A new modification of the Adomian decomposition method for solving boundary value problems for higher order nonlinear differential equations

• Published in: Applied mathematics and computation 2011-12, Vol.218 (8), p.4090-4118
• Main Authors: Duan, Jun-Sheng, Rach, Randolph
• Format: Article
• Language: English
• Subjects: Adomian decomposition method (ADM); Adomian polynomials; Boundary value problems; Boundary value problems (BVPs); Computation; Convergence; Decomposition; Exact sciences and technology; Mathematical analysis; Mathematical models; Mathematics; Nonlinear differential equations; Nonlinearity; Numerical analysis; Numerical analysis. Scientific computation; Ordinary differential equations; Partial differential equations, boundary value problems; Partial differential equations, initial value problems and time-dependant initial-boundary value problems; Recursion; Sciences and techniques of general use
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2011.09.037

In this paper we propose a new modified recursion scheme for the resolution of multi-order and multi-point boundary value problems for nonlinear ordinary and partial differential equations by the Adomian decomposition method (ADM). Our new approach, including Duan’s convergence parameter, provides a significant computational advantage by allowing for the acceleration of convergence and expansion of the interval of convergence during calculations of the solution components for nonlinear boundary value problems, in particular for such cases when one of the boundary points lies outside the interval of convergence of the usual decomposition series. We utilize the boundary conditions to derive an integral equation before establishing the recursion scheme for the solution components. Thus we can derive a modified recursion scheme without any undetermined coefficients when computing successive solution components, whereas several prior recursion schemes have done so. This modification also avoids solving a sequence of nonlinear algebraic equations for the undetermined coefficients fraught with multiple roots, which is required to complete calculation of the solution by several prior modified recursion schemes using the ADM.