A Step by Step Guide on Derivation and Analysis of a New Numerical Method for Solving Fourth-order Ordinary Differential Equations

Abstract

This manuscript presents a step by step guide on derivation and analysis of a new numerical method to solve initial value problem of fourth order ordinary differential equations. The method adopted hybrid techniques using power series as the basic function. Collocation of the fourth derivatives was done at both grid and off-grid points. The interpolation of the approximate function is also taken at the first four points. The complete derivation of the new technique is introduced and shown here, as well as the full analysis of the method. The discrete schemes and its first, second, and third derivatives were combined together and solved simultaneously to obtain the required 32 family of block integrators. The block integrators are then applied to solve problem. The method was tested on a linear system of equations of fourth order ordinary differential equation in order to check the practicability and reliability of the proposed method. The results are displaced in tables; it converges faster and uses smaller time for its computations. The basic properties of the method were examined, the method has order of accuracy p=10, the method is zero stable, consistence, convergence and absolutely stable. In future study, we will investigate the feasibility, convergence, and accuracy of the method by on some standard complex boundary value problems of fourth order ordinary differential equations. The extension of this new numerical method will be illustrated and comparison will also be made with some existing methods.

Three-step Four-point Optimized Hybrid Block Method for Direct Solution of General Third Order Differential Equations

“…Ukpebor [14]. 7 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 ( 1) 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0…”

A New Special 15-Step Block Method for Solving General Fourth Order Ordinary Differential Equations

“…In addition, in order to avoid order reduction at solving higher order ODEs, such authors as Kuboye [5], Jena, Mohanty and Mishra [6], Adoghe and Omole [7], Omar and Kuboye [8], Duromola [9], Adesanya, Momoh, Adamu and Tahir [10], Awoyemi, Kayode and Adoghe [11], Awoyemi [12] and Ukpebor, Omole and Adoghe [13] have solved equation (1) directly. Similarly, Omole and Ukpebor [14] and Kuboye, Elusakin and Quadri [15] have both formulated a 4-point and 5-point hybrids block formulae for the solution of system and linear fourth order initial value problems in ordinary differential equations using power series as a basis function via interpolation and collocation techniques with uniform order 4 and 7 respectively. Therefore, we are motivated to improve on Jena et al [6] who have formulated a 9-point block method of a uniform order 12 for the direct solution of equation ( 1) by proposing a new special 15-step block numerical method for the direct approximation of (1).…”