Numerical Treatment of Initial-Boundary Value Problems with Mixed Boundary Conditions

Abstract

In this paper, we extend the reliable modification of the Adomian Decomposition 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.

Semi-Implicit Scheme to Solve Allen-Cahn Equation with Different Boundary Conditions

“…Many methods can be used to solve nonlinear PDEs. In [3], the author use Adomian Decomposition 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. Allen-Cahn equation has been applied to many problems, such as image analysis [4] [5] the motion by mean curvature flows [6].…”