Iterative Laplace Transform Method for Solving Fractional Heat and Wave-Like Equations
Author Affiliations
- 1Department of Mathematics, University of Rajasthan, Jaipur-302004, Rajasthan, INDIA
- 2Department of Mathematics, University of Rajasthan, Jaipur-302004, Rajasthan, INDIA
Res. J. Mathematical & Statistical Sci., Volume 3, Issue (2), Pages 4-9, February,12 (2015)
Abstract
In this paper, we derive the closed form solutions of the fractional heat and wave like equations in terms of Mittag-Leffler functions by the use of iterative Laplace transform method. In the process the time-fractional derivatives are considered in Caputo sense for the said problem.
References
- Baleanu D., Diethelm K., Scalas E. and Trujillo J.J., Fractional Calculus, vol. 3 of Series on Complexity,Nonlinearity and Chaos, World Scientific, Singapore,(2012)
- Ortigueira M.D., Fractional calculus for scientists andengineers, Springer , (2011)
- Sabatier J., Agrawal O.P. and Tenreiro Machado J.A., Advances in Fractional Calculus: TheoreticalDevelopments and Applications in Physics andEngineering, Springer,(2007)
- Zhang S. and Zhang H.Q., Fractional sub-equation methodand its applications to nonlinear fractional PDEs, PhysicsLetters A, 375(7), 1069–1073 (2011)
- Lepik U., Solving fractional integral equations by the Haarwavelet method, Applied Mathematics andComputation,214(2), 468–478 (2009)
- Li Y., Solving a nonlinear fractional differential equationusing Chebyshev wavelets, Communications in NonlinearScience and Numerical Simulation, 15(9), 2284–2292,(2010)
- Rehman M. and Ali Khan R., The Legendre waveletmethod for solving fractional differential equations,Communications in Nonlinear Science and NumericalSimulation, 16 (11), 4163–4173 (2011)
- Wu J.L., A wavelet operational method for solvingfractional partial differential equations numerically, Applied Mathematics and Computation, 214(1), 31–40(2009)
- Jafari H., Khalique C.M., and Nazari M., Application ofthe Laplace decomposition method for solving linear andnonlinear fractional diffusion-wave equations, AppliedMathematics Letters, 24(11), 1799–1805 (2011)
- Ongun M.Y., The Laplace adomian decomposition methodfor solving a model for HIV infection of CD4+ cells, Mathematical and Computer Modelling, 53( 5-6), 597–603(2011)
- Liu Y. and Sun N, Numerical solution of fractionaldifferential equations using the generalized block pulseoperational matrix, Computers and Mathematics withApplications, 62(3), 1046–1054 (2011)
- Saadatmandi A. and Dehghan M., A new operationalmatrix for solving fractional-order differential equations, Computers and Mathematics with Applications, 59(3),1326–1336 (2010)
- Das S., Analytical solution of a fractional diffusionequation by variational iteration method, Computers andMathematics with Applications, 57(3), 483–487 (2009)
- Sweilam N.H., Khader M.M. and Al-Bar R.F., Numericalstudies for a multi-order fractional differential equation, Physics Letters A, 371(1-2), 26–33 (2007)
- Liu Y., Approximate solutions of fractional nonlinearequations using homotopy perturbation transformationmethod, Abstract and Applied Analysis,2012, Article ID752869, 14 ( 2012)
- Liu Y., Variational homotopy perturbation method forsolving fractional initial boundary value problems, Abstract and Applied Analysis,2012, Article ID 727031,10 (2012)
- Erturk V.S. and Momani S., Solving systems of fractionaldifferential equations using differential transform method, Journal of Computational and Applied Mathematics,215(1), 142–151 (2008)
- Ibrahim R.W., Fractional complex transforms for fractionaldifferential equations, Advances in Difference Equations, 192 (2012)
- Daftardar-Gejji V. and Jafari H., An iterative method forsolving nonlinear functional equations, Journal ofMathematical Analysis and Applications, 316(2), 753–763(2006)
- Jafari H., Iterative Methods for solving system of fractionaldifferential equations [Ph.D. thesis], Pune University,(2006)
- Bhalekar S. and Daftardar-Gejji V., Solving evolutionequations using a new iterative method, NumericalMethods for Partial Differential Equations, 26(4), 906–916(2010)
- Daftardar-Gejji V. and Bhalekar S., Solving fractionalboundary value problems with Dirichlet boundaryconditions using a new iterative method, ComputersandMathematics with Applications, 59(5), 1801–1809(2010)
- Jafari H., Nazari M., Baleanu D. and Khalique C.M., Anew approach for solving a system of fractional partialdifferential equations, Computers and Mathematics withApplications, 66(5), 838–843( 2013)
- Yan L., Numerical Solutions of Fractional Fokker-PlanckEquations Using Iterative Laplace Transform Method, Abstract and Applied Analysis,2013, Article ID 465160, 7pages, (2013)
- Caputo M, Elasticita e Dissipazione, Zani-Chelli, Bologna,( 1969)
- Podlubny I., Fractional Differential Equations, vol. 198,Academic Press, New York, NY, USA, (1999)
- Kilbas A.A., Srivastava H.M. and Trujillo J.J., Theory andApplications of Fractional Differential Equations, Elsevier,Amsterdam,(2006)
- Miller K.S. and Ross B., An Introduction to the FractionalCalculus and Fractional Differential Equations, John Wileyand Sons, New York ,USA,(1993)
- Wiman A., Uber de fundamental satz in der theorie derfunktionen E x α ( ), Acta Math., 29, 191-201, (1905)