Solution of Two-Point Linear and Nonlinear Boundary Value Problems with Neumann Boundary Conditions Using a New Modified Adomian Decomposition Method

Abstract

In this paper, we present the New Modified Adomian Decomposition Method which is a modification of the Modified Adomian Decomposition Method. The new method incorporates the inverse linear operator theorem into the modified Adomian decomposition method for the calculation of u0. Six linear and nonlinear boundary value problems with Neumann conditions are solved in order to test the method. The results show that the method is effective.

Utilization of the Modified Adomian Decomposition Method on the Bagley-Torvik Equation Amidst Dirichlet Boundary Conditions

“…The current section mainly makes use of MADM [2,9,12,36,37,42,48] to acquire the exact analytical solution of the Bagley-Torvik BVP where possible; moreover, when the acquisition of such an exact analytical solution is not feasible, a closed-form series solution of the governing model will be acquired. In fact, we will be making consideration to the BVP of the nonhomogeneous Bagley-Torvik equation endowed with Dirichlet boundary conditions as follows [51]…”

“…where L = D 2 = d 2 dx 2 . Furthermore, to determine the explicit exact analytical solution of the Bagley-Torvik BVP expressed in (8) -using the version expressed in (9) through the differential operator -the MADM [36]- [42] will be utilized. In fact, two algorithms based on MADM will be proposed in this section for the governing model in what follows.…”

“…However, we, in the current paper aim to make use of the modified Adomian decomposition method (MADM) [2,9,12,36,37,42,48] by proposing two different algorithms to treat the Bagley-Torvik equation with Dirichlet boundary condition. MADM is a semianalytical approach that was improved upon the classical ADM [3]- [38] to easily reveal exact analytical solutions whenever obtainable or closed-form series solutions whenever exact solutions are not feasible.…”