Efficient Decomposition Shooting Method for Solving Third-Order Boundary Value Problems

Abstract

The present paper proposes a mathematical method to numerically treat a class of third-order linear Boundary Value Problems (BVPs). This method is based on the combination of the Adomian Decomposition Method (ADM) and, the modified shooting method. A complete derivation of the proposed method has been provided, in addition to its numerical implementation and, validation via the utilization of the Runge-Kutta method and, other existing methods. The method has been applied to diverse test problems and turned out to perform remarkably. Lastly, the simulated numerical results have been graphically illustrated and, also supported by some absolute error comparison tables.

Computational Approach to Third-Order Nonlinear Boundary Value Problems via Efficient Decomposition Shooting Method

“…The shooting method [4] coupled with the ADM is an iterative method that has been widely utilized to solve various classes of BVPs [7,8,[20][21][22][23]. However, the iterative shooting method procedures for solving both cases of linear and nonlinear BVPs remain the same [9], except for the fact that the solution of the nonlinear model cannot be represented as a linear combination of the solutions of two IVPs. As such, the solution of the governing BVP (1)-( 2) is approximated by those of the IVPs, with t as a parameter.…”

“…Through their results, they concluded that the shooting method seems to be sufficiently convergent for the system, and the shooting method is preferable to obtain numerical solutions where other methods seem to be laborious in mathematical treatment. Subsequently, Attili and Syam [8] proposed a combination of the Adomian decomposition method (ADM) and the shooting method to numerically examine second-order linear and nonlinear BVPs using a special integration operator; the same method was equally extended to third-order linear BVPs by Alzahrani et al [9]. Additionally, we also cite [10][11][12][13][14][15][16][17][18][19] and the references therein for some relevant considerations through the ADM, and also refer the reader(s) to [20][21][22][23] for certain deliberations on the shooting approach.…”

“…In fact, we couple the efficient shooting method with the ADM, in what we called the "efficient decomposition shooting method" (EDSM); this study serves as an extension to the recent findings in [9] by deploying a superior version of the ADM. Stepwise, the approach starts by transforming the principal BVP into a system of IVPs, and thereafter, solves the resulting IVPs recurrently via the application of the coupled technique, which is easier to execute and more numerically friendly than both the shooting and ADM procedures individually. What is more, the application of this method is demonstrated on several test examples, alongside deploying the competing numerical Runge-Kutta method, among others, for validation of the obtained results.…”