Solving Directly Two Point Non Linear Boundary Value Problems Using Direct Adams Moulton Method

Abstract

<b>Problem statement:</b> In this study, a direct method of Adams Moulton type was developed&#13; for solving non linear two point Boundary Value Problems (BVPs) directly. Most of the existence&#13; researches involving BVPs will reduced the problem to a system of first order Ordinary Differential&#13; Equations (ODEs). This approach is very well established but it obviously will enlarge the systems of&#13; first order equations. However, the direct method in this research will solved the second order BVPs&#13; directly without reducing it to first order ODEs. <b>Approach:</b> Lagrange interpolation polynomial was&#13; applied in the derivation of the proposed method. The method was implemented using constant step&#13; size via shooting technique in order to determine the approximated solutions. The shooting technique&#13; will employ the Newton’s method for checking the convergent of the guessing values for the next&#13; iteration. <b>Results:</b> Numerical results confirmed that the direct method gave better accuracy and&#13; converged faster compared to the existing method. <b>Conclusion:</b> The proposed direct method is suitable&#13; for solving two point non linear boundary value problems

Direct Integration of Boundary Value Problems Using the Block Method via the Shooting Technique Combined with Steffensen’s Strategy

“…with the first derivative absent in the respective formula. In previous works [12,13], scholars employed Newton's method together with the nonlinear shooting technique in their implementation and demonstrated that their proposed iterative schemes required the knowledge of the partial derivative. Thus, our preference iterative scheme is Steffensen's method because we want to avoid the evaluation of the partial derivative throughout the procedure.…”

“…In the spirit of Lambert [14] and Fatunla [15], the following definition of order is referenced. Now, the proposed corrector formulas in Equation (13) and Equation 14may be written to satisfy Equation (17). Then, by applying the associated set of coefficients from Equation (19), we readily obtain that C 0 = C 1 = .…”

“…Concerning the concept of zero-stability as discussed by Ramos et al [16], the stability of the block method in Equation (13) and Equation 14is represented by their difference system as h → 0. Therefore, to analyze the zero-stability, we transform the corrector formulas in the appropriate matrix notation as:…”

New Generalized Algorithm for Developing k-Step Higher Derivative Block Methods for Solving Higher Order Ordinary Differential Equations

“…The main focus of numerical methods for solving differential equations in recent times has been placed on presenting numerical methods with an improved level of accuracy, ranging from first order ordinary differential equations [1][2][3] to higher order ordinary differential equations [4][5][6]. Numerical methods adopted to solve such ODEs have been developed using two popular approaches, i.e.…”

Solving Second-Order Delay Differential Equations by Direct Adams-Moulton Method

“…where ( ) is the initial function and is the delay term. The direct Adams-Moulton methods were studied by several researchers and the methods have shown their ability to solve first-, second-and higher-order ODEs [7,8] and boundary value problems [9] effectively and accurately. Hence, in this paper, we aim to propose the direct method of order five in the form of Adams-Moulton method for solving (1) using constant step size.…”