Abstract

A Runge-Kutta type method for directly solving special fourth-order ordinary differential equations (ODEs) which is denoted by RKFD method is constructed. The order conditions of RKFD method up to order five are derived; based on the order conditions, three-stage fourth- and fifth-order Runge-Kutta type methods are constructed. Zero-stability of the RKFD method is proven. Numerical results obtained are compared with the existing Runge-Kutta methods in the scientific literature after reducing the problems into a system of first-order ODEs and solving them. Numerical results are presented to illustrate the robustness and competency of the new methods in terms of accuracy and number of function evaluations.

1. Introduction

This paper deals with the numerical integration of the special fourth-order ordinary differential equations (ODEs) of the formwith initial conditions in which the first, second, and third derivatives do not appear explicitly. This type of problems often arises in many fields of applied science such as mechanics, astrophysics, quantum chemistry, and electronic and control engineering. The general approach for solving the higher-order ordinary differential equation (ODE) is by transforming it into an equivalent first-order system of differential equations and then applying the appropriate numerical methods to solve the resulting system (see [1–5]). However, the application of such technique takes a lot of computational time (see [6, 7]). Direct integration method is proposed to avoid such computational encumbrance and increase the efficiency of the method. Many authors have proposed several numerical methods for directly approximating the solutions for the higher-order ODEs; for example, Kayode [8] proposed zero-stable predictor-corrector methods for solving fourth-order ordinary differential equations. Majid and Suleiman [9] derived one point method to solve system of higher-order ODEs. Jain et al. [10] constructed finite difference method for solving fourth-order ODEs. Waeleh et al. [11] constructed a new block method for solving directly higher-order ODEs. Awoyemi and Idowu [12] proposed a hybrid collocations method for solving third-order ODEs. Hybrid linear multistep method with three steps to solve second-order ODEs was introduced by Jator [13], and all the methods discussed above are multistep in nature.

This paper primarily aims to construct a one-step method of orders four and five to solve special fourth-order ODEs directly; these new methods are self-starting in nature. The paper is organized as follows. In Section 2, the derivation of the order conditions of RKFD method is presented. In Section 3, the zero-stability of RKFD method is given. In Section 4, three-stage RKFD methods of order four and order five are constructed. In Section 5, numerical examples are given to show the effectiveness and competency of the new RKFD methods as compared with the well known Runge-Kutta methods from the scientific literature. Conclusions are given in Section 6.

2. Derivation of the RKFD Method

The general form of RKFD method with -stage for directly solving special fourth-order ODEs (1) can be written as follows:whereAll parameters , , , , , and of the RKFD method are used for and are supposed to be real. The RKFD method is an explicit method if for and is an implicit method if for . The RKFD method can be represented by Butcher tableau as follows: To determine the parameters of the RKFD method given in (3)–(8), the RKFD method expression is expanded using the Taylor series expansion. After performing some algebraic manipulations, this expansion is equated to the true solution that is given by the Taylor series expansion. The direct expansion of the local truncation error is used to derive the general order conditions for the RKFD method. This approach depends on the derivation of order conditions for the Runge-Kutta method proposed in Dormand [14]. The RKFD method in (3)–(6) can be written as follows:where the increment functions arewhere is given in (8).

The first few elementary differentials for the scalar equation areWe assume that the Taylor series increment function is .

The local truncation errors of , , , and can be obtained after substituting the exact solution of (1) into (11) as follows:The Taylor series increment functions of , , , and can be expressed as follows:Substituting (12) into (11), the increment functions , , , and for RKFD method become as follows:Using (11) and (14), the local truncation errors (13) can be written as follows:By offsetting (15) into (16) and expanding as a Taylor series expansion using computer algebra package MAPLE (see [15]), the local truncation errors or the order conditions for-stage fifth-order RKFD method can be written as follows.

The order conditions for are fourth order: fifth order:The order conditions for are third order: fourth order: fifth order:The order conditions for are second order: third order: fourth order: fifth order:The order conditions for are first order: second order: third order: fourth order: fifth order:

3. Zero-Stability of the RKFD Method

In this section, we discuss the convergence of the RKFD method by introducing the concept of zero-stability of the RKFD method. A good numerical method is a method in which the numerical approximation to the solution converges, and zero-stability is a significant criterion for convergence. The zero-stability concept for those numerical methods that are used for solving first- and second-order ODEs can be seen in Lambert [16], Dormand [14], and Butcher [4]. The RKFD method (3)–(8) can be expressed in the matrix form as follows:where is the identity matrix coefficients of , , , and , respectively, and is matrix coefficients of , , , and , respectively.

The characteristic polynomial of the RKFD method is denoted by which can be written as follows:Hence, .

We find that all the roots are . Generalizing the theorem proposed by Henrici [17] for solving special second-order ODEs, therefore, the RKFD method is zero-stable since all roots are less than or equal to the value of 1.

4. Construction of RKFD Methods

In this section, we proceed to construct explicit RKFD methods based on the order conditions which we have derived in Section 2.

4.1. A Three-Stage RKFD Method of Order Four

This section will focus on the derivation of a three-stage RKFD method of order four, where we use the algebraic order conditions (17), (19)-(20), (22)–(24), and (26)–(29), respectively. The resulting system of equations consists of 10 nonlinear equations with 14 unknown variables to be solved; solving the system simultaneously yields a solution with four free parameters , , , and as follows: Thus, these free parameters can be chosen by minimizing the local truncation error norms of the fifth-order conditions according to Dormand et al. [18]. However, we have another three free parameters , , and that do not appear in fourth-order conditions but they appear in the minimization of error equations for fifth-order conditions of .

The error norms and global error of fifth-order conditions are defined as follows:where , , , and are the local truncation error norms for , , , and , respectively, and is the global error.

Consequently, we find the error norms of , , and , respectively, as follows:Our goal is to choose the free parameters , , , and such that the error norms of fifth-order conditions have minimal value. By plotting the graph of versus and choosing a small value of in the interval , we find that is the optimal value which yields a minimum value for . Substituting the value of into and we get Also through plotting the graph of against and in the interval and choosing a small value of , we get that is the optimal value which gives and .