Editorial Analytical and Numerical Methods for...

EditorialAnalytical and Numerical Methods for SolvingPartial Differential Equations and Integral EquationsArising in Physical Models

Santanu Saha Ray1 Om P Agrawal2 R K Bera3 Shantanu Das4 and T Raja Sekhar5

1 Department of Mathematics National Institute of Technology Rourkela 769008 India2Department of Mechanical Engineering and Energy Processes Southern Illinois University Carbondale IL 62901 USA3Department of Science National Institute of Technical Teachersrsquo Training and Research Kolkata 700106 India4 Bhabha Atomic Research Centre Trombay Mumbai 400085 India5 Department of Mathematics Indian Institute of Technology Kharagpur 721302 India

Correspondence should be addressed to Santanu Saha Ray santanusaharayyahoocom

Received 15 December 2013 Accepted 15 December 2013 Published 9 January 2014

Copyright copy 2014 Santanu Saha Ray et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Mathematical modelling of real-life problems usually resultsin functional equations like ordinary or partial differentialequations integral and integrodifferential equations andstochastic equations Many mathematical formulations ofphysical phenomena contain integrodifferential equationsthese equations arise in many fields like fluid dynamicsbiological models and chemical kinetics Partial differentialequations (PDEs) have become a useful tool for describingthe natural phenomena of science and engineeringmodels Inaddition most physical phenomena of fluid dynamics quan-tum mechanics electricity ecological systems and manyother models are controlled within their domain of validityby PDEs Therefore it becomes increasingly important to befamiliar with all traditional and recently developed methodsfor solving PDEs and the implementations of these methodsLeaving aside quantummechanics which remains to date aninherently linear theory most real-world physical systemsincluding gas dynamics fluid mechanics elasticity relativityecology neurology and thermodynamics are modelled bynonlinear partial differential equations

The aim of this special issue is to bring together the lead-ing researchers of dynamics quantum mechanics ecologyand neurology area including applied mathematicians andallow them to share their original research work Analyticaland numerical methods with advanced mathematical andreal physical modelling recent developments of PDEs and

integral equations in physical systems are included in themain focus of the issue

Accordingly various papers on partial differential equa-tions and integral equations have been included in this specialissue after completing a heedful rigorous and peer-reviewprocess In particular the nonlinear hydroelastic waves prop-agating beneath an infinite ice sheet floating on an inviscidfluid of finite depth are investigated analytically in one of thepapers In this paper the approximate series solutions for thevelocity potential and the wave surface elevation are derivedrespectively by an analytic approximation technique namedhomotopy analysis method (HAM) and are presented for thesecond-order components

In another paper a domain decomposition method isproposed for the coupled stationary Navier-Stokes andDarcyequations with the Beavers-Joseph-Saffman interface condi-tion in order to improve the efficiency of the finite elementmethod The physical interface conditions are directly uti-lized to construct the boundary conditions on the interfaceand then decouple the Navier-Stokes and Darcy equationsNewton iteration is used to deal with the nonlinear systems

Another paper proposes a pressure-stabilized Lagrange-Galerkin method in a parallel domain decomposition systemin which the new stabilization strategy is proved to beeffective for large Reynolds number and Rayleigh numbersimulations The symmetry of the stiffness matrix enables

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 635235 3 pageshttpdxdoiorg1011552014635235

2 Abstract and Applied Analysis

the interface problems of the linear system to be solved bythe preconditioned conjugate method and an incompletebalanced domain preconditioner is applied to the flow-thermal coupled problems

One of the papers is of use of Sumudu transform onfractional derivatives for solving some interesting nonho-mogeneous fractional ordinary differential equations Thenspectral and spectral element methods have been discussedwith Legendre-Gauss-Lobatto nodal basis for general 2nd-order elliptic eigenvalue problems A priori and a posteriorierror estimates for spectral and spectral element methodshave been proposed In the another paper a generalizeddouble sinh-Gordon equation has many more applicationsin various fields such as fluid dynamics integrable quantumfield theory and kink dynamics has been solved by Exp-function method to obtain new exact solutions for thisgeneralized double sinh-Gordon equation A semianalyticalmethod called the optimal homotopy asymptotic method hasbeen also applied for solving the linear Fredholm integralequations of the first kind in another paper In one ofthe papers two strategies for inverting the open boundaryconditionswith adjointmethod are compared by carrying outsemi-idealized numerical experiments In the first strategythe open boundary curves are assumed to be partly spacevarying and are generated by linearly interpolating the valuesat feature points and in the second strategy the openboundary conditions are assumed to be fully space varyingand the values at every open boundary points are takenas control variables Another paper contains the use of arelatively new analytical method like homotopy decompo-sition method to solve the 2D and 3D Poisson equationsand biharmonic equations The method does not requirethe linearization or assumptions of weak nonlinearity thesolutions are generated in the form of general solution andit is more realistic compared to the method of simplifying thephysical problems

One of the papers has shown that a strong solution of theDegasperis-Procesi equation possesses persistence propertyin the sense that the solution with algebraically decayinginitial data and its spatial derivativemust retain this propertyIn another paper the fractional complex transformationhas been used to transform nonlinear partial differentialequations to nonlinear ordinary differential equations Theimproved (1198661015840119866)-expansion method has suggested solvingthe space and time fractional foam drainage and KdV equa-tions Integral equation has been one of the essential toolsfor various areas of applied mathematics For solving non-linear Fredholm integrodifferential equations the methodbased on hybrid functions approximate has been proposedin one of the papers The properties of hybrid of blockpulse functions and orthonormal Bernstein polynomials havebeen presented and utilized to reduce the problem to thesolution of nonlinear algebraic equations Another papercontains many numerical methods namely B-Spline waveletmethod Wavelet Galerkin method and quadrature methodfor solving Fredholm integral equations of second kind Apeer-review of different numerical methods for solving bothlinear and nonlinear Fredholm integral equations of secondkind has been presentedThis paper has more emphasized on

the importance of interdisciplinary effort for advancing thestudy on numerical methods for solving integral equationsAlso one of the papers has used a numericalmethod like func-tion approximation to determine the numerical solution ofsystem of linear Volterra integrodifferential equations usingBezier curves Two-dimensional Volterra integral equationshave also been solved usingmore recent semianalyticmethodlike the reduced differential transform method and alsocompared with the differential transform method One ofthe papers has presented a numerical method to achieve theapproximate solutions in a generalized expansion form oftwo-dimensional fractional-order Legendre functions (2D-FLFs) The operational matrices of integration and derivativefor 2D-FLFs have been derived

Then a mixed finite element method has been introducedfor an elliptic equation modelling of Darcy flow in porousmedia In present mixed finite element the approximatevelocity is continuous and the conservation law holds locallyIn order to assess the rotational potential vorticity-conservedequation with topography effect and dissipation effect themultiple-scale method has been studied to describe theRossby solitary waves in deep rotational fluids A one stepoptimal homotopy analysis method has been applied numer-ically to harmonic wave propagation in a nonlinear ther-moelasticity under influence of rotation thermal relaxationtimes and magnetic field The problem has been solved inone-dimensional elastic half-space model subjected initiallyto a prescribed harmonic displacement and the temperatureof the medium In one of the papers the analytical andmultishaped solitary wave solutions have been presentedfor extended reduced Ostrovsky equation The exact soli-tary (traveling) wave solutions are also expressed by threetypes of functions which are hyperbolic function solutiontrigonometric function solution and rational solution Inorder to classify the exact solutions including solitons andelliptic solutions of the generalized 119870(119898 119899) equation bythe complete discrimination system a polynomial methodhas been obtained To examine the possible approximatesolutions of both integer and noninteger systems of nonlineardifferential equations which describe tuberculosis diseasepopulation dynamics the relatively new analytical techniquelike homotopy decompositionmethod has been proposed Inone of the papers a relatively new operator called the tripleLaplace transform has been introduced and to make use ofthe operator some kind of third-order differential equationcalled Mboctara equations has been solved

Another paper investigates the effect of boundary slipon the transient pulsatile fluid flow through a vessel withbody acceleration To describe the non-Newtonian behaviorthe modified second-grade fluid model has been analyzedin which the viscosity and the normal stresses have beenrepresented in terms of the shear rate One of the papersproves the existence of global solutions for nonlinear waveequations with damping and source terms by constructinga stable set and also obtaining the asymptotic stability ofglobal solutions through the use of a difference inequality Inorder to assess the spatial dynamical behavior of a predator-prey system with Allee effect the bifurcation analyses havebeen used in which the exact Turing domain has been found

Abstract and Applied Analysis 3

in the parameters space According to the operator theorythe temperature dependence of the solution to the BCS gapequation has been connected with superconductivity Whenthe potential is a positive constant the BCS gap equationreduces to the simple gap equation The solution to the BCSgap equation has been indeed continuouswith respect to boththe temperature and the energy under a certain conditionwhen the potential is not a constant This study representsthat there is a unique nonnegative solution to the simple gapequation which is continuous and strictly decreasing and isof class 1198622 with respect to the temperature

At present the use of partial differential equation andintegral equation in real physical systems is commonlyencountered in the fields of science and engineering Analysisand numerical approximate of such physical models arerequired for efficient computational tools The present issuehas addressed recent trends and developments regardingthe analytical and numerical methods that may be used inthe dynamical system Eventually it may be expected thatthe present special issue would certainly helpful to explorethe researchers with their new arising problems and elevatethe efficiency and accuracy of the solution methods in usenowadays

Santanu Saha RayOm P Agrawal

R K BeraShantanu DasT Raja Sekhar

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the interface problems of the linear system to be solved bythe preconditioned conjugate method and an incompletebalanced domain preconditioner is applied to the flow-thermal coupled problems

One of the papers is of use of Sumudu transform onfractional derivatives for solving some interesting nonho-mogeneous fractional ordinary differential equations Thenspectral and spectral element methods have been discussedwith Legendre-Gauss-Lobatto nodal basis for general 2nd-order elliptic eigenvalue problems A priori and a posteriorierror estimates for spectral and spectral element methodshave been proposed In the another paper a generalizeddouble sinh-Gordon equation has many more applicationsin various fields such as fluid dynamics integrable quantumfield theory and kink dynamics has been solved by Exp-function method to obtain new exact solutions for thisgeneralized double sinh-Gordon equation A semianalyticalmethod called the optimal homotopy asymptotic method hasbeen also applied for solving the linear Fredholm integralequations of the first kind in another paper In one ofthe papers two strategies for inverting the open boundaryconditionswith adjointmethod are compared by carrying outsemi-idealized numerical experiments In the first strategythe open boundary curves are assumed to be partly spacevarying and are generated by linearly interpolating the valuesat feature points and in the second strategy the openboundary conditions are assumed to be fully space varyingand the values at every open boundary points are takenas control variables Another paper contains the use of arelatively new analytical method like homotopy decompo-sition method to solve the 2D and 3D Poisson equationsand biharmonic equations The method does not requirethe linearization or assumptions of weak nonlinearity thesolutions are generated in the form of general solution andit is more realistic compared to the method of simplifying thephysical problems

One of the papers has shown that a strong solution of theDegasperis-Procesi equation possesses persistence propertyin the sense that the solution with algebraically decayinginitial data and its spatial derivativemust retain this propertyIn another paper the fractional complex transformationhas been used to transform nonlinear partial differentialequations to nonlinear ordinary differential equations Theimproved (1198661015840119866)-expansion method has suggested solvingthe space and time fractional foam drainage and KdV equa-tions Integral equation has been one of the essential toolsfor various areas of applied mathematics For solving non-linear Fredholm integrodifferential equations the methodbased on hybrid functions approximate has been proposedin one of the papers The properties of hybrid of blockpulse functions and orthonormal Bernstein polynomials havebeen presented and utilized to reduce the problem to thesolution of nonlinear algebraic equations Another papercontains many numerical methods namely B-Spline waveletmethod Wavelet Galerkin method and quadrature methodfor solving Fredholm integral equations of second kind Apeer-review of different numerical methods for solving bothlinear and nonlinear Fredholm integral equations of secondkind has been presentedThis paper has more emphasized on

the importance of interdisciplinary effort for advancing thestudy on numerical methods for solving integral equationsAlso one of the papers has used a numericalmethod like func-tion approximation to determine the numerical solution ofsystem of linear Volterra integrodifferential equations usingBezier curves Two-dimensional Volterra integral equationshave also been solved usingmore recent semianalyticmethodlike the reduced differential transform method and alsocompared with the differential transform method One ofthe papers has presented a numerical method to achieve theapproximate solutions in a generalized expansion form oftwo-dimensional fractional-order Legendre functions (2D-FLFs) The operational matrices of integration and derivativefor 2D-FLFs have been derived

Then a mixed finite element method has been introducedfor an elliptic equation modelling of Darcy flow in porousmedia In present mixed finite element the approximatevelocity is continuous and the conservation law holds locallyIn order to assess the rotational potential vorticity-conservedequation with topography effect and dissipation effect themultiple-scale method has been studied to describe theRossby solitary waves in deep rotational fluids A one stepoptimal homotopy analysis method has been applied numer-ically to harmonic wave propagation in a nonlinear ther-moelasticity under influence of rotation thermal relaxationtimes and magnetic field The problem has been solved inone-dimensional elastic half-space model subjected initiallyto a prescribed harmonic displacement and the temperatureof the medium In one of the papers the analytical andmultishaped solitary wave solutions have been presentedfor extended reduced Ostrovsky equation The exact soli-tary (traveling) wave solutions are also expressed by threetypes of functions which are hyperbolic function solutiontrigonometric function solution and rational solution Inorder to classify the exact solutions including solitons andelliptic solutions of the generalized 119870(119898 119899) equation bythe complete discrimination system a polynomial methodhas been obtained To examine the possible approximatesolutions of both integer and noninteger systems of nonlineardifferential equations which describe tuberculosis diseasepopulation dynamics the relatively new analytical techniquelike homotopy decompositionmethod has been proposed Inone of the papers a relatively new operator called the tripleLaplace transform has been introduced and to make use ofthe operator some kind of third-order differential equationcalled Mboctara equations has been solved

Another paper investigates the effect of boundary slipon the transient pulsatile fluid flow through a vessel withbody acceleration To describe the non-Newtonian behaviorthe modified second-grade fluid model has been analyzedin which the viscosity and the normal stresses have beenrepresented in terms of the shear rate One of the papersproves the existence of global solutions for nonlinear waveequations with damping and source terms by constructinga stable set and also obtaining the asymptotic stability ofglobal solutions through the use of a difference inequality Inorder to assess the spatial dynamical behavior of a predator-prey system with Allee effect the bifurcation analyses havebeen used in which the exact Turing domain has been found













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