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

Abstract

An alternative block method for solving fifth-order initial value problems (IVPs) is proposed with an adaptive strategy of implementing variable step size. The derived method is designed to compute four solutions simultaneously without reducing the problem to a system of first-order IVPs. To validate the proposed method, the consistency and zero stability are also discussed. The improved performance of the developed method is demonstrated by comparing it with the existing methods and the results showed that the 4-point block method is suitable for solving fifth-order IVPs.

1. Introduction

Many natural processes or real-world problems can be translated into the language of mathematics [1–4]. The mathematical formulation of physical phenomena in science and engineering often leads to a differential equation, which can be categorized as an ordinary differential equation (ODE) and a partial differential equation (PDE). This formulation will explain the behavior of the phenomenon in detail. The search for solutions of real-world problems requires solving ODEs and thus has been an important aspect of mathematical study. For many interesting applications, an exact solution may be unattainable, or it may not give the answer in a convenient form. The reliability of numerical approximation techniques in solving such problems has been proven by many researchers as the role of numerical methods in engineering problems solving has increased dramatically in recent years. Thus a numerical approach has been chosen as an alternative tool for approximating the solutions consistent with the advancement in technology.

Commonly, the formulation of real-world problems will take the form of a higher order differential equation associated with its initial or boundary conditions [4]. In the literature, a mathematical model in the form of a fifth-order differential equation, known as Korteweg-de Vries (KdV) equation, has been used to describe several wave phenomena depending on the values of its parameters [2, 3, 5, 6]. The KdV equation is a PDE and researchers have tackled the problem analytically and numerically. 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. The fifth-order IVP with its initial conditions is defined as

Conventionally, (1) will be converted to a system of first-order ODEs by a simple change of variables. However, it will increase the computational cost in terms of function evaluation and thus will affect the computational time. This drawback is obviously seen when dealing with a higher order problem. Furthermore, [10] also has remarked that the block method is far more cost-effective when it is implemented in direct integration. Hence, several researchers [11–16] have shown an interest in the development of direct integration methods. A direct integration method of variable order and step size for solving systems of nonstiff higher order ODEs has been discussed in [11] whereby [12] has proposed an algorithm based on collocation of the differential system at selected grid points for direct solution of general second-order ODEs. In addition, [13] has used the Gaussian method in order to solve fourth-order differential equations directly. However, it requires a tedious computation as well, since it consists of higher order partial derivatives of Taylor series algorithm which supplies the starting values. Jator and Li [15] have proposed the linear multistep method (LMM) for solving general second-order IVPs directly. The method is self-starting, so it involves less computational time by avoiding incorporating subroutines to supply the starting values.

Thus far, a number of researchers have concerned themselves with developing a numerical method based on block features, and the characteristic feature of the block method is that in each application it generates a set of solutions concurrently [10]. Rosser [10] also has remarked that the implementation of block method in numerical computation will reduce the computational cost by reducing the number of function evaluations. Shampine and Watts [17] have constructed an -stable implicit one-step block method and Cash [18] has studied block methods based upon the Runge-Kutta method for the numerical solution of nonstiff IVPs. Furthermore [19] has used the self-starting LMM to solve second-order ODEs in a block-by-block fashion and recently [20] has constructed a predictor-corrector scheme 3-point block method with the implementation of variable step size. This research is an extension of the work in [20] in which the solution is computed at three points concurrently and it shows the satisfactory numerical results obtained when solving general higher order ODEs.

An increasing amount of literature is devoted to variable step size implementations of numerical methods [11, 21, 22]. The practicality of varying the step size for block method has been justified by [10]. This strategy is an attempt to reduce the computational cost as well as maintaining the accuracy. The Falkner method with variable step size implementation for the numerical solution of second-order IVPs has been employed in [21]. Although the implementation of the method involves varying the step size and solving directly, the computation is still tedious since the coefficients of the formulae must be calculated every time the step size is changed. On the contrary, the present work will store all the integration coefficients in the code in order to avoid the tedious calculations of the divided differences.

2. Methodology

2.1. Derivation of 4-Point Block Method

The basic approach of numerical methods for integration is performed by subdividing the interval of integration into certain subintervals. The proposed method was based on concurrent computation; hence the closed finite interval was subdivided into a series of blocks and each block contains four equal subintervals as illustrated in Figure 1.

Figure 1 

4-point block method.

Initially, (1) was integrated five times over the corresponding interval: , , , for first, second, third, and fourth point, respectively. The integration was started by replacing the function with the interpolating function which was generated from Lagrange polynomials. A set of points was interpolated for deriving predictor and corrector formulae, respectively. Let the Lagrange polynomial, , be written aswhereNine points were interpolated in (2) with set to be eight for deriving the corrector and thus one point less for the predictor formula. Then, the integration process was proceeded by substituting and in (2). Consistent with the number of interpolation points involved in deriving the formulae, predictor and corrector formulae were obtained in terms of variables and . The variables and refer to the distance ratio between current and previous point as a result of implementation variable step size strategy in the proposed method.

In this work, the selection of the next step size could be increased by a factor of or maintained by and for halving the current step size. This step size changing technique was limited in order to minimize the number of coefficients to be stored as well as reducing the computational storage. The increment of step size was also limited to doubling in order to control the accuracy of the computation [10]. The compact form of the 4-point block method is presented in where are the coefficients of the formulae to be calculated, is the number of points , is the number of times (1) will be integrated, and is the number of terms when the equation is integrated. The values of , and , were considered for deriving the predictor and corrector formulae, respectively. After further simplification, the associated corrector formulae of the 4-point block method when are represented below.   Integrate once:  Integrate twice:  Integrate thrice:  Integrate four times:  Integrate five times: