Research ArticleOpen AccessGoogle Scholar indexed
Numerical Treatment of Initial-Boundary Value Problems with Mixed Boundary Conditions
Department of Mathematics, King Abdulaziz University, Jeddah, KSA
Department of Mathematics, King Abdulaziz University, Jeddah, KSA
- 1 Department of Mathematics, King Abdulaziz University, Jeddah, KSA
- 2 Department of Mathematics, King Abdulaziz University, Jeddah, KSA
American Journal of Computational Mathematics·Volume 08 (2018)·Pages 153–174·Published 13 June 2018·DOI10.4236/ajcm.2018.82012
Copy link · social · email
Abstract
In this paper, we extend the reliable modification of the Adomian Decom-position Method coupled to the Lesnic’s approach to solve boundary value problems and initial boundary value problems with mixed boundary conditions for linear and nonlinear partial differential equations. The method is applied to different forms of heat and wave equations as illustrative examples to exhibit the effectiveness of the method. The method provides the solution in a rapidly convergent series with components that can be computed iteratively. The numerical results for the illustrative examples obtained show remarkable agreement with the exact solutions. We also provide some graphical representations for clear-cut comparisons between the solutions using Maple software.
KeywordsDecomposition MethodModified Adomian Decomposition MethodLinear and Nonlinear Partial Differential EquationsMixed Boundary ConditionsInitial-Boundary Value Problem
- Carslaw, H.S. and Jaeger, J.C. (1948) Conduction of Heat in Solids. Clarendon Press, Oxford.
- Carrier, G.F., Krook, M. and Pearson, C.E. (1966) Functions of a Complex Variable. McGraw-Hill, New York.
- Courant, R. and Hilbert, D. (1953) Methods of Mathematical Physics. Vol. 1, Interscience Publishers Inc., New York.
- Sneddon, I.N. (1966) Mixed Boundary Value Problems in Potential Theory. Wiley, New York.
- Tranter, C.J. (1951) Integral Transforms in Mathematical Physics. Wiley, New York.
- Fabrikant, V.I. (1991) Mixed Boundary Value Problems of Potential Theory and their Applications in Engineering. Kluwer, Boston.
- Sherwood, J.D. and Stone, H.A. (2001) Leakage through Filtercake into a Fluid Sampling Probe. Physics of Fluids, 13, 1151-1159. https://doi.org/10.1063/1.1360712
- Warrick, A.W., Broadbridge, P. and Lomen, D.O. (1992) Approximations for Diffusion from a Disc Source. Applied Mathematical Modelling, 16, 155-161. https://doi.org/10.1016/0307-904X(92)90067-D
- He, J.H. (2008) An Elementary Introduction to the Homotopy Perturbation Method. Computers & Mathematics with Applications, 57, 410-412. https://doi.org/10.1016/j.camwa.2008.06.003
- Wazwaz, A.M. (2006) The Modified Decomposition Method for Analytic Treatment of Differential Equations. Applied Mathematics and Computation, 173, 165-176. https://doi.org/10.1016/j.amc.2005.02.048
- Duan, J.S. and Rach, R. (2011) A New Modification of the Adomian Decomposition Method for Solving Boundary Value Problems for Higher Order Nonlinear Differential Equations. Applied Mathematics and Computation, 218, 4090-4118. https://doi.org/10.1016/j.amc.2011.09.037
- Adomian, G. and Rach, R. (1983) Inversion of Nonlinear Stochastic Operators. Journal of Mathematical Analysis and Applications, 91, 39-46. https://doi.org/10.1016/0022-247X(83)90090-2
- Adomian, G. (1986) Nonlinear Stochastic Operator Equations. Academic, Orlando.
- Adomian, G. (1989) Nonlinear Stochastic Systems Theory and Applications to Physics. Kluwer Academic, Dordrecht. https://doi.org/10.1007/978-94-009-2569-4
- Adomian, G., Rach, R. and Meyers, R. (1991) An Efficient Methodology for the Physical Sciences. Kybernetes, 20, 24-34. https://doi.org/10.1108/eb005909
- Adomian, G. and Rach, R. (1993) Analytic Solution of Nonlinear Boundary Value Problems in Several Dimensions by Decomposition. Journal of Mathematical Analysis and Applications, 174, 118-137. https://doi.org/10.1006/jmaa.1993.1105