A Multiderivative Collocation Method for 5th Order Ordinary Differential Equations

Abstract

<b>Problem statement:</b> The conventional methods of solving higher order differential equations have been by reducing them to systems of first order equations. This approach is cumbersome and increases computational time. <b>Approach:</b> To address this problem, a numerical algorithm for direct solution of 5th order initial value problems in ordinary differential equations (odes), using power series as basis function, is proposed in this research. Collocation of the differential system is taken at selected grid points to reduce the number of functions to be evaluated per iteration. A number of predictors and their derivatives having the same order of accuracy with the main method are proposed. <b>Results:</b> The approach yields a multiderivative method of order six. Numerical examples solved show increased efficiency of the method with increased number of iterations, converging to the theoretical solutions. <b>Conclusion/Recommendations:</b> The new mutiderivative method is efficient to solve linear and nonlinear fifth order odes without reduction to system of lower order equations

A 4-Point Block Method for Solving Higher Order Ordinary Differential Equations Directly

“…To illustrate the technique proposed in the preceding sections, two test problems are solved and the results obtained compared with the method proposed in [8,9,25] and the ODE solver in MATLAB ode45. The work done by [25] involved the solving of the first-order ODEs using 4-point block method using variable step size.…”

“…It is also noted that in certain cases by using different approaches the KdV might be transformed into a higher order ODE [7]. To date, there are a number of studies that have proposed solving fifth-order ODE directly [8,9]. Hence, the purpose of the present paper is to solve directly the fifth-order IVPs with the implementation of a variable step size strategy.…”

Numerical Solution of Higher Order Ordinary Differential Equations by Direct Block Code

“…Thus, several researchers have concerned themselves with study to solve Eq. 1 directly such as Awoyemi (2003); Majid (2004); Awoyemi (2005); Majid and Suleiman (2006); Jator (2010); Jain et al (1977) Kayode and Awoyemi (2010). Awoyemi (2005) has proposed a multiderivative collocation method for direct solution of fourth order IVPs of ODEs while Majid and Suleiman (2006) have introduced a direct integration implicit variable step method for solving higher order systems of ODEs.…”

An Embedded Explicit Hybrid Method for Ordinary Differential Equations

“…Problems of the form (1) can be reduced to firstorder systems of twice the dimension and solved by using Runge Kutta methods (see for example Babatola et al, 2008) or second derivative multistep methods (Parand and Hojjati, 2008). However, this approach is cumbersome and increases computational time (Kayode and Awoyemi, 2010). Thus, it is more efficient to solve the problems directly using Runge Kutta-Nystrom (RKN) methods, multistep methods or block methods (Ken et al, 2008).…”