Research Article Analytic Solution of a Class of...
Transcript of Research Article Analytic Solution of a Class of...
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013, Article ID 109582, 5 pageshttp://dx.doi.org/10.1155/2013/109582
Research ArticleAnalytic Solution of a Class of Fractional Differential Equations
Yue Hu1 and Zuodong Yang2
1 School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, Henan 454000, China2 Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Nanjing, Jiangsu 210023, China
Correspondence should be addressed to Yue Hu; [email protected]
Received 8 July 2013; Revised 23 October 2013; Accepted 13 November 2013
Academic Editor: Shawn X. Wang
Copyright Β© 2013 Y. Hu and Z. Yang. This 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 the analytic solution of a class of fractional differential equations with variable coefficients by operatorial methods.Weobtain three theorems which extend the Garraβs results to the general case.
1. Introduction
Recently, Garra [1] studied the analytic solution of a class offractional differential equations with variable coefficients byusing operatorial methods.
The proof of his main results is strongly based onoperatorial properties of Caputo fractional derivative and thefollowing theorem by Miller and Ross [2].
TheoremM.-R. (Theorem 1.1 in [1]). Let π(π‘) be any functionof the form π‘ππ(π‘) or π‘π(ln π‘)π(π‘), where π > β1, and
π (π‘) =β
βπ=0
πππ‘π (1)
has a radius of convergence π > 0 and 0 < π < π . Then
π·πΌπ‘π·π½
π‘π (π‘) = π·
πΌ+π½
π‘π (π‘) (2)
holds for all π‘ β (0, π), provided(1) π½ < π + 1 and πΌ are arbitrary, or(2) π½ β₯ π + 1 and πΌ are arbitrary and ππ = 0 for π =0, 1, . . . , π β 1, whereπ = βπ½β (ceiling of the number).
Garraβs main results are as follows.
Theorem Garra (Theorem 1.2 in [1]). Consider the followingboundary value problem (BVP):
π·πΏπ₯
π (π₯, π‘) = π·πΌ
π‘π (π₯, π‘) ,
π (0, π‘) = π (π‘) ,(3)
in the half plane π‘ > 0, with analytic boundary condition π(π‘)such that the conditions of Theorem 1.1 (Theorem M.-R.) aresatisfied. The operatorial solution of BVP (3) is given by
π (π₯, π‘) =β
βπ=0
π₯ππ·πΌππ‘
(π!)2π (π‘) , (4)
where π·πΏπ₯
is the Laguerre derivative and π·πΌπ‘denotes Caputo
fractional derivative. On the basis of the previous result, Garraproved (Example 1 in [1]) that if π(π‘) = π‘π, π β N, then theanalytic solution of the BVP (3) is given by
π (π₯, π‘) =β
βπ=0
π₯ππ·πΌππ‘π‘π
(π!)2=[π/πΌ]
βπ=0
Ξ (π + 1)
Ξ (π + 1 β πΌπ)β π₯ππ‘πβπΌπ
(π!)2,
(5)
where Ξ(πΌ) = β«β0πβπ‘π‘πΌβ1ππ‘.
Motivated by the results, in the present note, we extendGarraβs results to the general case.
2. Preliminaries
Definition 1 (see [3]). For every positive integer π, theoperatorπ·ππΏ
π₯
:= π·π₯, . . . , π·π₯π·π₯π· (containing π+1 ordinaryderivatives) is called the π-order Laguerre derivatives and theππΏ-exponential function is defined by
ππ (π₯) :=β
βπ=0
π₯π
(π!)π+1. (6)
2 Abstract and Applied Analysis
In [4], the following result is proved.
Lemma 2. Let π be an arbitrary real or complex number. Thefunction ππ(ππ₯) is an eigenfunction of the operator π·ππΏ
π₯
; thatis,
π·ππΏπ₯
ππ = πππ (ππ₯) . (7)
For π = 0, we have π·0πΏ := π·. Thus, (7) leads to theclassical property of the exponential functionπ·πππ₯ = ππππ₯.
Similarly, the spectral properties can be obtained [3] byusing the general Laguerre derivatives
π·1πΏ + ππ· = π· (π₯π· + π) (8)
(here π is a real or complex constant) or more generally theoperator
π·(π₯π· + π)π =π
βπ=0
(ππ)π·ππΏπ
πβπ (9)
(π β N) and the corresponding eigenfunctions
β
βπ=0
π₯π
π! (π + π)!(10)
orβ
βπ=0
π₯π
π!((π + π)!)π. (11)
Throughout this paper, we use the Caputo fractionalderivatives as in [1].
Definition 3 (see [5]). The Caputo derivative of fractionalorder πΌ of function π(π‘) is defined as
π·πΌπ (π‘) :=
{{{{{{{{{
1
Ξ (π β πΌ)β«π‘
0
π(π) (π)
(π‘ β π)πΌ+1βπππ, π β 1 < πΌ < π
πππ
ππ‘π, πΌ = π,
(12)
whereπ := βπΌβ.
Lemma 4 (see [5]). Let π(π‘) = π‘π for some π β₯ 0. Then
π·πΌπ (π‘)
=
{{{{{{{
0, ππ π β 0, 1, . . . , π β 1,Ξ (π + 1)
Ξ (π + 1 β πΌ)π‘πβπΌ, ππ π β N, π β₯ π ππ
π β N, π > π β 1,
(13)
whereπ =: βπΌβ.
Lemma 5. Let π > 0. Let πΌ > 0, π½ > 0, and πΌ + π½ β€ π. Then
π·πΌ+π½π‘π = π·πΌπ·π½π‘π (π‘ > 0) . (14)
Proof. This follows immediately from Lemma 4.
Remark 6. In general, for πΌ, π½ > 0, π·πΌπ·π½ = π·πΌ+π½ is not true.For example,
π·0.6π·0.6π‘ =Ξ (2)
Ξ (0.8)π‘β0.2, (15)
but
π·0.6+0.6π‘ = π·1.2π‘ = 0. (16)
Lemma 7. Suppose πΌ > 0, and let π(π‘) = ββπ=0πππ‘π have a
radius of convergence π > 0 and 0 < π < π . If [πΌπ] = βπΌ(π +1)β β 1 for some π β N, then
π·(π+1)πΌπ (π‘) = π·πΌπ·ππΌπ (π‘) . (17)
Proof. Let βπΌ(π + 1)β = π. Thenπ β 1 < πΌ(π + 1) β€ 0.By lemmas 4 and 5, we have
π·(π+1)πΌπ (π‘) = π·(π+1)πΌ(
β
βπ=0
πππ‘π)
= π·(π+1)πΌ(β
βπ=π
πππ‘π) .
(18)
Since [πΌπ] = βπΌ(π + 1)β β 1 = π β 1, together withDefinition 3, we have
π·πΌπ·πΌπ(πβ1
βπ=0
πππ‘π) = 0, (19)
and therefore (18) implies that
π·(π+1)πΌπ (π‘) = π·πΌπ·πΌπ(
β
βπ=0
πππ‘π) = π·πΌπ·πΌππ (π‘) . (20)
This completes the proof.
3. Main Results
We first study the following BVP in the plane π‘ > 0:
π·ππΏπ₯
π (π₯, π‘) = π·πΌ
π‘π (π₯, π‘) ,
π (0, π‘) = π (π‘) .(21)
Theorem 8. Let π(π‘) = π‘π with π > 0 and assume that π/πΌ is apositive integer. Then the operatorial form solution of BVP (21)is
π (π₯, π‘) =π/πΌ
βπ=0
π₯ππ·πΌππ‘π
(π!)π+1
=π/πΌ
βπ=0
Ξ (π + 1)
Ξ (π + 1 β πΌπ)β π₯π
(π!)π+1π‘πβπΌπ.
(22)
Abstract and Applied Analysis 3
Proof. Let π/πΌ = π. By Lemmas 4 and 5, we have
π·ππΏπ₯
π (π₯, π‘) =π
βπ=1
ππ+1π·πΌππ‘π‘π
(π!)π+1=π
βπ=1
π₯πβ1π·πΌππ‘π‘π
[(π β 1)!]π+1
=πβ1
βπ=0
π·πΌ(π+1)π‘
π‘π
(π!)π+1= π·πΌπ‘(πβ1
βπ=0
π₯ππ·πΌππ‘π
(π!)π+1)
= π·πΌπ‘(π
βπ=0
π₯ππ·πΌππ‘π
(π!)π+1) = π·πΌ
π‘π (π₯, π‘) .
(23)
In the fifth previous equality, we use the fact that
π·πΌπ‘(π₯ππ·πΌππ‘π
(π!)π+1) = π·πΌ
π‘(π₯ππ·ππ‘π
(π!)π+1) = 0. (24)
Remark 9. We point out that the result of Example 1 in [1] isincorrect. A counterexample is as follows. Let π(π‘) = π‘ andπΌ = 0.6. By Lemma 5, we have
π·πΏπ₯
([1/0.6]
βπ=0
π₯ππ·πΌππ‘
(π!)2π‘)
= π·π₯π₯π·π₯ (π‘ +π₯π·0.6π‘
(1!)2π‘) = π·0.6
π‘(π‘) .
(25)
On the other hand, we have
π·0.6π‘([1/0.6]
βπ=0
π₯ππ·πΌππ‘
(π!)2π‘)
= π·0.6π‘(π‘ +
π₯π·0.6π‘
(1!)2π‘) = π·0.6
π‘(π‘) + π₯π·
0.6
π‘(π·0.6π‘π‘) .
(26)
Note that by Remark 6, we have
π·0.6+0.6π‘
π‘ = 0,
π·0.6π‘π·0.6π‘π‘ =
1
Ξ (0.8)π‘β0.2 ΜΈ= 0.
(27)
Hence,
π·πΏπ₯
([1/0.6]
βπ=0
π₯ππ·πΌππ‘
(π!)2π‘) ΜΈ=π·0.6
π‘([1/0.6]
βπ=0
π₯ππ·πΌππ‘
(π!)2π‘) . (28)
Theorem 10. Let π(π‘) satisfy the assumptions of Lemma 7. If[πΌπ] = βπΌ(π + 1)β β 1, (π = 1, 2, . . .), then the operatorial formsolution of BVP (21) is given by
π (π₯, π‘) = ππ (π₯π·πΌ
π‘) π (π‘) =
β
βπ=0
π₯ππ·πΌππ‘
(π!)π+1π (π‘) . (29)
Proof. By Lemmas 2 and 7, we have
π·ππΏπ₯
π (π₯, π‘) = π·ππΏπ₯
(ππ (π₯π·πΌ
π‘) π (π‘))
= π·πΌπ‘ππ (π₯π·
πΌ
π‘) π (π‘) = π·
πΌ
π‘π (π₯, π‘) .
(30)
This completes the proof.
The following generalization of the Theorem 10 can beproved similarly.
Theorem 11. Let π be a real or complex constant and π β N.Consider the following BVP:
π·π₯(π₯π·π₯ + π)ππ (π₯, π‘) = π·
πΌ
π‘π (π₯, π‘) ,
π (0, π‘) = π (π‘) ,(31)
in the half plane π‘ > 0, with analytic boundary condition π(π‘)such that the conditions of Lemma 7 are satisfied. If [πΌπ] =βπΌ(π + 1)β β 1, (π = 1, 2, . . .), then the operatorial solution ofBVP (31) is given by
π (π₯, π‘) =β
βπ=0
π₯ππ·πΌππ‘
π!((π + π)!)ππ (π‘) . (32)
Proof. Using spectral properties of Laguerre derivative,together with Lemma 7, we have
π·π₯(π₯π·π₯ + π)ππ (π₯, π‘) = π·π₯(π₯π·π₯ + π)
π
Γ (β
βπ=0
π₯ππ·πΌππ‘
π!((π + π)!)ππ (π‘))
=β
βπ=1
π(π + π)ππ₯πβ1π·πΌππ‘
π!((π + π)!)ππ (π‘)
=β
βπ=0
π₯ππ·πΌ(π+1)
π!((π + π)!)π+1π (π‘)
= π·πΌ(β
βπ=0
π₯ππ·πΌπ
π!((π + π)!)ππ (π‘))
= π·πΌπ (π₯, π‘) .
(33)
This completes the proof.
Remark 12. Conditions [πΌπ] = βπΌ(π + 1)β β 1 (π = 1, 2, . . .)are very harsh. However, if we remove them,Theorems 10 and11 no longer remain valid. It should be noted that the mainresult in [1] (Theorem 1.2 in [1]) is incorrect in general case. Acounterexample is as follows. Let π(π‘) = ππ‘ and πΌ = 0.6. Onehas
π·πΏπ₯
(β
βπ=0
π₯ππ·πΌππ‘ππ‘
(π!)2) = π·π₯π₯π·π₯ (
π₯
(1!)2π·0.6π‘ππ‘ +
π₯2
(2!)2π·1.2π‘ππ‘
+π₯3
(3!)2π·1.8π‘ππ‘ + β β β )
= π·0.6π‘ππ‘ +
π₯
(1!)2π·1.2π‘ππ‘
+π₯2
(2!)2π·1.8π‘ππ‘ + β β β .
(34)
4 Abstract and Applied Analysis
On the other hand, we have
π·πΌπ‘(β
βπ=0
π₯ππ·πΌππ‘ππ‘
(π!)2)
= π·0.6π‘(ππ‘ +
π₯
(1!)2π·0.6π‘ππ‘ +
π₯2
(2!)2π·1.2π‘ππ‘ + β β β ) .
(35)
Set π₯ = 1 in (34) and (35). By Definition 3 and Lemma 4,we deduce that all terms in (34) are positive and cannotcontain negative exponent of variable π‘. For example,
π·0.6π‘ππ‘ = π·0.6
π‘(1 + π‘ +
π‘2
2!+ β β β ) = π·0.6
π‘(π‘ +
π‘2
2!+ β β β )
=1
Ξ (1.4)π‘0.4 +
1
Ξ (2.4)π‘1.4 + β β β ,
π·1.8π‘ππ‘ = π·1.8
π‘(1 + π‘ +
π‘2
2!+ β β β ) = π·1.8
π‘(π‘2
2!+π‘3
3!+ β β β )
=1
Ξ (1.2)π‘0.2 +
1
Ξ (2.2)π‘1.2 + β β β ,
and so forth.(36)
Similarly, all terms in (35) are also positive, except thatsome terms contain negative exponent of variable π‘. Forexample,
π·0.6π‘π·0.6π‘ππ‘ = π·0.6
π‘π·0.6π‘(1 + π‘ +
π‘2
2!+ β β β )
= π·0.6π‘π·0.6π‘(π‘ +
π‘2
2!+ β β β )
=1
Ξ (0.8)π‘β0.2 + β β β ,
π·0.6π‘π·6.6π‘ππ‘ = π·0.6
π‘π·6.6π‘(1 + π‘ +
π‘2
2!+ β β β )
= π·0.6π‘π·6.6π‘(π‘ +
π‘2
2!+ β β β )
=1
Ξ (0.8)π‘β0.2 + β β β ,
and so forth.
(37)
Thus, we conclude that
π·πΏπ₯
(β
βπ=0
π₯ππ·0.6ππ‘ππ‘
(π!)2) ΜΈ=π·0.6
π‘(β
βπ=0
π₯ππ·0.6ππ‘ππ‘
(π!)2) . (38)
4. Conclusion and Discussion
In this paper, we point out that Garraβs results are incorrectand give some necessary counterexamples. In addition, we
established three theorems (Theorems 8, 10, and 11) whichcorrect and extend the corresponding results of [1].
Different from integer-order derivative, there are manykinds of definitions for fractional derivatives, includingRiemann-Liouville, Caputo, Grunwald-Letnikov, Weyl,Jumarie, Hadamard, Davison and Essex, Riesz, Erdelyi-Kober, and Coimbra (see [1, 6β8]). These definitions aregenerally not equivalent to each other. Every derivative hasits own serviceable range. In other words, all these fractionalderivatives definitions have their own advantages anddisadvantages. For example, the Caputo derivative is veryuseful when dealing with real-world problem, since it allowstraditional initial and boundary conditions to be included inthe formulation of the problem and the Laplace transformof Caputo fractional derivative is a natural generalizationof the corresponding well-known Laplace transform ofinteger-order derivative. So, the Caputo fractional-ordersystem is often used in modelling and analysis. However, thefunctions that are not differentiable do not have fractionalderivative, which reduces the field of application of Caputoderivative (see [1, 8, 9]).
When solving fractional-order systems, the law of expo-nents (semigroup property) is the most important. Unlikeinteger-order derivative, for πΌ > 0 and π½ > 0, πΌ derivative oftheπ½ derivative of a function is, in general, not equal to theπΌ+π½ derivative of such function. About the semigroup propertyof the fractional derivatives, under suitable assumptions offractional order, there have existed some studies, but only afew studies provide valuable judgment methods (see [1, 9]).
Fortunately, if we define Ξ© as the class of all functionsπ which are infinitely differentiable everywhere and aresuch that π and all its derivatives are of order π‘βπ for allπ, π = 1, 2, . . ., then, for all functions of class Ξ©, Weylfractional derivatives possess the semigroup property [1].This has brought us great convenience for studying Weylfractional differential equations.Wewill considered this topicin a forthcoming paper.
Acknowledgments
The authors are grateful to the anonymous reviewers forseveral comments and suggestions which contributed tothe improvement of this paper. This work is supportedby the National Natural Science Foundation of China (no.11171092) and the Natural Science Foundation of JiangsuHigher Education Institutions of China (no. 08KJB110005).
References
[1] R. Garra, βAnalytic solution of a class of fractional differentialequations with variable coefficients by operatorial methods,βCommunications in Nonlinear Science and Numerical Simula-tion, vol. 17, no. 4, pp. 1549β1554, 2012.
[2] K. S. Miller and B. Ross, An Introduction to the FractionalCalculus and Fractional Differential Equations, John Wiley &Sons, New York, NY, USA, 1993.
[3] G. Bretti and P. E. Ricci, βLaguerre-type special functions andpopulation dynamics,β Applied Mathematics and Computation,vol. 187, no. 1, pp. 89β100, 2007.
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[4] G. Dattoli and P. E. Ricci, βLaguerre-type exponentials, andthe relevant πΏ-circular and πΏ-hyperbolic functions,β GeorgianMathematical Journal, vol. 10, no. 3, pp. 481β494, 2003.
[5] K. Diethelm, The Analysis of Fractional Differential Equations,Springer, Berlin, Germany, 2010.
[6] I. Podlubny, Fractional Differential Equations, vol. 198, Aca-demic Press, New York, NY, USA, 1999.
[7] P. L. Butzer and U. Westphal, An Introduction to FractionalCalculus, World Scientific, Singapore, 2000.
[8] A. Atangana and A. Secer, βA note on fractional order deriva-tives and table of fractional derivatives of some special func-tions,β Abstract and Applied Analysis, vol. 2013, Article ID279681, 8 pages, 2013.
[9] C. Li andW. Deng, βRemarks on fractional derivatives,βAppliedMathematics andComputation, vol. 187, no. 2, pp. 777β784, 2007.
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