1 minus 1205762)2 (1 +
2120573)2
sdot 10038171003817100381710038171003817120601101584010158401003817100381710038171003817100381721198712(minus120587120587) + 10038171003817100381710038171003817120595101584010158401003817100381710038171003817100381721198712(minus120587120587) (66)
Now we can easily obtain an estimate for Ψ119905(119909 119905) from(1) This fact together with (60) and (66) gives the necessaryestimate (51) for the solution
From the obtained estimates it also follows that in theconstructed by us formal solution of the inverse problem allthe series converge they can be term-by-term differentiatedand the series obtained during differentiation also convergein sense of the metrics 1198712
From (47) and (54) by using the Holderrsquos inequality it iseasy to justify the inequality
max(119909119905)isinΩ
|Φ (119909 119905)|2 le 11986210038171003817100381710038171003817120601101584010158401003817100381710038171003817100381721198712(minus120587120587) + 10038171003817100381710038171003817120595101584010158401003817100381710038171003817100381721198712(minus120587120587) (67)
which justifies the continuity ofΦ(119909 119905) in the closed domainΩFrom the representation of the solution in the form of
series (37) (38) and inequalities (53) (54) it is easy to justifythe estimates 1003816100381610038161003816Φ119909119909 (119909 119905)1003816100381610038161003816 + 1003816100381610038161003816Φ119905 (119909 119905)1003816100381610038161003816 + 1003816100381610038161003816119891 (119909)1003816100381610038161003816
le 119862infinsum119896=1
100381610038161003816100381612058211989611989410038161003816100381610038162 10038161003816100381610038161206011198961198941003816100381610038161003816 + 10038161003816100381610038161205951198961198941003816100381610038161003816 (68)
Let 120601(119909) 120595(119909) isin 1198624[minus120587 120587] and let the functions 120601(119909) 120595(119909)and 12060110158401015840(119909) 12059510158401015840(119909) satisfy the boundary conditions (13) thenthe number series in the right-hand side of (68) convergesTherefore in such case the constructed by us formal solutiongives the regular solution of the inverse problem (1)-(3) (46)
The uniqueness of this constructed solution follows fromLemma 4
The theorem is completely proved
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This paper was published with the support of the ScienceCommittee of the Ministry of Education and Science of theRepublic of Kazakhstan under Project AP05133271 and TargetProgram BR05236656
References
[1] B Ahmad A Alsaedi M Kirane and R G TapdigogluldquoAn inverse problem for space and time fractional evolution
equations with an involution perturbationrdquo Quaestiones Math-ematicae vol 40 no 2 pp 151ndash160 2017
[2] A Cabada and A F Tojo ldquoEquations with involutionsrdquo inProceedings of the Workshop on Differential Equations p 240MallaMoravka CzechRepublicMarch 2014 httpusersmathcascz
[3] A Ashyralyev and A Sarsenbi ldquoWell-posedness of a parabolicequation with nonlocal boundary conditionrdquo Boundary ValueProblems vol 2015 no 1 2015
[4] M Kirane and N Al-Salti ldquoInverse problems for a nonlocalwave equation with an involution perturbationrdquo Journal ofNonlinear Science and Applications vol 9 no 3 pp 1243ndash12512016
[5] A Ashyralyev and A Sarsenbi ldquoWell-Posedness of a ParabolicEquation with Involutionrdquo Numerical Functional Analysis andOptimization vol 38 no 10 pp 1295ndash1304 2017
[6] I Orazov and M A Sadybekov ldquoOne nonlocal problem ofdetermination of the temperature and density of heat sourcesrdquoRussian Mathematics vol 56 no 2 pp 60ndash64 2012
[7] I Orazov and M A Sadybekov ldquoOn a class of problems ofdetermining the temperature and density of heat sources giveninitial and final temperaturerdquo Siberian Mathematical Journalvol 53 no 1 pp 146ndash151 2012
[8] N I Ivanchov ldquoInverse problems for the heat-conductionequation with nonlocal boundary conditionsrdquoUkrainianMath-ematical Journal vol 45 no 8 pp 1186ndash1192 1993
[9] I A Kaliev and M M Sabitova ldquoProblems of determiningthe temperature and density of heat sources from the initialand final temperaturesrdquo Journal of Applied and IndustrialMathematics vol 4 no 3 pp 332ndash339 2010
[10] I A KalievM FMugafarov andOV Fattahova ldquoInverse prob-lem for forwardbackward parabolic equation with generalizedconjugation conditionsrdquo Ufa Mathematical Journal vol 3 no2 pp 33ndash41 2011
[11] M Kirane and S A Malik ldquoDetermination of an unknownsource term and the temperature distribution for the linearheat equation involving fractional derivative in timerdquo AppliedMathematics and Computation vol 218 no 1 pp 163ndash170 2011
[12] F Kanca andM I Ismailov ldquoThe inverse problem of finding thetime-dependent diffusion coefficient of the heat equation fromintegral overdetermination datardquo Inverse Problems in Scienceand Engineering vol 20 no 4 pp 463ndash476 2012
[13] M Kirane S A Malik and M Al-Gwaiz ldquoAn inverse sourceproblem for a two dimensional time fractional diffusionequation with nonlocal boundary conditionsrdquo MathematicalMethods in the Applied Sciences vol 36 no 9 pp 1056ndash10692013
[14] A Ashyralyev and Y A Sharifov ldquoExistence and uniquenessof solutions for nonlinear impulsive differential equations withtwo-point and integral boundary conditionsrdquo Advances inDifference Equations vol 2013 no 1 p 173 2013
[15] F Kanca ldquoInverse coefficient problem of the parabolic equa-tion with periodic boundary and integral overdeterminationconditionsrdquo Abstract and Applied Analysis vol 2013 Article ID659804 7 pages 2013
[16] D Lesnic S A Yousefi and M Ivanchov ldquoDetermination of atime-dependent diffusivity from nonlocal conditionsrdquo AppliedMathematics and Computation vol 41 no 1-2 pp 301ndash3202013
[17] L Miller and M Yamamoto ldquoCoefficient inverse problem for afractional diffusion equationrdquo Inverse Problems vol 29 no 7 p075013 2013
8 Advances in Mathematical Physics
[18] G Li D Zhang X Jia and M Yamamoto ldquoSimultaneousinversion for the space-dependent diffusion coefficient andthe fractional order in the time-fractional diffusion equationrdquoInverse Problems vol 29 no 6 Article ID 065014 2013
[19] A B Kostin ldquoCounterexamples in inverse problems forparabolic elliptic and hyperbolic equationsrdquo ComputationalMathematics and Mathematical Physics vol 54 no 5 pp 797ndash810 2014
[20] A Ashyralyev and A Hanalyev ldquoWell-posedness of nonlocalparabolic differential problems with dependent operatorsrdquoTheScientific World Journal vol 2014 Article ID 519814 2014
[21] I Orazov and M A Sadybekov ldquoOn an inverse problemof mathematical modeling of the extraction process of poly-disperse porous materialsrdquo in Proceedings of the ADVANCE-MENTS IN MATHEMATICAL SCIENCES Proceedings of theInternational Conference on Advancements in MathematicalSciences p 020005 Antalya Turkey
[22] I Orazov and M A Sadybekov ldquoOne-dimensional diffusionproblem with not strengthened regular boundary conditionsrdquoin Proceedings of the 41ST International Conference ldquoApplicationsof Mathematics in Engineering And EconomicSrdquo AMEE rsquo15 p040007 Sozopol Bulgaria
[23] N H Tuan D N D Hai L D Long V T Nguyen and MKirane ldquoOn a RieszndashFeller space fractional backward diffusionproblemwith a nonlinear sourcerdquo Journal of Computational andApplied Mathematics vol 312 pp 103ndash126 2017
[24] M Sadybekov G Oralsyn and M Ismailov ldquoAn inverseproblem of finding the time-dependent heat transfer coefficientfrom an integral conditionrdquo International Journal of Pure andApplied Mathematics vol 113 no 4 pp 139ndash149 2017
[25] M Kirane andMNH Tuan ldquoIdentification and regularizationfor unknown source for a time-fractional diffusion equationrdquoComputersMathematics withApplications vol 73 no 6 pp 931ndash950 2017
[26] B T Torebek and R Tapdigoglu ldquoSome inverse problems forthe nonlocal heat equation with Caputo fractional derivativerdquoMathematical Methods in the Applied Sciences vol 40 no 18pp 6468ndash6479 2017
[27] M Kirane B Samet and B T Torebek ldquoDetermination ofan unknown source term temperature distribution for thesub-diffusion equation at the initial and final datardquo ElectronicJournal of Differential Equations vol 2017 2017
[28] O Taki-Eddine and B Abdelfatah ldquoOn determining the coef-ficient in a parabolic equation with nonlocal boundary andintegral conditionrdquo Electronic Journal of Mathematical Analysisand Applications vol 6 no 1 pp 94ndash102 2018
[29] A M Sarsenbi ldquoUnconditional bases related to a nonclassicalsecond-order differential operatorrdquo Journal of Differential Equa-tions vol 46 no 4 pp 509ndash514 2010
[30] V PKurdyumov andA PKhromov ldquoTheRiesz bases consistingof eigen and associated functions for a functional differentialoperator with variable structurerdquo Russian Mathematics vol 54no 2 pp 33ndash45 2010
[31] A M Sarsenbi and A A Tengaeva ldquoOn the basis propertiesof the root functions of two generalized spectral problemsrdquoJournal of Differential Equations vol 48 no 2 pp 306ndash3082012
[32] M A Sadybekov and A M Sarsenbi ldquoCriterion for the basisproperty of the eigenfunction system of a multiple differen-tiation operator with an involutionrdquo Journal of DifferentialEquations vol 48 no 8 pp 1112ndash1118 2012
[33] A Kopzhassarova and A Sarsenbi ldquoBasis properties of eigen-functions of second-order differential operators with invo-lutionrdquo Abstract and Applied Analysis vol 2012 Article ID576843 6 pages 2012
[34] AAKopzhassarova A L Lukashov andAM Sarsenbi ldquoSpec-tral properties of non-self-adjoint perturbations for a spectralproblem with involutionrdquo Abstract and Applied Analysis vol2012 Article ID 590781 5 pages 2012
[35] A Sarsenbi and M Sadybekov ldquoEigenfunctions of a fourthorder operator pencilrdquo in Proceedings of the 2nd InternationalConference on Analysis and Applied Mathematics ICAAM 2014pp 241ndash245 kaz September 2014
[36] L V Kritskov and A M Sarsenbi ldquoSpectral properties of anonlocal problem for a second-order differential equation withan involutionrdquo Journal of Differential Equations vol 51 no 8pp 984ndash990 2015
[37] L V Kritskov and A M Sarsenbi ldquoBasicity in Lp of rootfunctions for differential equations with involutionrdquo ElectronicJournal of Differential Equations vol 2015 2015
[38] M A Sadybekov A Sarsenbi and A Tengayeva ldquoDescriptionof spectral properties of a generalized spectral problem withinvolution for differentiation operator of the second orderrdquo inProceedings of the 3rd International Conference on Analysis andApplied Mathematics ICAAM 2016 kaz September 2016
[39] A G Baskakov I A Krishtal and E Y Romanova ldquoSpectralanalysis of a differential operator with an involutionrdquo Journal ofEvolution Equations (JEE) vol 17 no 2 pp 669ndash684 2017
[40] L V Kritskov and A M Sarsenbi ldquoRiesz basis property ofsystem of root functions of second-order differential operatorwith involutionrdquo Journal of Differential Equations vol 53 no 1pp 33ndash46 2017
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4 Advances in Mathematical Physics
We apply the operator 119889119889119905 to 119906119896119894(119905) Then using (8) byintegrating by parts we obtain the following problem
1199061015840119896119894 (119905) + 120582119896119894119906119896119894 (119905) = 119891119896119894 0 lt 119905 lt 119879 119894 = 1 2 (21)
119906119896119894 (0) = 120601119896119894 minus 120590119896119894 119894 = 1 2 (22)
119906119896119894 (119879) = 120595119896119894 minus 120590119896119894 119894 = 1 2 (23)
Here we use the following notations
120601119896119894 = int120587minus120587120601 (119909)119883119896119894 (119909) 119889119909
120595119896119894 = int120587minus120587120595 (119909)119883119896119894 (119909) 119889119909
120590119896119894 = 120574int120587minus120587119891 (120585) 119889120585int120587
minus120587119883119896119894 (119909) 119889119909
(24)
It is easy to see that the function 1198961(119905) = (120582119896119894)minus1119891119896119894is a partial solution of the inhomogeneous equation (21)Therefore we obtain a unique solution of theCauchy problem(21)-(22) in the following form
119906119896119894 (119905) = (120601119896119894 minus 120590119896119894 minus 119891119896119894120582119896119894) 119890minus120582119896119894119905 + 119891119896119894120582119896119894 (25)
Since |120576| lt 1 119886 gt 0 we have 120582119896119894 gt 0 for all 119896 119894Consequently
1 minus 119890minus120582119896119894119879 ge 1 minus 119890minus12058202119879 gt 0for all the values of the indexes 119896 119894 (26)
Therefore using the condition of rdquofinal overdetermina-tionrdquo (23) we get
119891119896119894 = 120582119896119894120595119896119894 minus 120601119896119894119890minus120582119896119894119879
1 minus 119890minus120582119896119894119879 minus 120582119896119894120590119896119894 (27)
Lemma 4 If 119886 gt 0 then the generalized solution(119906(119909 119905) 119891(119909)) of the inverse problem (8)-(11) is unique
Proof Suppose that there are two generalized solutionsof the inverse problem (8)-(11) (1199061(119909 119905) 1198911(119909)) and(1199062(119909 119905) 1198912(119909)) Denote
119906 (119909 119905) = 1199061 (119909 119905) minus 1199062 (119909 119905) 119891 (119909) = 1198911 (119909) minus 1198912 (119909) (28)
Then the functions 119906(119909 119905) and 119891(119909) satisfy (8) the boundaryconditions (11) and the homogeneous conditions (9) and (10)
119906 (119909 0) = minus120574int120587minus120587119891 (120585) 119889120585 119909 isin [minus120587 120587]
119906 (119909 119879) = minus120574int120587minus120587119891 (120585) 119889120585 119909 isin [minus120587 120587]
(29)
Therefore by using the notations (20) from (27) we find
119891119896119894 = minus120582119896119894120590119896119894 equiv minus120582119896119894120574int120587minus120587119891 (120585) 119889120585int120587
minus120587119883119896119894 (119909) 119889119909 (30)
Since
120582119896119894 int120587minus120587119883119896119894 (119909) 119889119909 = int120587
minus120587minus11988310158401015840119896119894 (119909) + 12057611988310158401015840119896119894 (minus119909) 119889119909
= (1 minus 120576) 1198831015840119896119894 (minus120587) minus 1198831015840119896119894 (120587)= 119886 (1 minus 120576)119883119896119894 (120587)
(31)
this gives
119891119896119894 = minus119886 (1 minus 120576) (120574int120587minus120587119891 (120585) 119889120585)119883119896119894 (120587) (32)
Since1198831198961(120587) = 0 then for 119894 = 1 from (32) we have 1198911198961 =0For the generalized solution of the problem (8)-(11) it
is necessary that 119891 isin 1198712(minus120587 120587) From this by virtue ofParsevalrsquos equality for orthogonal bases we have that
infinsum119896=1
1003816100381610038161003816119891119896210038161003816100381610038162 le 100381710038171003817100381711989110038171003817100381710038172 lt infin (33)
is necessarySince 119886(1 minus 120576) = 0
1198831198962 (120587) = ]119896 cos ((119896 + 120575119896) 120587) lim119896rarrinfin
]119896 = 1radic120587lim119896rarrinfin
120575119896 = 0(34)
then (32) and inequality (33) are possible only at(120574 int120587minus120587119891(120585)119889120585) = 0 Hence we obtain 1198911198962 = 0
Therefore using this result from (25) and (27) we find
119906119896119894 (119905) equiv int120587minus120587119906 (119909 119905) 119883119896119894 (119909) 119889119909 = 0
119891119896119894 equiv int120587minus120587119891 (119909)119883119896119894 (119909) 119889119909 = 0
(35)
for all values of the indexes 119896 isin N for 119894 = 1 and 119896 isin N0 for 119894 =2 Further by the completeness of system (17) in 1198712(minus120587 120587)we obtain
119906 (119909 119905) equiv 0119891 (119909) equiv 0
forall (119909 119905) isin Ω(36)
The uniqueness of the generalized solution of the inverseproblem (8)-(11) is proved
Corollary 5 If 119886 gt 0 then the regular solution (119906(119909 119905) 119891(119909))of the inverse problem (8)-(11) is unique
Advances in Mathematical Physics 5
5 Construction of a Formal Solution ofthe Problem
As the eigenfunctions system (17) forms the orthonormalbasis in 1198712(minus120587 120587) then the unknown functions 119906(119909 119905) and119891(119909) can be represented as
119906 (119909 119905) = infinsum119896=1
1199061198961 (119905) 1198831198961 (119909) + infinsum119896=0
1199061198962 (119905) 1198831198962 (119909) (37)
119891 (119909) = infinsum119896=1
11989111989611198831198961 (119909) + infinsum119896=0
11989111989621198831198962 (119909) (38)
where 1199061198961(119905) and 1199061198962(119905) are unknown functions 1198911198961 and 1198911198962are unknown constants
Substituting (37) and (38) into (8) we obtain the inverseproblems (21)-(23) If the constants 120590119896119894 are assumed to begiven then the solutions of these inverse problems existare unique and are represented by formulas (25) and (27)Substituting (25) and (27) into series (37) and (38) we obtaina formal solution of the inverse problem (8)-(11)
Let 120601(119909) 120595(119909) isin 11988222 (minus120587 120587) and satisfy the boundaryconditions (13) Since 120582119896119894 gt 0 then
infinsum119896=1
1003816100381610038161003816100381610038161003816100381610038161003816120582119896119894120595119896119894 minus 120601119896119894119890minus1205821198961198941198791 minus 119890minus120582119896119894119879
10038161003816100381610038161003816100381610038161003816100381610038162
le 11986210038171003817100381710038171206011003817100381710038171003817211988222(minus120587120587)
+ 10038171003817100381710038171205951003817100381710038171003817211988222(minus120587120587)
119894 = 1 2(39)
Therefore from the analysis of formula (27) it is easy tosee that in order for the formal solution (37) of the problem(8)-(11) to be a generalized solution it is necessary that
lim119896rarrinfin
120582119896119894120590119896119894 = 0 119894 = 1 2 (40)
holdsAs above we calculate
120582119896119894120590119896119894 equiv 119886 (1 minus 120576) (120574int120587minus120587119891 (120585) 119889120585)119883119896119894 (120587) (41)
where1198831198961(120587) = 0 and1198831198962 (120587) = ]119896 cos ((119896 + 120575119896) 120587) lim119896rarrinfin
]119896 = 1radic120587lim119896rarrinfin
120575119896 = 0(42)
Thus (40) holds if and only if 120590119896119894 = 0 for all values of theindexes 119896 119894 This is possible only in the case
int120587minus120587119891 (120585) 119889120585 = 0 (43)
Thus condition (43) is a necessary condition for theexistence of the generalized solution of the inverse problem
(8)-(11) In this case problems (8)-(11) and (1)-(4) coincideIndeed from (43) and (1) we have
0 = int120587minus120587119891 (120585) 119889120585
= int120587minus120587Φ119905 (120585 119905) 119889120585
minus int120587minus120587Φ120585120585 (120585 119905) + 120576Φ120585120585 (minus120585 119905) 119889120585
(44)
For the first integral we apply condition (4) and calculatethe second integral Then we obtain
0 = (1 minus 120576)sdot [Φ119909 (minus120587 119905) minus Φ119909 (120587 119905) + 1120574 (1 minus 120576)Φ (minus120587 119905)] (45)
This means that the boundary conditions (4) and (11) coin-cide Hence the problems (8)-(11) and (1)-(4) also coincide
Thus in what follows we shall consider problem (1)-(3)with the boundary condition
Φ119909 (minus120587 119905) minus Φ119909 (120587 119905) minus 119886Φ (minus120587 119905) = 0 (46)Thus in what follows we will consider the inverse
problem (1)-(3) (46)Similarly as before the formal solution of this problem
can be constructed in the form of series
Φ (119909 119905) = infinsum119896=1
Φ1198961 (119905) 1198831198961 (119909) + infinsum119896=0
Φ1198962 (119905) 1198831198962 (119909) (47)
119891 (119909) = infinsum119896=1
11989111989611198831198961 (119909) + infinsum119896=0
11989111989621198831198962 (119909) (48)
where
Φ119896119894 (119905) = (120601119896119894 minus 119891119896119894120582119896119894) 119890minus120582119896119894119905 + 119891119896119894120582119896119894 (49)
119891119896119894 = 120582119896119894120595119896119894 minus 120601119896119894119890minus120582119896119894119879
1 minus 119890minus120582119896119894119879 (50)
In order to complete our study it is necessary as in theFourier method to justify the smoothness of the resultingformal solutions and the convergence of all appearing series
6 Main Results
Here we present the existence and uniqueness results for ourinverse problem
Theorem 6 (let 119886 gt 0) (A) Let 120601(119909) 120595(119909) isin 11988222 (minus120587 120587)and let the functions120601(119909) 120595(119909) satisfy the boundary conditions(13) Then for a real number 120576 such that |120576| lt 1 the inverseproblem (1)-(3) (46) has a unique generalized solution whichis stable in norm1003817100381710038171003817Φ119905100381710038171003817100381721198712(Ω) + 1003817100381710038171003817Φ119909119909100381710038171003817100381721198712(Ω) + 1003817100381710038171003817119891100381710038171003817100381721198712(minus120587120587)
le 11986210038171003817100381710038171206011003817100381710038171003817211988222(minus120587120587)
+ 10038171003817100381710038171205951003817100381710038171003817211988222(minus120587120587)
(51)
where the constant 119862 does not depend on 120601(119909) and 120595(119909)
6 Advances in Mathematical Physics
(B) Let 120601(119909) 120595(119909) isin 1198624[minus120587 120587] and the functions120601(119909) 120595(119909) and 12060110158401015840(119909) 12059510158401015840(119909) satisfy the boundary conditions(13) then for a real number 120576 such that |120576| lt 1 the inverseproblem (1)-(3) (46) has a unique regular solution
Proof Since |120576| lt 1 119886 gt 0 we have 120582119896119894 gt 0 for all 119896 119894Consequently there exists a constant 120573 gt 0 such that for allthe values of the indexes 119896 119894 we have
1 minus 119890minus120582119896119894119879 ge 1 minus 119890minus12058202119879 ge 120573 gt 0 (52)
Since 120582119896119894 gt 0 then 119890minus120582119896119894119879 le 119890minus120582119896119894119905 lt 1 Therefore fromrepresentations (49) and (50) we get estimates
10038161003816100381610038161198911198961198941003816100381610038161003816 le10038161003816100381610038161205821198961198941003816100381610038161003816120573 10038161003816100381610038161206011198961198941003816100381610038161003816 + 10038161003816100381610038161205951198961198941003816100381610038161003816 (53)
1003816100381610038161003816Φ119896119894 (119905)1003816100381610038161003816 le (1 + 2120573) 10038161003816100381610038161206011198961198941003816100381610038161003816 + 10038161003816100381610038161205951198961198941003816100381610038161003816 (54)
where constant 120573 does not depend on the indexes 119896 119894 and onthe functions 120601(119909) and 120595(119909)
As proved in Lemma 3 the eigenfunctions system (17)forms the orthonormal basis in 1198712(minus120587 120587)Therefore we havean expansion
120601 (119909) = infinsum119896=1
12060111989611198831198961 (119909) + infinsum119896=0
12060111989621198831198962 (119909) (55)
Taking into account (12) we obtain
minus12060110158401015840 (119909) + 12057612060110158401015840 (minus119909) = infinsum119896=1
120582119896112060111989611198831198961 (119909)
+ infinsum119896=0
120582119896212060111989621198831198962 (119909) (56)
From this by Parsevalrsquos equality it is easy to obtain anestimate
infinsum119896=1
1003816100381610038161003816120582119896110038161003816100381610038162 1003816100381610038161003816120601119896110038161003816100381610038162 +infinsum119896=1
1003816100381610038161003816120582119896210038161003816100381610038162 1003816100381610038161003816120601119896210038161003816100381610038162
le 2 (1 + 1205762) 10038171003817100381710038171003817120601101584010158401003817100381710038171003817100381721198712(minus120587120587) (57)
Similarly for the coefficients of the expansion
120595 (119909) = infinsum119896=1
12059511989611198831198961 (119909) + infinsum119896=0
12059511989621198831198962 (119909) (58)
we have the inequality
infinsum119896=1
1003816100381610038161003816120582119896110038161003816100381610038162 1003816100381610038161003816120595119896110038161003816100381610038162 +infinsum119896=1
1003816100381610038161003816120582119896210038161003816100381610038162 1003816100381610038161003816120595119896210038161003816100381610038162
le 2 (1 + 1205762) 10038171003817100381710038171003817120595101584010158401003817100381710038171003817100381721198712(minus120587120587) (59)
Now from (53) (57) and (59) we see that the series (48)converges and the estimate
1003817100381710038171003817119891100381710038171003817100381721198712(minus120587120587) le 21 + 12057621205732 10038171003817100381710038171003817120601101584010158401003817100381710038171003817100381721198712(minus120587120587) + 10038171003817100381710038171003817120595101584010158401003817100381710038171003817100381721198712(minus120587120587) (60)
holdsTaking into account (12) from (47) we obtain
minus Φ119909119909 (119909 119905) + 120576Φ119909119909 (minus119909 119905)= infinsum119896=1
1205821198961Φ1198961 (119905) 1198831198961 (119909)
+ infinsum119896=0
1205821198962Φ1198962 (119905) 1198831198962 (119909) (61)
Substituting the value (minus119909) instead of 119909 we findminus Φ119909119909 (minus119909 119905) + 120576Φ119909119909 (119909 119905)= minusinfinsum119896=1
1205821198961Φ1198961 (119905) 1198831198961 (119909)
+ infinsum119896=0
1205821198962Φ1198962 (119905) 1198831198962 (119909) (62)
Here we use the properties 1198831198961(minus119909) = minus1198831198961(119909) 1198831198962(minus119909) =1198831198962(119909) obtained from the explicit form (12) of the eigenfunc-tions
Multiplying (62) by 120576 and summing with (61) we have
(minus1 + 1205762)Φ119909119909 (119909 119905)= (1 minus 120576) infinsum
119896=1
1205821198961Φ1198961 (119905) 1198831198961 (119909)
+ (1 + 120576) infinsum119896=0
1205821198962Φ1198962 (119905) 1198831198962 (119909) (63)
Since |120576| lt 1 from this by Parsevalrsquos equality we find
1003817100381710038171003817Φ119909119909 (sdot 119905)100381710038171003817100381721198712(minus120587120587) = 1(1 + 120576)2
infinsum119896=1
1003816100381610038161003816120582119896110038161003816100381610038162 1003816100381610038161003816Φ1198961 (119905)10038161003816100381610038162
+ 1(1 minus 120576)2
infinsum119896=0
1003816100381610038161003816120582119896210038161003816100381610038162 1003816100381610038161003816Φ1198962 (119905)10038161003816100381610038162(64)
for each 119905 isin [0 119879] Integrating this equation with respect to119905 isin [0 119879] taking (54) into account we have1003817100381710038171003817Φ119909119909100381710038171003817100381721198712(Ω) le 8119879
(1 minus 1205762)2 (1 +2120573)2
sdot 2sum119894=1
infinsum119896=1
100381610038161003816100381612058211989611989410038161003816100381610038162 100381610038161003816100381612060111989611989410038161003816100381610038162 + 100381610038161003816100381612059511989611989410038161003816100381610038162 (65)
Advances in Mathematical Physics 7
Hence using (57) and (59) we obtain
1003817100381710038171003817Φ119909119909100381710038171003817100381721198712(Ω) le 16119879 1 + 1205762(1 minus 1205762)2 (1 +
2120573)2
sdot 10038171003817100381710038171003817120601101584010158401003817100381710038171003817100381721198712(minus120587120587) + 10038171003817100381710038171003817120595101584010158401003817100381710038171003817100381721198712(minus120587120587) (66)
Now we can easily obtain an estimate for Ψ119905(119909 119905) from(1) This fact together with (60) and (66) gives the necessaryestimate (51) for the solution
From the obtained estimates it also follows that in theconstructed by us formal solution of the inverse problem allthe series converge they can be term-by-term differentiatedand the series obtained during differentiation also convergein sense of the metrics 1198712
From (47) and (54) by using the Holderrsquos inequality it iseasy to justify the inequality
max(119909119905)isinΩ
|Φ (119909 119905)|2 le 11986210038171003817100381710038171003817120601101584010158401003817100381710038171003817100381721198712(minus120587120587) + 10038171003817100381710038171003817120595101584010158401003817100381710038171003817100381721198712(minus120587120587) (67)
which justifies the continuity ofΦ(119909 119905) in the closed domainΩFrom the representation of the solution in the form of
series (37) (38) and inequalities (53) (54) it is easy to justifythe estimates 1003816100381610038161003816Φ119909119909 (119909 119905)1003816100381610038161003816 + 1003816100381610038161003816Φ119905 (119909 119905)1003816100381610038161003816 + 1003816100381610038161003816119891 (119909)1003816100381610038161003816
le 119862infinsum119896=1
100381610038161003816100381612058211989611989410038161003816100381610038162 10038161003816100381610038161206011198961198941003816100381610038161003816 + 10038161003816100381610038161205951198961198941003816100381610038161003816 (68)
Let 120601(119909) 120595(119909) isin 1198624[minus120587 120587] and let the functions 120601(119909) 120595(119909)and 12060110158401015840(119909) 12059510158401015840(119909) satisfy the boundary conditions (13) thenthe number series in the right-hand side of (68) convergesTherefore in such case the constructed by us formal solutiongives the regular solution of the inverse problem (1)-(3) (46)
The uniqueness of this constructed solution follows fromLemma 4
The theorem is completely proved
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This paper was published with the support of the ScienceCommittee of the Ministry of Education and Science of theRepublic of Kazakhstan under Project AP05133271 and TargetProgram BR05236656
References
[1] B Ahmad A Alsaedi M Kirane and R G TapdigogluldquoAn inverse problem for space and time fractional evolution
equations with an involution perturbationrdquo Quaestiones Math-ematicae vol 40 no 2 pp 151ndash160 2017
[2] A Cabada and A F Tojo ldquoEquations with involutionsrdquo inProceedings of the Workshop on Differential Equations p 240MallaMoravka CzechRepublicMarch 2014 httpusersmathcascz
[3] A Ashyralyev and A Sarsenbi ldquoWell-posedness of a parabolicequation with nonlocal boundary conditionrdquo Boundary ValueProblems vol 2015 no 1 2015
[4] M Kirane and N Al-Salti ldquoInverse problems for a nonlocalwave equation with an involution perturbationrdquo Journal ofNonlinear Science and Applications vol 9 no 3 pp 1243ndash12512016
[5] A Ashyralyev and A Sarsenbi ldquoWell-Posedness of a ParabolicEquation with Involutionrdquo Numerical Functional Analysis andOptimization vol 38 no 10 pp 1295ndash1304 2017
[6] I Orazov and M A Sadybekov ldquoOne nonlocal problem ofdetermination of the temperature and density of heat sourcesrdquoRussian Mathematics vol 56 no 2 pp 60ndash64 2012
[7] I Orazov and M A Sadybekov ldquoOn a class of problems ofdetermining the temperature and density of heat sources giveninitial and final temperaturerdquo Siberian Mathematical Journalvol 53 no 1 pp 146ndash151 2012
[8] N I Ivanchov ldquoInverse problems for the heat-conductionequation with nonlocal boundary conditionsrdquoUkrainianMath-ematical Journal vol 45 no 8 pp 1186ndash1192 1993
[9] I A Kaliev and M M Sabitova ldquoProblems of determiningthe temperature and density of heat sources from the initialand final temperaturesrdquo Journal of Applied and IndustrialMathematics vol 4 no 3 pp 332ndash339 2010
[10] I A KalievM FMugafarov andOV Fattahova ldquoInverse prob-lem for forwardbackward parabolic equation with generalizedconjugation conditionsrdquo Ufa Mathematical Journal vol 3 no2 pp 33ndash41 2011
[11] M Kirane and S A Malik ldquoDetermination of an unknownsource term and the temperature distribution for the linearheat equation involving fractional derivative in timerdquo AppliedMathematics and Computation vol 218 no 1 pp 163ndash170 2011
[12] F Kanca andM I Ismailov ldquoThe inverse problem of finding thetime-dependent diffusion coefficient of the heat equation fromintegral overdetermination datardquo Inverse Problems in Scienceand Engineering vol 20 no 4 pp 463ndash476 2012
[13] M Kirane S A Malik and M Al-Gwaiz ldquoAn inverse sourceproblem for a two dimensional time fractional diffusionequation with nonlocal boundary conditionsrdquo MathematicalMethods in the Applied Sciences vol 36 no 9 pp 1056ndash10692013
[14] A Ashyralyev and Y A Sharifov ldquoExistence and uniquenessof solutions for nonlinear impulsive differential equations withtwo-point and integral boundary conditionsrdquo Advances inDifference Equations vol 2013 no 1 p 173 2013
[15] F Kanca ldquoInverse coefficient problem of the parabolic equa-tion with periodic boundary and integral overdeterminationconditionsrdquo Abstract and Applied Analysis vol 2013 Article ID659804 7 pages 2013
[16] D Lesnic S A Yousefi and M Ivanchov ldquoDetermination of atime-dependent diffusivity from nonlocal conditionsrdquo AppliedMathematics and Computation vol 41 no 1-2 pp 301ndash3202013
[17] L Miller and M Yamamoto ldquoCoefficient inverse problem for afractional diffusion equationrdquo Inverse Problems vol 29 no 7 p075013 2013
8 Advances in Mathematical Physics
[18] G Li D Zhang X Jia and M Yamamoto ldquoSimultaneousinversion for the space-dependent diffusion coefficient andthe fractional order in the time-fractional diffusion equationrdquoInverse Problems vol 29 no 6 Article ID 065014 2013
[19] A B Kostin ldquoCounterexamples in inverse problems forparabolic elliptic and hyperbolic equationsrdquo ComputationalMathematics and Mathematical Physics vol 54 no 5 pp 797ndash810 2014
[20] A Ashyralyev and A Hanalyev ldquoWell-posedness of nonlocalparabolic differential problems with dependent operatorsrdquoTheScientific World Journal vol 2014 Article ID 519814 2014
[21] I Orazov and M A Sadybekov ldquoOn an inverse problemof mathematical modeling of the extraction process of poly-disperse porous materialsrdquo in Proceedings of the ADVANCE-MENTS IN MATHEMATICAL SCIENCES Proceedings of theInternational Conference on Advancements in MathematicalSciences p 020005 Antalya Turkey
[22] I Orazov and M A Sadybekov ldquoOne-dimensional diffusionproblem with not strengthened regular boundary conditionsrdquoin Proceedings of the 41ST International Conference ldquoApplicationsof Mathematics in Engineering And EconomicSrdquo AMEE rsquo15 p040007 Sozopol Bulgaria
[23] N H Tuan D N D Hai L D Long V T Nguyen and MKirane ldquoOn a RieszndashFeller space fractional backward diffusionproblemwith a nonlinear sourcerdquo Journal of Computational andApplied Mathematics vol 312 pp 103ndash126 2017
[24] M Sadybekov G Oralsyn and M Ismailov ldquoAn inverseproblem of finding the time-dependent heat transfer coefficientfrom an integral conditionrdquo International Journal of Pure andApplied Mathematics vol 113 no 4 pp 139ndash149 2017
[25] M Kirane andMNH Tuan ldquoIdentification and regularizationfor unknown source for a time-fractional diffusion equationrdquoComputersMathematics withApplications vol 73 no 6 pp 931ndash950 2017
[26] B T Torebek and R Tapdigoglu ldquoSome inverse problems forthe nonlocal heat equation with Caputo fractional derivativerdquoMathematical Methods in the Applied Sciences vol 40 no 18pp 6468ndash6479 2017
[27] M Kirane B Samet and B T Torebek ldquoDetermination ofan unknown source term temperature distribution for thesub-diffusion equation at the initial and final datardquo ElectronicJournal of Differential Equations vol 2017 2017
[28] O Taki-Eddine and B Abdelfatah ldquoOn determining the coef-ficient in a parabolic equation with nonlocal boundary andintegral conditionrdquo Electronic Journal of Mathematical Analysisand Applications vol 6 no 1 pp 94ndash102 2018
[29] A M Sarsenbi ldquoUnconditional bases related to a nonclassicalsecond-order differential operatorrdquo Journal of Differential Equa-tions vol 46 no 4 pp 509ndash514 2010
[30] V PKurdyumov andA PKhromov ldquoTheRiesz bases consistingof eigen and associated functions for a functional differentialoperator with variable structurerdquo Russian Mathematics vol 54no 2 pp 33ndash45 2010
[31] A M Sarsenbi and A A Tengaeva ldquoOn the basis propertiesof the root functions of two generalized spectral problemsrdquoJournal of Differential Equations vol 48 no 2 pp 306ndash3082012
[32] M A Sadybekov and A M Sarsenbi ldquoCriterion for the basisproperty of the eigenfunction system of a multiple differen-tiation operator with an involutionrdquo Journal of DifferentialEquations vol 48 no 8 pp 1112ndash1118 2012
[33] A Kopzhassarova and A Sarsenbi ldquoBasis properties of eigen-functions of second-order differential operators with invo-lutionrdquo Abstract and Applied Analysis vol 2012 Article ID576843 6 pages 2012
[34] AAKopzhassarova A L Lukashov andAM Sarsenbi ldquoSpec-tral properties of non-self-adjoint perturbations for a spectralproblem with involutionrdquo Abstract and Applied Analysis vol2012 Article ID 590781 5 pages 2012
[35] A Sarsenbi and M Sadybekov ldquoEigenfunctions of a fourthorder operator pencilrdquo in Proceedings of the 2nd InternationalConference on Analysis and Applied Mathematics ICAAM 2014pp 241ndash245 kaz September 2014
[36] L V Kritskov and A M Sarsenbi ldquoSpectral properties of anonlocal problem for a second-order differential equation withan involutionrdquo Journal of Differential Equations vol 51 no 8pp 984ndash990 2015
[37] L V Kritskov and A M Sarsenbi ldquoBasicity in Lp of rootfunctions for differential equations with involutionrdquo ElectronicJournal of Differential Equations vol 2015 2015
[38] M A Sadybekov A Sarsenbi and A Tengayeva ldquoDescriptionof spectral properties of a generalized spectral problem withinvolution for differentiation operator of the second orderrdquo inProceedings of the 3rd International Conference on Analysis andApplied Mathematics ICAAM 2016 kaz September 2016
[39] A G Baskakov I A Krishtal and E Y Romanova ldquoSpectralanalysis of a differential operator with an involutionrdquo Journal ofEvolution Equations (JEE) vol 17 no 2 pp 669ndash684 2017
[40] L V Kritskov and A M Sarsenbi ldquoRiesz basis property ofsystem of root functions of second-order differential operatorwith involutionrdquo Journal of Differential Equations vol 53 no 1pp 33ndash46 2017
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Advances in Mathematical Physics 5
5 Construction of a Formal Solution ofthe Problem
As the eigenfunctions system (17) forms the orthonormalbasis in 1198712(minus120587 120587) then the unknown functions 119906(119909 119905) and119891(119909) can be represented as
119906 (119909 119905) = infinsum119896=1
1199061198961 (119905) 1198831198961 (119909) + infinsum119896=0
1199061198962 (119905) 1198831198962 (119909) (37)
119891 (119909) = infinsum119896=1
11989111989611198831198961 (119909) + infinsum119896=0
11989111989621198831198962 (119909) (38)
where 1199061198961(119905) and 1199061198962(119905) are unknown functions 1198911198961 and 1198911198962are unknown constants
Substituting (37) and (38) into (8) we obtain the inverseproblems (21)-(23) If the constants 120590119896119894 are assumed to begiven then the solutions of these inverse problems existare unique and are represented by formulas (25) and (27)Substituting (25) and (27) into series (37) and (38) we obtaina formal solution of the inverse problem (8)-(11)
Let 120601(119909) 120595(119909) isin 11988222 (minus120587 120587) and satisfy the boundaryconditions (13) Since 120582119896119894 gt 0 then
infinsum119896=1
1003816100381610038161003816100381610038161003816100381610038161003816120582119896119894120595119896119894 minus 120601119896119894119890minus1205821198961198941198791 minus 119890minus120582119896119894119879
10038161003816100381610038161003816100381610038161003816100381610038162
le 11986210038171003817100381710038171206011003817100381710038171003817211988222(minus120587120587)
+ 10038171003817100381710038171205951003817100381710038171003817211988222(minus120587120587)
119894 = 1 2(39)
Therefore from the analysis of formula (27) it is easy tosee that in order for the formal solution (37) of the problem(8)-(11) to be a generalized solution it is necessary that
lim119896rarrinfin
120582119896119894120590119896119894 = 0 119894 = 1 2 (40)
holdsAs above we calculate
120582119896119894120590119896119894 equiv 119886 (1 minus 120576) (120574int120587minus120587119891 (120585) 119889120585)119883119896119894 (120587) (41)
where1198831198961(120587) = 0 and1198831198962 (120587) = ]119896 cos ((119896 + 120575119896) 120587) lim119896rarrinfin
]119896 = 1radic120587lim119896rarrinfin
120575119896 = 0(42)
Thus (40) holds if and only if 120590119896119894 = 0 for all values of theindexes 119896 119894 This is possible only in the case
int120587minus120587119891 (120585) 119889120585 = 0 (43)
Thus condition (43) is a necessary condition for theexistence of the generalized solution of the inverse problem
(8)-(11) In this case problems (8)-(11) and (1)-(4) coincideIndeed from (43) and (1) we have
0 = int120587minus120587119891 (120585) 119889120585
= int120587minus120587Φ119905 (120585 119905) 119889120585
minus int120587minus120587Φ120585120585 (120585 119905) + 120576Φ120585120585 (minus120585 119905) 119889120585
(44)
For the first integral we apply condition (4) and calculatethe second integral Then we obtain
0 = (1 minus 120576)sdot [Φ119909 (minus120587 119905) minus Φ119909 (120587 119905) + 1120574 (1 minus 120576)Φ (minus120587 119905)] (45)
This means that the boundary conditions (4) and (11) coin-cide Hence the problems (8)-(11) and (1)-(4) also coincide
Thus in what follows we shall consider problem (1)-(3)with the boundary condition
Φ119909 (minus120587 119905) minus Φ119909 (120587 119905) minus 119886Φ (minus120587 119905) = 0 (46)Thus in what follows we will consider the inverse
problem (1)-(3) (46)Similarly as before the formal solution of this problem
can be constructed in the form of series
Φ (119909 119905) = infinsum119896=1
Φ1198961 (119905) 1198831198961 (119909) + infinsum119896=0
Φ1198962 (119905) 1198831198962 (119909) (47)
119891 (119909) = infinsum119896=1
11989111989611198831198961 (119909) + infinsum119896=0
11989111989621198831198962 (119909) (48)
where
Φ119896119894 (119905) = (120601119896119894 minus 119891119896119894120582119896119894) 119890minus120582119896119894119905 + 119891119896119894120582119896119894 (49)
119891119896119894 = 120582119896119894120595119896119894 minus 120601119896119894119890minus120582119896119894119879
1 minus 119890minus120582119896119894119879 (50)
In order to complete our study it is necessary as in theFourier method to justify the smoothness of the resultingformal solutions and the convergence of all appearing series
6 Main Results
Here we present the existence and uniqueness results for ourinverse problem
Theorem 6 (let 119886 gt 0) (A) Let 120601(119909) 120595(119909) isin 11988222 (minus120587 120587)and let the functions120601(119909) 120595(119909) satisfy the boundary conditions(13) Then for a real number 120576 such that |120576| lt 1 the inverseproblem (1)-(3) (46) has a unique generalized solution whichis stable in norm1003817100381710038171003817Φ119905100381710038171003817100381721198712(Ω) + 1003817100381710038171003817Φ119909119909100381710038171003817100381721198712(Ω) + 1003817100381710038171003817119891100381710038171003817100381721198712(minus120587120587)
le 11986210038171003817100381710038171206011003817100381710038171003817211988222(minus120587120587)
+ 10038171003817100381710038171205951003817100381710038171003817211988222(minus120587120587)
(51)
where the constant 119862 does not depend on 120601(119909) and 120595(119909)
6 Advances in Mathematical Physics
(B) Let 120601(119909) 120595(119909) isin 1198624[minus120587 120587] and the functions120601(119909) 120595(119909) and 12060110158401015840(119909) 12059510158401015840(119909) satisfy the boundary conditions(13) then for a real number 120576 such that |120576| lt 1 the inverseproblem (1)-(3) (46) has a unique regular solution
Proof Since |120576| lt 1 119886 gt 0 we have 120582119896119894 gt 0 for all 119896 119894Consequently there exists a constant 120573 gt 0 such that for allthe values of the indexes 119896 119894 we have
1 minus 119890minus120582119896119894119879 ge 1 minus 119890minus12058202119879 ge 120573 gt 0 (52)
Since 120582119896119894 gt 0 then 119890minus120582119896119894119879 le 119890minus120582119896119894119905 lt 1 Therefore fromrepresentations (49) and (50) we get estimates
10038161003816100381610038161198911198961198941003816100381610038161003816 le10038161003816100381610038161205821198961198941003816100381610038161003816120573 10038161003816100381610038161206011198961198941003816100381610038161003816 + 10038161003816100381610038161205951198961198941003816100381610038161003816 (53)
1003816100381610038161003816Φ119896119894 (119905)1003816100381610038161003816 le (1 + 2120573) 10038161003816100381610038161206011198961198941003816100381610038161003816 + 10038161003816100381610038161205951198961198941003816100381610038161003816 (54)
where constant 120573 does not depend on the indexes 119896 119894 and onthe functions 120601(119909) and 120595(119909)
As proved in Lemma 3 the eigenfunctions system (17)forms the orthonormal basis in 1198712(minus120587 120587)Therefore we havean expansion
120601 (119909) = infinsum119896=1
12060111989611198831198961 (119909) + infinsum119896=0
12060111989621198831198962 (119909) (55)
Taking into account (12) we obtain
minus12060110158401015840 (119909) + 12057612060110158401015840 (minus119909) = infinsum119896=1
120582119896112060111989611198831198961 (119909)
+ infinsum119896=0
120582119896212060111989621198831198962 (119909) (56)
From this by Parsevalrsquos equality it is easy to obtain anestimate
infinsum119896=1
1003816100381610038161003816120582119896110038161003816100381610038162 1003816100381610038161003816120601119896110038161003816100381610038162 +infinsum119896=1
1003816100381610038161003816120582119896210038161003816100381610038162 1003816100381610038161003816120601119896210038161003816100381610038162
le 2 (1 + 1205762) 10038171003817100381710038171003817120601101584010158401003817100381710038171003817100381721198712(minus120587120587) (57)
Similarly for the coefficients of the expansion
120595 (119909) = infinsum119896=1
12059511989611198831198961 (119909) + infinsum119896=0
12059511989621198831198962 (119909) (58)
we have the inequality
infinsum119896=1
1003816100381610038161003816120582119896110038161003816100381610038162 1003816100381610038161003816120595119896110038161003816100381610038162 +infinsum119896=1
1003816100381610038161003816120582119896210038161003816100381610038162 1003816100381610038161003816120595119896210038161003816100381610038162
le 2 (1 + 1205762) 10038171003817100381710038171003817120595101584010158401003817100381710038171003817100381721198712(minus120587120587) (59)
Now from (53) (57) and (59) we see that the series (48)converges and the estimate
1003817100381710038171003817119891100381710038171003817100381721198712(minus120587120587) le 21 + 12057621205732 10038171003817100381710038171003817120601101584010158401003817100381710038171003817100381721198712(minus120587120587) + 10038171003817100381710038171003817120595101584010158401003817100381710038171003817100381721198712(minus120587120587) (60)
holdsTaking into account (12) from (47) we obtain
minus Φ119909119909 (119909 119905) + 120576Φ119909119909 (minus119909 119905)= infinsum119896=1
1205821198961Φ1198961 (119905) 1198831198961 (119909)
+ infinsum119896=0
1205821198962Φ1198962 (119905) 1198831198962 (119909) (61)
Substituting the value (minus119909) instead of 119909 we findminus Φ119909119909 (minus119909 119905) + 120576Φ119909119909 (119909 119905)= minusinfinsum119896=1
1205821198961Φ1198961 (119905) 1198831198961 (119909)
+ infinsum119896=0
1205821198962Φ1198962 (119905) 1198831198962 (119909) (62)
Here we use the properties 1198831198961(minus119909) = minus1198831198961(119909) 1198831198962(minus119909) =1198831198962(119909) obtained from the explicit form (12) of the eigenfunc-tions
Multiplying (62) by 120576 and summing with (61) we have
(minus1 + 1205762)Φ119909119909 (119909 119905)= (1 minus 120576) infinsum
119896=1
1205821198961Φ1198961 (119905) 1198831198961 (119909)
+ (1 + 120576) infinsum119896=0
1205821198962Φ1198962 (119905) 1198831198962 (119909) (63)
Since |120576| lt 1 from this by Parsevalrsquos equality we find
1003817100381710038171003817Φ119909119909 (sdot 119905)100381710038171003817100381721198712(minus120587120587) = 1(1 + 120576)2
infinsum119896=1
1003816100381610038161003816120582119896110038161003816100381610038162 1003816100381610038161003816Φ1198961 (119905)10038161003816100381610038162
+ 1(1 minus 120576)2
infinsum119896=0
1003816100381610038161003816120582119896210038161003816100381610038162 1003816100381610038161003816Φ1198962 (119905)10038161003816100381610038162(64)
for each 119905 isin [0 119879] Integrating this equation with respect to119905 isin [0 119879] taking (54) into account we have1003817100381710038171003817Φ119909119909100381710038171003817100381721198712(Ω) le 8119879
(1 minus 1205762)2 (1 +2120573)2
sdot 2sum119894=1
infinsum119896=1
100381610038161003816100381612058211989611989410038161003816100381610038162 100381610038161003816100381612060111989611989410038161003816100381610038162 + 100381610038161003816100381612059511989611989410038161003816100381610038162 (65)
Advances in Mathematical Physics 7
Hence using (57) and (59) we obtain
1003817100381710038171003817Φ119909119909100381710038171003817100381721198712(Ω) le 16119879 1 + 1205762(1 minus 1205762)2 (1 +
2120573)2
sdot 10038171003817100381710038171003817120601101584010158401003817100381710038171003817100381721198712(minus120587120587) + 10038171003817100381710038171003817120595101584010158401003817100381710038171003817100381721198712(minus120587120587) (66)
Now we can easily obtain an estimate for Ψ119905(119909 119905) from(1) This fact together with (60) and (66) gives the necessaryestimate (51) for the solution
From the obtained estimates it also follows that in theconstructed by us formal solution of the inverse problem allthe series converge they can be term-by-term differentiatedand the series obtained during differentiation also convergein sense of the metrics 1198712
From (47) and (54) by using the Holderrsquos inequality it iseasy to justify the inequality
max(119909119905)isinΩ
|Φ (119909 119905)|2 le 11986210038171003817100381710038171003817120601101584010158401003817100381710038171003817100381721198712(minus120587120587) + 10038171003817100381710038171003817120595101584010158401003817100381710038171003817100381721198712(minus120587120587) (67)
which justifies the continuity ofΦ(119909 119905) in the closed domainΩFrom the representation of the solution in the form of
series (37) (38) and inequalities (53) (54) it is easy to justifythe estimates 1003816100381610038161003816Φ119909119909 (119909 119905)1003816100381610038161003816 + 1003816100381610038161003816Φ119905 (119909 119905)1003816100381610038161003816 + 1003816100381610038161003816119891 (119909)1003816100381610038161003816
le 119862infinsum119896=1
100381610038161003816100381612058211989611989410038161003816100381610038162 10038161003816100381610038161206011198961198941003816100381610038161003816 + 10038161003816100381610038161205951198961198941003816100381610038161003816 (68)
Let 120601(119909) 120595(119909) isin 1198624[minus120587 120587] and let the functions 120601(119909) 120595(119909)and 12060110158401015840(119909) 12059510158401015840(119909) satisfy the boundary conditions (13) thenthe number series in the right-hand side of (68) convergesTherefore in such case the constructed by us formal solutiongives the regular solution of the inverse problem (1)-(3) (46)
The uniqueness of this constructed solution follows fromLemma 4
The theorem is completely proved
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This paper was published with the support of the ScienceCommittee of the Ministry of Education and Science of theRepublic of Kazakhstan under Project AP05133271 and TargetProgram BR05236656
References
[1] B Ahmad A Alsaedi M Kirane and R G TapdigogluldquoAn inverse problem for space and time fractional evolution
equations with an involution perturbationrdquo Quaestiones Math-ematicae vol 40 no 2 pp 151ndash160 2017
[2] A Cabada and A F Tojo ldquoEquations with involutionsrdquo inProceedings of the Workshop on Differential Equations p 240MallaMoravka CzechRepublicMarch 2014 httpusersmathcascz
[3] A Ashyralyev and A Sarsenbi ldquoWell-posedness of a parabolicequation with nonlocal boundary conditionrdquo Boundary ValueProblems vol 2015 no 1 2015
[4] M Kirane and N Al-Salti ldquoInverse problems for a nonlocalwave equation with an involution perturbationrdquo Journal ofNonlinear Science and Applications vol 9 no 3 pp 1243ndash12512016
[5] A Ashyralyev and A Sarsenbi ldquoWell-Posedness of a ParabolicEquation with Involutionrdquo Numerical Functional Analysis andOptimization vol 38 no 10 pp 1295ndash1304 2017
[6] I Orazov and M A Sadybekov ldquoOne nonlocal problem ofdetermination of the temperature and density of heat sourcesrdquoRussian Mathematics vol 56 no 2 pp 60ndash64 2012
[7] I Orazov and M A Sadybekov ldquoOn a class of problems ofdetermining the temperature and density of heat sources giveninitial and final temperaturerdquo Siberian Mathematical Journalvol 53 no 1 pp 146ndash151 2012
[8] N I Ivanchov ldquoInverse problems for the heat-conductionequation with nonlocal boundary conditionsrdquoUkrainianMath-ematical Journal vol 45 no 8 pp 1186ndash1192 1993
[9] I A Kaliev and M M Sabitova ldquoProblems of determiningthe temperature and density of heat sources from the initialand final temperaturesrdquo Journal of Applied and IndustrialMathematics vol 4 no 3 pp 332ndash339 2010
[10] I A KalievM FMugafarov andOV Fattahova ldquoInverse prob-lem for forwardbackward parabolic equation with generalizedconjugation conditionsrdquo Ufa Mathematical Journal vol 3 no2 pp 33ndash41 2011
[11] M Kirane and S A Malik ldquoDetermination of an unknownsource term and the temperature distribution for the linearheat equation involving fractional derivative in timerdquo AppliedMathematics and Computation vol 218 no 1 pp 163ndash170 2011
[12] F Kanca andM I Ismailov ldquoThe inverse problem of finding thetime-dependent diffusion coefficient of the heat equation fromintegral overdetermination datardquo Inverse Problems in Scienceand Engineering vol 20 no 4 pp 463ndash476 2012
[13] M Kirane S A Malik and M Al-Gwaiz ldquoAn inverse sourceproblem for a two dimensional time fractional diffusionequation with nonlocal boundary conditionsrdquo MathematicalMethods in the Applied Sciences vol 36 no 9 pp 1056ndash10692013
[14] A Ashyralyev and Y A Sharifov ldquoExistence and uniquenessof solutions for nonlinear impulsive differential equations withtwo-point and integral boundary conditionsrdquo Advances inDifference Equations vol 2013 no 1 p 173 2013
[15] F Kanca ldquoInverse coefficient problem of the parabolic equa-tion with periodic boundary and integral overdeterminationconditionsrdquo Abstract and Applied Analysis vol 2013 Article ID659804 7 pages 2013
[16] D Lesnic S A Yousefi and M Ivanchov ldquoDetermination of atime-dependent diffusivity from nonlocal conditionsrdquo AppliedMathematics and Computation vol 41 no 1-2 pp 301ndash3202013
[17] L Miller and M Yamamoto ldquoCoefficient inverse problem for afractional diffusion equationrdquo Inverse Problems vol 29 no 7 p075013 2013
8 Advances in Mathematical Physics
[18] G Li D Zhang X Jia and M Yamamoto ldquoSimultaneousinversion for the space-dependent diffusion coefficient andthe fractional order in the time-fractional diffusion equationrdquoInverse Problems vol 29 no 6 Article ID 065014 2013
[19] A B Kostin ldquoCounterexamples in inverse problems forparabolic elliptic and hyperbolic equationsrdquo ComputationalMathematics and Mathematical Physics vol 54 no 5 pp 797ndash810 2014
[20] A Ashyralyev and A Hanalyev ldquoWell-posedness of nonlocalparabolic differential problems with dependent operatorsrdquoTheScientific World Journal vol 2014 Article ID 519814 2014
[21] I Orazov and M A Sadybekov ldquoOn an inverse problemof mathematical modeling of the extraction process of poly-disperse porous materialsrdquo in Proceedings of the ADVANCE-MENTS IN MATHEMATICAL SCIENCES Proceedings of theInternational Conference on Advancements in MathematicalSciences p 020005 Antalya Turkey
[22] I Orazov and M A Sadybekov ldquoOne-dimensional diffusionproblem with not strengthened regular boundary conditionsrdquoin Proceedings of the 41ST International Conference ldquoApplicationsof Mathematics in Engineering And EconomicSrdquo AMEE rsquo15 p040007 Sozopol Bulgaria
[23] N H Tuan D N D Hai L D Long V T Nguyen and MKirane ldquoOn a RieszndashFeller space fractional backward diffusionproblemwith a nonlinear sourcerdquo Journal of Computational andApplied Mathematics vol 312 pp 103ndash126 2017
[24] M Sadybekov G Oralsyn and M Ismailov ldquoAn inverseproblem of finding the time-dependent heat transfer coefficientfrom an integral conditionrdquo International Journal of Pure andApplied Mathematics vol 113 no 4 pp 139ndash149 2017
[25] M Kirane andMNH Tuan ldquoIdentification and regularizationfor unknown source for a time-fractional diffusion equationrdquoComputersMathematics withApplications vol 73 no 6 pp 931ndash950 2017
[26] B T Torebek and R Tapdigoglu ldquoSome inverse problems forthe nonlocal heat equation with Caputo fractional derivativerdquoMathematical Methods in the Applied Sciences vol 40 no 18pp 6468ndash6479 2017
[27] M Kirane B Samet and B T Torebek ldquoDetermination ofan unknown source term temperature distribution for thesub-diffusion equation at the initial and final datardquo ElectronicJournal of Differential Equations vol 2017 2017
[28] O Taki-Eddine and B Abdelfatah ldquoOn determining the coef-ficient in a parabolic equation with nonlocal boundary andintegral conditionrdquo Electronic Journal of Mathematical Analysisand Applications vol 6 no 1 pp 94ndash102 2018
[29] A M Sarsenbi ldquoUnconditional bases related to a nonclassicalsecond-order differential operatorrdquo Journal of Differential Equa-tions vol 46 no 4 pp 509ndash514 2010
[30] V PKurdyumov andA PKhromov ldquoTheRiesz bases consistingof eigen and associated functions for a functional differentialoperator with variable structurerdquo Russian Mathematics vol 54no 2 pp 33ndash45 2010
[31] A M Sarsenbi and A A Tengaeva ldquoOn the basis propertiesof the root functions of two generalized spectral problemsrdquoJournal of Differential Equations vol 48 no 2 pp 306ndash3082012
[32] M A Sadybekov and A M Sarsenbi ldquoCriterion for the basisproperty of the eigenfunction system of a multiple differen-tiation operator with an involutionrdquo Journal of DifferentialEquations vol 48 no 8 pp 1112ndash1118 2012
[33] A Kopzhassarova and A Sarsenbi ldquoBasis properties of eigen-functions of second-order differential operators with invo-lutionrdquo Abstract and Applied Analysis vol 2012 Article ID576843 6 pages 2012
[34] AAKopzhassarova A L Lukashov andAM Sarsenbi ldquoSpec-tral properties of non-self-adjoint perturbations for a spectralproblem with involutionrdquo Abstract and Applied Analysis vol2012 Article ID 590781 5 pages 2012
[35] A Sarsenbi and M Sadybekov ldquoEigenfunctions of a fourthorder operator pencilrdquo in Proceedings of the 2nd InternationalConference on Analysis and Applied Mathematics ICAAM 2014pp 241ndash245 kaz September 2014
[36] L V Kritskov and A M Sarsenbi ldquoSpectral properties of anonlocal problem for a second-order differential equation withan involutionrdquo Journal of Differential Equations vol 51 no 8pp 984ndash990 2015
[37] L V Kritskov and A M Sarsenbi ldquoBasicity in Lp of rootfunctions for differential equations with involutionrdquo ElectronicJournal of Differential Equations vol 2015 2015
[38] M A Sadybekov A Sarsenbi and A Tengayeva ldquoDescriptionof spectral properties of a generalized spectral problem withinvolution for differentiation operator of the second orderrdquo inProceedings of the 3rd International Conference on Analysis andApplied Mathematics ICAAM 2016 kaz September 2016
[39] A G Baskakov I A Krishtal and E Y Romanova ldquoSpectralanalysis of a differential operator with an involutionrdquo Journal ofEvolution Equations (JEE) vol 17 no 2 pp 669ndash684 2017
[40] L V Kritskov and A M Sarsenbi ldquoRiesz basis property ofsystem of root functions of second-order differential operatorwith involutionrdquo Journal of Differential Equations vol 53 no 1pp 33ndash46 2017
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Submit your manuscripts atwwwhindawicom
6 Advances in Mathematical Physics
(B) Let 120601(119909) 120595(119909) isin 1198624[minus120587 120587] and the functions120601(119909) 120595(119909) and 12060110158401015840(119909) 12059510158401015840(119909) satisfy the boundary conditions(13) then for a real number 120576 such that |120576| lt 1 the inverseproblem (1)-(3) (46) has a unique regular solution
Proof Since |120576| lt 1 119886 gt 0 we have 120582119896119894 gt 0 for all 119896 119894Consequently there exists a constant 120573 gt 0 such that for allthe values of the indexes 119896 119894 we have
1 minus 119890minus120582119896119894119879 ge 1 minus 119890minus12058202119879 ge 120573 gt 0 (52)
Since 120582119896119894 gt 0 then 119890minus120582119896119894119879 le 119890minus120582119896119894119905 lt 1 Therefore fromrepresentations (49) and (50) we get estimates
10038161003816100381610038161198911198961198941003816100381610038161003816 le10038161003816100381610038161205821198961198941003816100381610038161003816120573 10038161003816100381610038161206011198961198941003816100381610038161003816 + 10038161003816100381610038161205951198961198941003816100381610038161003816 (53)
1003816100381610038161003816Φ119896119894 (119905)1003816100381610038161003816 le (1 + 2120573) 10038161003816100381610038161206011198961198941003816100381610038161003816 + 10038161003816100381610038161205951198961198941003816100381610038161003816 (54)
where constant 120573 does not depend on the indexes 119896 119894 and onthe functions 120601(119909) and 120595(119909)
As proved in Lemma 3 the eigenfunctions system (17)forms the orthonormal basis in 1198712(minus120587 120587)Therefore we havean expansion
120601 (119909) = infinsum119896=1
12060111989611198831198961 (119909) + infinsum119896=0
12060111989621198831198962 (119909) (55)
Taking into account (12) we obtain
minus12060110158401015840 (119909) + 12057612060110158401015840 (minus119909) = infinsum119896=1
120582119896112060111989611198831198961 (119909)
+ infinsum119896=0
120582119896212060111989621198831198962 (119909) (56)
From this by Parsevalrsquos equality it is easy to obtain anestimate
infinsum119896=1
1003816100381610038161003816120582119896110038161003816100381610038162 1003816100381610038161003816120601119896110038161003816100381610038162 +infinsum119896=1
1003816100381610038161003816120582119896210038161003816100381610038162 1003816100381610038161003816120601119896210038161003816100381610038162
le 2 (1 + 1205762) 10038171003817100381710038171003817120601101584010158401003817100381710038171003817100381721198712(minus120587120587) (57)
Similarly for the coefficients of the expansion
120595 (119909) = infinsum119896=1
12059511989611198831198961 (119909) + infinsum119896=0
12059511989621198831198962 (119909) (58)
we have the inequality
infinsum119896=1
1003816100381610038161003816120582119896110038161003816100381610038162 1003816100381610038161003816120595119896110038161003816100381610038162 +infinsum119896=1
1003816100381610038161003816120582119896210038161003816100381610038162 1003816100381610038161003816120595119896210038161003816100381610038162
le 2 (1 + 1205762) 10038171003817100381710038171003817120595101584010158401003817100381710038171003817100381721198712(minus120587120587) (59)
Now from (53) (57) and (59) we see that the series (48)converges and the estimate
1003817100381710038171003817119891100381710038171003817100381721198712(minus120587120587) le 21 + 12057621205732 10038171003817100381710038171003817120601101584010158401003817100381710038171003817100381721198712(minus120587120587) + 10038171003817100381710038171003817120595101584010158401003817100381710038171003817100381721198712(minus120587120587) (60)
holdsTaking into account (12) from (47) we obtain
minus Φ119909119909 (119909 119905) + 120576Φ119909119909 (minus119909 119905)= infinsum119896=1
1205821198961Φ1198961 (119905) 1198831198961 (119909)
+ infinsum119896=0
1205821198962Φ1198962 (119905) 1198831198962 (119909) (61)
Substituting the value (minus119909) instead of 119909 we findminus Φ119909119909 (minus119909 119905) + 120576Φ119909119909 (119909 119905)= minusinfinsum119896=1
1205821198961Φ1198961 (119905) 1198831198961 (119909)
+ infinsum119896=0
1205821198962Φ1198962 (119905) 1198831198962 (119909) (62)
Here we use the properties 1198831198961(minus119909) = minus1198831198961(119909) 1198831198962(minus119909) =1198831198962(119909) obtained from the explicit form (12) of the eigenfunc-tions
Multiplying (62) by 120576 and summing with (61) we have
(minus1 + 1205762)Φ119909119909 (119909 119905)= (1 minus 120576) infinsum
119896=1
1205821198961Φ1198961 (119905) 1198831198961 (119909)
+ (1 + 120576) infinsum119896=0
1205821198962Φ1198962 (119905) 1198831198962 (119909) (63)
Since |120576| lt 1 from this by Parsevalrsquos equality we find
1003817100381710038171003817Φ119909119909 (sdot 119905)100381710038171003817100381721198712(minus120587120587) = 1(1 + 120576)2
infinsum119896=1
1003816100381610038161003816120582119896110038161003816100381610038162 1003816100381610038161003816Φ1198961 (119905)10038161003816100381610038162
+ 1(1 minus 120576)2
infinsum119896=0
1003816100381610038161003816120582119896210038161003816100381610038162 1003816100381610038161003816Φ1198962 (119905)10038161003816100381610038162(64)
for each 119905 isin [0 119879] Integrating this equation with respect to119905 isin [0 119879] taking (54) into account we have1003817100381710038171003817Φ119909119909100381710038171003817100381721198712(Ω) le 8119879
(1 minus 1205762)2 (1 +2120573)2
sdot 2sum119894=1
infinsum119896=1
100381610038161003816100381612058211989611989410038161003816100381610038162 100381610038161003816100381612060111989611989410038161003816100381610038162 + 100381610038161003816100381612059511989611989410038161003816100381610038162 (65)
Advances in Mathematical Physics 7
Hence using (57) and (59) we obtain
1003817100381710038171003817Φ119909119909100381710038171003817100381721198712(Ω) le 16119879 1 + 1205762(1 minus 1205762)2 (1 +
2120573)2
sdot 10038171003817100381710038171003817120601101584010158401003817100381710038171003817100381721198712(minus120587120587) + 10038171003817100381710038171003817120595101584010158401003817100381710038171003817100381721198712(minus120587120587) (66)
Now we can easily obtain an estimate for Ψ119905(119909 119905) from(1) This fact together with (60) and (66) gives the necessaryestimate (51) for the solution
From the obtained estimates it also follows that in theconstructed by us formal solution of the inverse problem allthe series converge they can be term-by-term differentiatedand the series obtained during differentiation also convergein sense of the metrics 1198712
From (47) and (54) by using the Holderrsquos inequality it iseasy to justify the inequality
max(119909119905)isinΩ
|Φ (119909 119905)|2 le 11986210038171003817100381710038171003817120601101584010158401003817100381710038171003817100381721198712(minus120587120587) + 10038171003817100381710038171003817120595101584010158401003817100381710038171003817100381721198712(minus120587120587) (67)
which justifies the continuity ofΦ(119909 119905) in the closed domainΩFrom the representation of the solution in the form of
series (37) (38) and inequalities (53) (54) it is easy to justifythe estimates 1003816100381610038161003816Φ119909119909 (119909 119905)1003816100381610038161003816 + 1003816100381610038161003816Φ119905 (119909 119905)1003816100381610038161003816 + 1003816100381610038161003816119891 (119909)1003816100381610038161003816
le 119862infinsum119896=1
100381610038161003816100381612058211989611989410038161003816100381610038162 10038161003816100381610038161206011198961198941003816100381610038161003816 + 10038161003816100381610038161205951198961198941003816100381610038161003816 (68)
Let 120601(119909) 120595(119909) isin 1198624[minus120587 120587] and let the functions 120601(119909) 120595(119909)and 12060110158401015840(119909) 12059510158401015840(119909) satisfy the boundary conditions (13) thenthe number series in the right-hand side of (68) convergesTherefore in such case the constructed by us formal solutiongives the regular solution of the inverse problem (1)-(3) (46)
The uniqueness of this constructed solution follows fromLemma 4
The theorem is completely proved
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This paper was published with the support of the ScienceCommittee of the Ministry of Education and Science of theRepublic of Kazakhstan under Project AP05133271 and TargetProgram BR05236656
References
[1] B Ahmad A Alsaedi M Kirane and R G TapdigogluldquoAn inverse problem for space and time fractional evolution
equations with an involution perturbationrdquo Quaestiones Math-ematicae vol 40 no 2 pp 151ndash160 2017
[2] A Cabada and A F Tojo ldquoEquations with involutionsrdquo inProceedings of the Workshop on Differential Equations p 240MallaMoravka CzechRepublicMarch 2014 httpusersmathcascz
[3] A Ashyralyev and A Sarsenbi ldquoWell-posedness of a parabolicequation with nonlocal boundary conditionrdquo Boundary ValueProblems vol 2015 no 1 2015
[4] M Kirane and N Al-Salti ldquoInverse problems for a nonlocalwave equation with an involution perturbationrdquo Journal ofNonlinear Science and Applications vol 9 no 3 pp 1243ndash12512016
[5] A Ashyralyev and A Sarsenbi ldquoWell-Posedness of a ParabolicEquation with Involutionrdquo Numerical Functional Analysis andOptimization vol 38 no 10 pp 1295ndash1304 2017
[6] I Orazov and M A Sadybekov ldquoOne nonlocal problem ofdetermination of the temperature and density of heat sourcesrdquoRussian Mathematics vol 56 no 2 pp 60ndash64 2012
[7] I Orazov and M A Sadybekov ldquoOn a class of problems ofdetermining the temperature and density of heat sources giveninitial and final temperaturerdquo Siberian Mathematical Journalvol 53 no 1 pp 146ndash151 2012
[8] N I Ivanchov ldquoInverse problems for the heat-conductionequation with nonlocal boundary conditionsrdquoUkrainianMath-ematical Journal vol 45 no 8 pp 1186ndash1192 1993
[9] I A Kaliev and M M Sabitova ldquoProblems of determiningthe temperature and density of heat sources from the initialand final temperaturesrdquo Journal of Applied and IndustrialMathematics vol 4 no 3 pp 332ndash339 2010
[10] I A KalievM FMugafarov andOV Fattahova ldquoInverse prob-lem for forwardbackward parabolic equation with generalizedconjugation conditionsrdquo Ufa Mathematical Journal vol 3 no2 pp 33ndash41 2011
[11] M Kirane and S A Malik ldquoDetermination of an unknownsource term and the temperature distribution for the linearheat equation involving fractional derivative in timerdquo AppliedMathematics and Computation vol 218 no 1 pp 163ndash170 2011
[12] F Kanca andM I Ismailov ldquoThe inverse problem of finding thetime-dependent diffusion coefficient of the heat equation fromintegral overdetermination datardquo Inverse Problems in Scienceand Engineering vol 20 no 4 pp 463ndash476 2012
[13] M Kirane S A Malik and M Al-Gwaiz ldquoAn inverse sourceproblem for a two dimensional time fractional diffusionequation with nonlocal boundary conditionsrdquo MathematicalMethods in the Applied Sciences vol 36 no 9 pp 1056ndash10692013
[14] A Ashyralyev and Y A Sharifov ldquoExistence and uniquenessof solutions for nonlinear impulsive differential equations withtwo-point and integral boundary conditionsrdquo Advances inDifference Equations vol 2013 no 1 p 173 2013
[15] F Kanca ldquoInverse coefficient problem of the parabolic equa-tion with periodic boundary and integral overdeterminationconditionsrdquo Abstract and Applied Analysis vol 2013 Article ID659804 7 pages 2013
[16] D Lesnic S A Yousefi and M Ivanchov ldquoDetermination of atime-dependent diffusivity from nonlocal conditionsrdquo AppliedMathematics and Computation vol 41 no 1-2 pp 301ndash3202013
[17] L Miller and M Yamamoto ldquoCoefficient inverse problem for afractional diffusion equationrdquo Inverse Problems vol 29 no 7 p075013 2013
8 Advances in Mathematical Physics
[18] G Li D Zhang X Jia and M Yamamoto ldquoSimultaneousinversion for the space-dependent diffusion coefficient andthe fractional order in the time-fractional diffusion equationrdquoInverse Problems vol 29 no 6 Article ID 065014 2013
[19] A B Kostin ldquoCounterexamples in inverse problems forparabolic elliptic and hyperbolic equationsrdquo ComputationalMathematics and Mathematical Physics vol 54 no 5 pp 797ndash810 2014
[20] A Ashyralyev and A Hanalyev ldquoWell-posedness of nonlocalparabolic differential problems with dependent operatorsrdquoTheScientific World Journal vol 2014 Article ID 519814 2014
[21] I Orazov and M A Sadybekov ldquoOn an inverse problemof mathematical modeling of the extraction process of poly-disperse porous materialsrdquo in Proceedings of the ADVANCE-MENTS IN MATHEMATICAL SCIENCES Proceedings of theInternational Conference on Advancements in MathematicalSciences p 020005 Antalya Turkey
[22] I Orazov and M A Sadybekov ldquoOne-dimensional diffusionproblem with not strengthened regular boundary conditionsrdquoin Proceedings of the 41ST International Conference ldquoApplicationsof Mathematics in Engineering And EconomicSrdquo AMEE rsquo15 p040007 Sozopol Bulgaria
[23] N H Tuan D N D Hai L D Long V T Nguyen and MKirane ldquoOn a RieszndashFeller space fractional backward diffusionproblemwith a nonlinear sourcerdquo Journal of Computational andApplied Mathematics vol 312 pp 103ndash126 2017
[24] M Sadybekov G Oralsyn and M Ismailov ldquoAn inverseproblem of finding the time-dependent heat transfer coefficientfrom an integral conditionrdquo International Journal of Pure andApplied Mathematics vol 113 no 4 pp 139ndash149 2017
[25] M Kirane andMNH Tuan ldquoIdentification and regularizationfor unknown source for a time-fractional diffusion equationrdquoComputersMathematics withApplications vol 73 no 6 pp 931ndash950 2017
[26] B T Torebek and R Tapdigoglu ldquoSome inverse problems forthe nonlocal heat equation with Caputo fractional derivativerdquoMathematical Methods in the Applied Sciences vol 40 no 18pp 6468ndash6479 2017
[27] M Kirane B Samet and B T Torebek ldquoDetermination ofan unknown source term temperature distribution for thesub-diffusion equation at the initial and final datardquo ElectronicJournal of Differential Equations vol 2017 2017
[28] O Taki-Eddine and B Abdelfatah ldquoOn determining the coef-ficient in a parabolic equation with nonlocal boundary andintegral conditionrdquo Electronic Journal of Mathematical Analysisand Applications vol 6 no 1 pp 94ndash102 2018
[29] A M Sarsenbi ldquoUnconditional bases related to a nonclassicalsecond-order differential operatorrdquo Journal of Differential Equa-tions vol 46 no 4 pp 509ndash514 2010
[30] V PKurdyumov andA PKhromov ldquoTheRiesz bases consistingof eigen and associated functions for a functional differentialoperator with variable structurerdquo Russian Mathematics vol 54no 2 pp 33ndash45 2010
[31] A M Sarsenbi and A A Tengaeva ldquoOn the basis propertiesof the root functions of two generalized spectral problemsrdquoJournal of Differential Equations vol 48 no 2 pp 306ndash3082012
[32] M A Sadybekov and A M Sarsenbi ldquoCriterion for the basisproperty of the eigenfunction system of a multiple differen-tiation operator with an involutionrdquo Journal of DifferentialEquations vol 48 no 8 pp 1112ndash1118 2012
[33] A Kopzhassarova and A Sarsenbi ldquoBasis properties of eigen-functions of second-order differential operators with invo-lutionrdquo Abstract and Applied Analysis vol 2012 Article ID576843 6 pages 2012
[34] AAKopzhassarova A L Lukashov andAM Sarsenbi ldquoSpec-tral properties of non-self-adjoint perturbations for a spectralproblem with involutionrdquo Abstract and Applied Analysis vol2012 Article ID 590781 5 pages 2012
[35] A Sarsenbi and M Sadybekov ldquoEigenfunctions of a fourthorder operator pencilrdquo in Proceedings of the 2nd InternationalConference on Analysis and Applied Mathematics ICAAM 2014pp 241ndash245 kaz September 2014
[36] L V Kritskov and A M Sarsenbi ldquoSpectral properties of anonlocal problem for a second-order differential equation withan involutionrdquo Journal of Differential Equations vol 51 no 8pp 984ndash990 2015
[37] L V Kritskov and A M Sarsenbi ldquoBasicity in Lp of rootfunctions for differential equations with involutionrdquo ElectronicJournal of Differential Equations vol 2015 2015
[38] M A Sadybekov A Sarsenbi and A Tengayeva ldquoDescriptionof spectral properties of a generalized spectral problem withinvolution for differentiation operator of the second orderrdquo inProceedings of the 3rd International Conference on Analysis andApplied Mathematics ICAAM 2016 kaz September 2016
[39] A G Baskakov I A Krishtal and E Y Romanova ldquoSpectralanalysis of a differential operator with an involutionrdquo Journal ofEvolution Equations (JEE) vol 17 no 2 pp 669ndash684 2017
[40] L V Kritskov and A M Sarsenbi ldquoRiesz basis property ofsystem of root functions of second-order differential operatorwith involutionrdquo Journal of Differential Equations vol 53 no 1pp 33ndash46 2017
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
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Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Advances in Mathematical Physics 7
Hence using (57) and (59) we obtain
1003817100381710038171003817Φ119909119909100381710038171003817100381721198712(Ω) le 16119879 1 + 1205762(1 minus 1205762)2 (1 +
2120573)2
sdot 10038171003817100381710038171003817120601101584010158401003817100381710038171003817100381721198712(minus120587120587) + 10038171003817100381710038171003817120595101584010158401003817100381710038171003817100381721198712(minus120587120587) (66)
Now we can easily obtain an estimate for Ψ119905(119909 119905) from(1) This fact together with (60) and (66) gives the necessaryestimate (51) for the solution
From the obtained estimates it also follows that in theconstructed by us formal solution of the inverse problem allthe series converge they can be term-by-term differentiatedand the series obtained during differentiation also convergein sense of the metrics 1198712
From (47) and (54) by using the Holderrsquos inequality it iseasy to justify the inequality
max(119909119905)isinΩ
|Φ (119909 119905)|2 le 11986210038171003817100381710038171003817120601101584010158401003817100381710038171003817100381721198712(minus120587120587) + 10038171003817100381710038171003817120595101584010158401003817100381710038171003817100381721198712(minus120587120587) (67)
which justifies the continuity ofΦ(119909 119905) in the closed domainΩFrom the representation of the solution in the form of
series (37) (38) and inequalities (53) (54) it is easy to justifythe estimates 1003816100381610038161003816Φ119909119909 (119909 119905)1003816100381610038161003816 + 1003816100381610038161003816Φ119905 (119909 119905)1003816100381610038161003816 + 1003816100381610038161003816119891 (119909)1003816100381610038161003816
le 119862infinsum119896=1
100381610038161003816100381612058211989611989410038161003816100381610038162 10038161003816100381610038161206011198961198941003816100381610038161003816 + 10038161003816100381610038161205951198961198941003816100381610038161003816 (68)
Let 120601(119909) 120595(119909) isin 1198624[minus120587 120587] and let the functions 120601(119909) 120595(119909)and 12060110158401015840(119909) 12059510158401015840(119909) satisfy the boundary conditions (13) thenthe number series in the right-hand side of (68) convergesTherefore in such case the constructed by us formal solutiongives the regular solution of the inverse problem (1)-(3) (46)
The uniqueness of this constructed solution follows fromLemma 4
The theorem is completely proved
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This paper was published with the support of the ScienceCommittee of the Ministry of Education and Science of theRepublic of Kazakhstan under Project AP05133271 and TargetProgram BR05236656
References
[1] B Ahmad A Alsaedi M Kirane and R G TapdigogluldquoAn inverse problem for space and time fractional evolution
equations with an involution perturbationrdquo Quaestiones Math-ematicae vol 40 no 2 pp 151ndash160 2017
[2] A Cabada and A F Tojo ldquoEquations with involutionsrdquo inProceedings of the Workshop on Differential Equations p 240MallaMoravka CzechRepublicMarch 2014 httpusersmathcascz
[3] A Ashyralyev and A Sarsenbi ldquoWell-posedness of a parabolicequation with nonlocal boundary conditionrdquo Boundary ValueProblems vol 2015 no 1 2015
[4] M Kirane and N Al-Salti ldquoInverse problems for a nonlocalwave equation with an involution perturbationrdquo Journal ofNonlinear Science and Applications vol 9 no 3 pp 1243ndash12512016
[5] A Ashyralyev and A Sarsenbi ldquoWell-Posedness of a ParabolicEquation with Involutionrdquo Numerical Functional Analysis andOptimization vol 38 no 10 pp 1295ndash1304 2017
[6] I Orazov and M A Sadybekov ldquoOne nonlocal problem ofdetermination of the temperature and density of heat sourcesrdquoRussian Mathematics vol 56 no 2 pp 60ndash64 2012
[7] I Orazov and M A Sadybekov ldquoOn a class of problems ofdetermining the temperature and density of heat sources giveninitial and final temperaturerdquo Siberian Mathematical Journalvol 53 no 1 pp 146ndash151 2012
[8] N I Ivanchov ldquoInverse problems for the heat-conductionequation with nonlocal boundary conditionsrdquoUkrainianMath-ematical Journal vol 45 no 8 pp 1186ndash1192 1993
[9] I A Kaliev and M M Sabitova ldquoProblems of determiningthe temperature and density of heat sources from the initialand final temperaturesrdquo Journal of Applied and IndustrialMathematics vol 4 no 3 pp 332ndash339 2010
[10] I A KalievM FMugafarov andOV Fattahova ldquoInverse prob-lem for forwardbackward parabolic equation with generalizedconjugation conditionsrdquo Ufa Mathematical Journal vol 3 no2 pp 33ndash41 2011
[11] M Kirane and S A Malik ldquoDetermination of an unknownsource term and the temperature distribution for the linearheat equation involving fractional derivative in timerdquo AppliedMathematics and Computation vol 218 no 1 pp 163ndash170 2011
[12] F Kanca andM I Ismailov ldquoThe inverse problem of finding thetime-dependent diffusion coefficient of the heat equation fromintegral overdetermination datardquo Inverse Problems in Scienceand Engineering vol 20 no 4 pp 463ndash476 2012
[13] M Kirane S A Malik and M Al-Gwaiz ldquoAn inverse sourceproblem for a two dimensional time fractional diffusionequation with nonlocal boundary conditionsrdquo MathematicalMethods in the Applied Sciences vol 36 no 9 pp 1056ndash10692013
[14] A Ashyralyev and Y A Sharifov ldquoExistence and uniquenessof solutions for nonlinear impulsive differential equations withtwo-point and integral boundary conditionsrdquo Advances inDifference Equations vol 2013 no 1 p 173 2013
[15] F Kanca ldquoInverse coefficient problem of the parabolic equa-tion with periodic boundary and integral overdeterminationconditionsrdquo Abstract and Applied Analysis vol 2013 Article ID659804 7 pages 2013
[16] D Lesnic S A Yousefi and M Ivanchov ldquoDetermination of atime-dependent diffusivity from nonlocal conditionsrdquo AppliedMathematics and Computation vol 41 no 1-2 pp 301ndash3202013
[17] L Miller and M Yamamoto ldquoCoefficient inverse problem for afractional diffusion equationrdquo Inverse Problems vol 29 no 7 p075013 2013
8 Advances in Mathematical Physics
[18] G Li D Zhang X Jia and M Yamamoto ldquoSimultaneousinversion for the space-dependent diffusion coefficient andthe fractional order in the time-fractional diffusion equationrdquoInverse Problems vol 29 no 6 Article ID 065014 2013
[19] A B Kostin ldquoCounterexamples in inverse problems forparabolic elliptic and hyperbolic equationsrdquo ComputationalMathematics and Mathematical Physics vol 54 no 5 pp 797ndash810 2014
[20] A Ashyralyev and A Hanalyev ldquoWell-posedness of nonlocalparabolic differential problems with dependent operatorsrdquoTheScientific World Journal vol 2014 Article ID 519814 2014
[21] I Orazov and M A Sadybekov ldquoOn an inverse problemof mathematical modeling of the extraction process of poly-disperse porous materialsrdquo in Proceedings of the ADVANCE-MENTS IN MATHEMATICAL SCIENCES Proceedings of theInternational Conference on Advancements in MathematicalSciences p 020005 Antalya Turkey
[22] I Orazov and M A Sadybekov ldquoOne-dimensional diffusionproblem with not strengthened regular boundary conditionsrdquoin Proceedings of the 41ST International Conference ldquoApplicationsof Mathematics in Engineering And EconomicSrdquo AMEE rsquo15 p040007 Sozopol Bulgaria
[23] N H Tuan D N D Hai L D Long V T Nguyen and MKirane ldquoOn a RieszndashFeller space fractional backward diffusionproblemwith a nonlinear sourcerdquo Journal of Computational andApplied Mathematics vol 312 pp 103ndash126 2017
[24] M Sadybekov G Oralsyn and M Ismailov ldquoAn inverseproblem of finding the time-dependent heat transfer coefficientfrom an integral conditionrdquo International Journal of Pure andApplied Mathematics vol 113 no 4 pp 139ndash149 2017
[25] M Kirane andMNH Tuan ldquoIdentification and regularizationfor unknown source for a time-fractional diffusion equationrdquoComputersMathematics withApplications vol 73 no 6 pp 931ndash950 2017
[26] B T Torebek and R Tapdigoglu ldquoSome inverse problems forthe nonlocal heat equation with Caputo fractional derivativerdquoMathematical Methods in the Applied Sciences vol 40 no 18pp 6468ndash6479 2017
[27] M Kirane B Samet and B T Torebek ldquoDetermination ofan unknown source term temperature distribution for thesub-diffusion equation at the initial and final datardquo ElectronicJournal of Differential Equations vol 2017 2017
[28] O Taki-Eddine and B Abdelfatah ldquoOn determining the coef-ficient in a parabolic equation with nonlocal boundary andintegral conditionrdquo Electronic Journal of Mathematical Analysisand Applications vol 6 no 1 pp 94ndash102 2018
[29] A M Sarsenbi ldquoUnconditional bases related to a nonclassicalsecond-order differential operatorrdquo Journal of Differential Equa-tions vol 46 no 4 pp 509ndash514 2010
[30] V PKurdyumov andA PKhromov ldquoTheRiesz bases consistingof eigen and associated functions for a functional differentialoperator with variable structurerdquo Russian Mathematics vol 54no 2 pp 33ndash45 2010
[31] A M Sarsenbi and A A Tengaeva ldquoOn the basis propertiesof the root functions of two generalized spectral problemsrdquoJournal of Differential Equations vol 48 no 2 pp 306ndash3082012
[32] M A Sadybekov and A M Sarsenbi ldquoCriterion for the basisproperty of the eigenfunction system of a multiple differen-tiation operator with an involutionrdquo Journal of DifferentialEquations vol 48 no 8 pp 1112ndash1118 2012
[33] A Kopzhassarova and A Sarsenbi ldquoBasis properties of eigen-functions of second-order differential operators with invo-lutionrdquo Abstract and Applied Analysis vol 2012 Article ID576843 6 pages 2012
[34] AAKopzhassarova A L Lukashov andAM Sarsenbi ldquoSpec-tral properties of non-self-adjoint perturbations for a spectralproblem with involutionrdquo Abstract and Applied Analysis vol2012 Article ID 590781 5 pages 2012
[35] A Sarsenbi and M Sadybekov ldquoEigenfunctions of a fourthorder operator pencilrdquo in Proceedings of the 2nd InternationalConference on Analysis and Applied Mathematics ICAAM 2014pp 241ndash245 kaz September 2014
[36] L V Kritskov and A M Sarsenbi ldquoSpectral properties of anonlocal problem for a second-order differential equation withan involutionrdquo Journal of Differential Equations vol 51 no 8pp 984ndash990 2015
[37] L V Kritskov and A M Sarsenbi ldquoBasicity in Lp of rootfunctions for differential equations with involutionrdquo ElectronicJournal of Differential Equations vol 2015 2015
[38] M A Sadybekov A Sarsenbi and A Tengayeva ldquoDescriptionof spectral properties of a generalized spectral problem withinvolution for differentiation operator of the second orderrdquo inProceedings of the 3rd International Conference on Analysis andApplied Mathematics ICAAM 2016 kaz September 2016
[39] A G Baskakov I A Krishtal and E Y Romanova ldquoSpectralanalysis of a differential operator with an involutionrdquo Journal ofEvolution Equations (JEE) vol 17 no 2 pp 669ndash684 2017
[40] L V Kritskov and A M Sarsenbi ldquoRiesz basis property ofsystem of root functions of second-order differential operatorwith involutionrdquo Journal of Differential Equations vol 53 no 1pp 33ndash46 2017
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
8 Advances in Mathematical Physics
[18] G Li D Zhang X Jia and M Yamamoto ldquoSimultaneousinversion for the space-dependent diffusion coefficient andthe fractional order in the time-fractional diffusion equationrdquoInverse Problems vol 29 no 6 Article ID 065014 2013
[19] A B Kostin ldquoCounterexamples in inverse problems forparabolic elliptic and hyperbolic equationsrdquo ComputationalMathematics and Mathematical Physics vol 54 no 5 pp 797ndash810 2014
[20] A Ashyralyev and A Hanalyev ldquoWell-posedness of nonlocalparabolic differential problems with dependent operatorsrdquoTheScientific World Journal vol 2014 Article ID 519814 2014
[21] I Orazov and M A Sadybekov ldquoOn an inverse problemof mathematical modeling of the extraction process of poly-disperse porous materialsrdquo in Proceedings of the ADVANCE-MENTS IN MATHEMATICAL SCIENCES Proceedings of theInternational Conference on Advancements in MathematicalSciences p 020005 Antalya Turkey
[22] I Orazov and M A Sadybekov ldquoOne-dimensional diffusionproblem with not strengthened regular boundary conditionsrdquoin Proceedings of the 41ST International Conference ldquoApplicationsof Mathematics in Engineering And EconomicSrdquo AMEE rsquo15 p040007 Sozopol Bulgaria
[23] N H Tuan D N D Hai L D Long V T Nguyen and MKirane ldquoOn a RieszndashFeller space fractional backward diffusionproblemwith a nonlinear sourcerdquo Journal of Computational andApplied Mathematics vol 312 pp 103ndash126 2017
[24] M Sadybekov G Oralsyn and M Ismailov ldquoAn inverseproblem of finding the time-dependent heat transfer coefficientfrom an integral conditionrdquo International Journal of Pure andApplied Mathematics vol 113 no 4 pp 139ndash149 2017
[25] M Kirane andMNH Tuan ldquoIdentification and regularizationfor unknown source for a time-fractional diffusion equationrdquoComputersMathematics withApplications vol 73 no 6 pp 931ndash950 2017
[26] B T Torebek and R Tapdigoglu ldquoSome inverse problems forthe nonlocal heat equation with Caputo fractional derivativerdquoMathematical Methods in the Applied Sciences vol 40 no 18pp 6468ndash6479 2017
[27] M Kirane B Samet and B T Torebek ldquoDetermination ofan unknown source term temperature distribution for thesub-diffusion equation at the initial and final datardquo ElectronicJournal of Differential Equations vol 2017 2017
[28] O Taki-Eddine and B Abdelfatah ldquoOn determining the coef-ficient in a parabolic equation with nonlocal boundary andintegral conditionrdquo Electronic Journal of Mathematical Analysisand Applications vol 6 no 1 pp 94ndash102 2018
[29] A M Sarsenbi ldquoUnconditional bases related to a nonclassicalsecond-order differential operatorrdquo Journal of Differential Equa-tions vol 46 no 4 pp 509ndash514 2010
[30] V PKurdyumov andA PKhromov ldquoTheRiesz bases consistingof eigen and associated functions for a functional differentialoperator with variable structurerdquo Russian Mathematics vol 54no 2 pp 33ndash45 2010
[31] A M Sarsenbi and A A Tengaeva ldquoOn the basis propertiesof the root functions of two generalized spectral problemsrdquoJournal of Differential Equations vol 48 no 2 pp 306ndash3082012
[32] M A Sadybekov and A M Sarsenbi ldquoCriterion for the basisproperty of the eigenfunction system of a multiple differen-tiation operator with an involutionrdquo Journal of DifferentialEquations vol 48 no 8 pp 1112ndash1118 2012
[33] A Kopzhassarova and A Sarsenbi ldquoBasis properties of eigen-functions of second-order differential operators with invo-lutionrdquo Abstract and Applied Analysis vol 2012 Article ID576843 6 pages 2012
[34] AAKopzhassarova A L Lukashov andAM Sarsenbi ldquoSpec-tral properties of non-self-adjoint perturbations for a spectralproblem with involutionrdquo Abstract and Applied Analysis vol2012 Article ID 590781 5 pages 2012
[35] A Sarsenbi and M Sadybekov ldquoEigenfunctions of a fourthorder operator pencilrdquo in Proceedings of the 2nd InternationalConference on Analysis and Applied Mathematics ICAAM 2014pp 241ndash245 kaz September 2014
[36] L V Kritskov and A M Sarsenbi ldquoSpectral properties of anonlocal problem for a second-order differential equation withan involutionrdquo Journal of Differential Equations vol 51 no 8pp 984ndash990 2015
[37] L V Kritskov and A M Sarsenbi ldquoBasicity in Lp of rootfunctions for differential equations with involutionrdquo ElectronicJournal of Differential Equations vol 2015 2015
[38] M A Sadybekov A Sarsenbi and A Tengayeva ldquoDescriptionof spectral properties of a generalized spectral problem withinvolution for differentiation operator of the second orderrdquo inProceedings of the 3rd International Conference on Analysis andApplied Mathematics ICAAM 2016 kaz September 2016
[39] A G Baskakov I A Krishtal and E Y Romanova ldquoSpectralanalysis of a differential operator with an involutionrdquo Journal ofEvolution Equations (JEE) vol 17 no 2 pp 669ndash684 2017
[40] L V Kritskov and A M Sarsenbi ldquoRiesz basis property ofsystem of root functions of second-order differential operatorwith involutionrdquo Journal of Differential Equations vol 53 no 1pp 33ndash46 2017
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
