Runge-Kutta Type Methods for Directly Solving Special Fourth-Order Ordinary Differential Equations

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 fourthand 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.


Introduction
This paper deals with the numerical integration of the special fourth-order ordinary differential equations (ODEs) of the form with 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][2][3][4][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 fourthorder ordinary differential equations. Majid and Suleiman [9] derived one point method to solve system of higherorder 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 higherorder ODEs. Awoyemi and Idowu [12] proposed a hybrid collocations method for solving third-order ODEs. Hybrid linear multistep method with three steps to solve secondorder 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 2 Mathematical Problems in Engineering 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.
All parameters , , , , , and of the RKFD method are used for , = 1, 2, . . . , and are supposed to be real. The RKFD method is an explicit method if = 0 for ≤ and is an implicit method if ̸ = 0 for ≤ . The RKFD method can be represented by Butcher tableau as follows: (9) 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 are where is given in (8).
The first few elementary differentials for the scalar equation are We assume that the Taylor series increment function is Δ.

Mathematical Problems in Engineering
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 forstage 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:

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)   is matrix coefficients of , ℎ , ℎ 2 , and ℎ 3 , respectively. The characteristic polynomial of the RKFD method is denoted by ( ) which can be written as follows: Hence, ( ) = ( − 1) 4 . We find that all the roots are = 1, 1, 1, 1. Generalizing the theorem proposed by Henrici [17] for solving special second-order ODEs, therefore, the RKFD method is zerostable since all roots are less than or equal to the value of 1.

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.

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 3 , 1 , 1 , and 3 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 21 , 31 , and 32 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 (5) , (5) , (5) , and (5) are the local truncation error norms for , , , and , respectively, and (5) is the global error.
Consequently, we find the error norms of , , and , respectively, as follows: 6

Mathematical Problems in Engineering
Our goal is to choose the free parameters 3 , 1 , 1 , and 3 such that the error norms of fifth-order conditions have minimal value. By plotting the graph of ‖ (5) ‖ 2 versus 3 and choosing a small value of 3 in the interval [0.7, 3], we find that 3 = 17/20 is the optimal value which yields a minimum value for ‖ (5) ‖ 2 = 3.787878788 × 10 −4 . Substituting the value of 3 = 17/20 into ‖ (5) ‖ 2 and ‖ (5) ‖ 2 we get Also through plotting the graph of ‖ Therefore, the error equation of the fifth-order condition of is as follows: Consequently, the global error is Thus, these free parameters can be chosen by minimizing the local truncation error norms of the sixth-order conditions. The error norms and the global error of the sixth-order conditions are given by where (6) , (6) , (6) , and (6) are the local truncation error norms for , , , and of the RKFD method, respectively. (6) is the global error. The error equation of sixth-order condition for with respect to the free parameter 1 is as follows: The error equation ‖ (6) ‖ 2 has a minimum value equal to zero at 1 = 19/1080 ≈ 0.01759259259 which leads to 2 = 13/1080 − 11 √ 6/2160 and 3 = 13/1080 + 11 √ 6/2160. The truncation error norms of the sixth-order condition of , , , and are calculated as follows: Also, the global error can be written as Now, minimizing the error coefficients in (43) and (44) with respect to the free parameters 21 , 31 , we obtain 21 = 4059/187793 and 31 = −1502/532215 which gives 32 = 1826/569317. Thus, the error equations for , , , and are computed and given by and global error norm is Therefore, the parameters of the three-stage fifth-order RKFD method denoted by RKFD5 can be represented in Butcher tableau as follows:

Numerical Examples
In this section, some numerical examples will be solved to show the efficiency of the new RKFD methods of order four (i) RKFD5: the three-stage fifth-order RKFD method derived in this paper.
(ii) RKFD4: the three-stage fourth-order RKFD method derived in this paper.

Example 1.
The homogeneous linear problem is as follows: The exact solution is given by ( ) = e sin( ). The problem is integrated in the interval [0, 10].

Example 2.
The nonhomogeneous nonlinear problem is as follows: The exact solution is given by ( ) = arcsin( ). The problem is integrated in the interval [0, /4].
Example 5. The linear system is as follows: The problem is integrated in the interval [0, 2].
Example 6. The nonlinear system is as follows: ( V) = 16 + 1 The exact solution is given by The problem is integrated in the interval [0, 2].
The problem is integrated in the interval [0, 2].

Conclusion
This paper deals with Runge-Kutta type method denoted by RKFD method for directly solving special fourth-order ODEs of the form ( V) ( ) = ( , ). First, we derived the order conditions for RKFD method, which were then used to construct three-stage fourth-and fifth-order RKFD methods. The methods are denoted by RKFD5 and RKFD4, respectively. We also proved that the RKFD method is zero-stable. From the numerical results, we observed that the new RKFD methods are more competent as compared with the existing Runge-Kutta methods in the scientific literature. From the numerical results, we conclude that the new RKFD methods are computationally more efficient in solving special fourthorder ODEs and outperformed the existing methods in terms of error precision and number of function evaluations.