Development and Implementation of a Tenth-Order Hybrid Block Method for Solving Fifth-Order Boundary Value Problems

Abstract

A hybrid convergent method of tenth-order is presented in this work for directly solving fifth-order boundary value problems in ordinary differential equations. A unique direct block approach is obtained by combining multiple Finite Difference Formulas which are derived via the collocation technique. The proposed method is fully analyzed and the existence and uniqueness of the discrete solution is established. Different numerical examples are considered and the results are compared with those provided by existing works in the literature. The comparison shows the good performance of the present method over some cited works in the literature, confirming the competitiveness and superiority of the new numerical integrator.

Advancing Numerical Methods of Block Multi-Derivative Approaches for ODEs of Various Orders

“…The area in the complex z-plane in which the numerical method exhibits the behaviours of the real solution is known as the region of absolute stability (see Ramos and Momoh [9]). The method's behaviour is studied here by considering its application to test equations.…”

“…Awoyemi [8], Ramos and Momoh [9] successfully applied numerical algorithms as integrators to solve higher order-order initial value problems directly without reducing it to first order (ODEs).…”

“…Two major classes of numerical methods are usually adopted for the numerical solution of equation (1). The classes are Runge-Kutta and Linear multistep method see Ramos and Momoh [9]. More work has been done on applying RK methods for solving first-order ODEs, and their direct application to higher-order ODEs still needs to be improved.…”

“…The following are some of recent work on block method for directly solving initial value problems of ODEs: Olabode and Momoh [6], [7], Adeyefa [14], Ramos and Momoh [9], Duromola et el., [15], Olabode et el., [16], Kashkari [17], Ogunfeyitimi and Ikhile [18]. This work aims to make available a single method that can handle the first-order initial value problems of ODEs and the direct solution of second and third-order initial value problems of ODEs.…”

Derivation of Embedded Diagonally Implicit Methods for Directly Solving Fourth-order ODEs

“…However, according to several researchers (see [1,13,14]) this technique has a drawback as it consumes a lot of computing time and human effort. Consequently, several researchers have turned their attention to direct integration methods for solving higher-order ODEs, as these methods have demonstrated features of accuracy and speed [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30]. Nevertheless, all of the methods mentioned above are above are multistep.…”