Solving nonlocal initial-boundary value problems for linear and nonlinear parabolic and hyperbolic partial differential equations by the Adomian decomposition method

Abstract

In this paper, we present a new approach to solve nonlocal initial-boundary value problems for linear and nonlinear parabolic and hyperbolic partial differential equations subject to initial and nonlocal boundary conditions of integral type. We first transform the given nonlocal initial-boundary value problems of integral type for the linear and nonlinear parabolic and hyperbolic partial differential equations into local Dirichlet initial-boundary value problems, and then use a relatively new modified Adomian decomposition method (ADM). Furthermore we investigate the Fourier–Adomian method, which also does not require any a priori assumptions on the solution, for the solution of nonlocal initial-boundary value problems combined with our new approach. Several examples are presented to demonstrate the efficiency of the ADM.