Two Embedded Pairs of Runge-Kutta Type Methods for Direct Solution of Special Fourth-Order Ordinary Differential Equations

We present two pairs of embedded Runge-Kutta type methods for direct solution of fourth-order ordinary differential equations (ODEs) of the form y = f(x, y) denoted as RKFD methods. The first pair, which we will call RKFD5(4), has orders 5 and 4, and the second one has orders 6 and 5 and we will call it RKFD6(5). The techniques used in the derivation of the methods are that the higher order methods are very precise and the lower order methods give the best error estimate. Based on these pairs, we have developed variable step codes and we have used them to solve a set of special fourth-order problems. Numerical results show the robustness and the efficiency of the new RKFD pairs as compared with the well-known embedded Runge-Kutta pairs in the scientific literature after reducing the problems into a system of first-order ordinary differential equations (ODEs) and solving them.


Introduction
This paper deals with embedded RKFD methods for directly solving special fourth-order ordinary differential equations (ODEs) of the form  (V) () =  (, ) ,  ≥  0 (1) with initial conditions in which the first, second, and third derivatives do not appear explicitly.This type of problems can be found in various fields of applied science and engineering such as beam theory [1,2], fluid dynamics [3], neural networks [4], and electric circuits [5].Traditionally, the fourth-order ordinary differential equations are transformed to a first-order system of ordinary differential equations, so that standard numerical methods can be applied (see [6][7][8][9][10][11]).However, several researchers (see [1,12,13]) observed the drawback of this technique as it wastes a lot of computing time and human effort.Therefore, direct integration methods have attracted significant attention from several authors for solving higher order ODEs, because these direct methods demonstrated the features in accuracy and speed (see [14][15][16][17][18][19][20][21][22][23]).However, all the methods discussed above are multistep methods in nature.This paper primarily aims to construct a one-step method to solve special fourthorder ODEs directly; this new method is self-starting in nature.

Mathematical Problems in Engineering
The general form of RKFD method with -stage for solving special fourth-order ODEs (1) can be expressed as follows [24]: where for  = 2, 3, . . ., .
The parameters   ,    ,    ,    ,   , and   of the RKFD method are to be determined for  = 1, 2, . . .,  and  = 1, 2, . . .,  and supposed to be real.The RKFD method is an explicit method if   = 0 for  ≤  and is an implicit method if   ̸ = 0 for some  such that  ≤ .To determine the parameters of the RKFD method given in (3)-( 4), the RKFD method expression is expanded using the Taylor series expansion.After doing some algebraic simplifications, this expansion is equated to the true solution that is given by the Taylor series expansion.The direct expansion of the truncation error is used to derive the order conditions for the RKFD method [25].A good deal of algebraic and numerical calculations are required for the above operation which were carried out using algebra package Maple [26].Algebraic order conditions for the RKFD method can be obtained from the direct expansion of the local truncation error.
In this paper we will derive embedded Runge-Kutta pairs for direct integration of special fourth-order ODEs.Embedded pairs of RK type methods have a built-in local truncation error estimate; as a result, the step size can be controlled at virtually no extra cost, and hence an efficient variable step size code can be developed.
In recent years, the construction of embedded Runge-Kutta method is an effective research area yielding continuous development to the existing codes.The present paper is primarily dedicated as an extra work in this research area.This technique involves two Runge-Kutta formulae of orders  and V ( > V, usually  = V + 1) (see [25,[27][28][29][30][31][32][33]).We are interested in deriving the effective embedded (V) pairs of RKFD methods that provide a cheap error estimation for variable step size codes.They depend on the methods (, , ,   ,   ,   ) of order  and (, , b, b , b , b ) of order V. Butcher tableau of embedded RKFD pair can be written as follows: The method will compute  +1 ,   +1 ,   +1 , and   +1 to approximate ( +1 ),   ( +1 ),   ( +1 ), and   ( +1 ), where  +1 is the computed solution and ( +1 ) is the exact solution.
The remainder of this paper is organized as follows.In Section 2, we present the order conditions of RKFD method as well as the basic concepts and notations which are used for embedded method.In Section 3, we present the construction of the new embedded RKFD pairs of orders 5(4) and 6(5), respectively.In Section 4, we carry out the numerical experiments to show the efficiency of the new embedded RKFD pairs when compared with the well-known Runge-Kutta pairs from the scientific literature.Conclusions of the paper are given in Section 5.

The Order Conditions of RKFD Method
The order conditions of RKFD method up to fifth order have been derived using Taylor series expansion by Hussain et al. [24].We use the same approach to derive the order conditions up to seventh order and for convenience we will present the algebraic order conditions of the RKFD method given in [24] together with the seventh order method in this paper.Next, we give the order conditions up to order 7.
The order conditions for  are as follows: Fourth order: Fifth order: Sixth order: Seventh order: The order conditions for   are as follows: Third order: Fourth order: Fifth order: Sixth order: Seventh order: The order conditions for   are as follows: Second order: Third order: Fourth order: Fifth order: Sixth order: Seventh order: The order conditions for   are as follows: First order: Second order: Third order: Fourth order: Fifth order: Sixth order: Seventh order: The following strategies are utilized for developing efficient embedded pairs.
The local error estimation, EST, can be used to control the step size ℎ by the standard formula as given in [35][36][37][38] where 0.9 is a safety factor, the local error estimation at each step is represented by EST, and TOL is the maximum allowable local error which is the precision required.
If EST ≤ TOL, then the step is accepted and we applied the procedure of performing local extrapolation (or higher order mode) meaning that the more accurate approximation will be used to advance the integration.If EST > TOL, then the step is rejected and the step size ℎ will be updated using formula (37).

Construction of Embedded Explicit RKFD Pairs
The construction of embedded explicit RKFD pairs will be discussed in this section.In specific, we will derive two embedded RKFD pairs of orders 5(4) and 6( 5) with three and four stages per step, respectively.
3.1.The Derivation of Embedded RKFD5(4) Pair.This section will focus on the derivation of embedded RKFD5(4) pair with three stages.The authors in [24] derived three-stage fifthorder RKFD method and the solution is given as follows: Now, based on the above solution for values of  and , we derive a three-stage fourth-order embedded formula.Solving the algebraic conditions ( 6), ( 10)-( 11), ( 16)- (18), and ( 23)-( 26) simultaneously gives a solution for b in terms of b1 and b3 and the solution of b  in terms of b 3 which are given as follows: while the values of b  and b  ,  = 1, 2, 3, are the same as the fifth-order method.Our objective now is to select the values of the free parameters b1 , b3 , and b 3 and thus the values of  (6) ,  (6) ,   (6) ,   (6) , ‖τ (5) ‖ 2 , and ‖τ  (5) ‖ 2 are as small as possible.Using the above solution, we get  (6) = 1.200 is a constant value; however,   (6) and ‖τ (5) ‖ 2 are functions in terms of b 3 , by plotting the graph of ‖τ (5) ‖ 2 versus b 3 (see Figure 1).From the numerical experiment and for the accurate result, we choose b 3 = 1/10 using Minimize command in Maple Software, which gives the error norm ‖τ (5) ‖ 2 = 1.14566368 × 10 −3 and   (6) = 0.4726195824.
Consequently, we denote this pair as RKFD5(4) method and it can be written in Butcher tableau as follows:  5) Pair.In this section a four-stage embedded RKFD6(5) pair will be derived.
Finally, all the parameters of the embedded pair are represented in the Butcher tableau and denoted as RKFD6(5) method as follows: (56)

Numerical Experiments
In this section, we present some problems which involve special fourth-order ODEs of the form  (V) () = (, ()) in order to test the performance of the two new embedded pairs derived in Section 3. The numerical results obtained are compared with the well-known embedded Runge-Kutta methods which are chosen from the scientific literature after transforming the same problems to a system of first-order differential equations and solving them.The methods chosen in the numerical experiments are as follows: (i) RKFD6( 5): the new embedded Runge-Kutta pair of orders 6(5) derived in Section 3 in this paper.
Problem 1.We consider that the homogeneous linear problem is as follows: The exact solution is given by () = e  sin().The problem is integrated in the interval [0, 5].
Problem 2. We consider that the homogeneous nonlinear problem is as follows: The exact solution is given by () = √ 1 + .The problem is integrated in the interval [0, 10].
Problem 3. We consider that the inhomogeneous nonlinear problem is as follows: The exact solution is given by () = sin().The problem is integrated in the interval [0, 8].
Problem 4. We consider that the linear system is as follows: ( The problem is integrated in the interval [0, 3]. In the numerical experiments, we have computed for each method and problem the maximum global error and the number of function evaluations used in the integration.In Figures 3-6, we plot these values in double logarithmic     scale.From Table 1 and Figures 3-6, we observed that the new RKFD6(5) and RKFD5(4) methods are more efficient to solve directly fourth-order differential equations compared with RK6(5)V and RK5(4)D methods in terms of function evaluations and the total number of steps.This is due to the fact that when using RK6(5)V and RK5(4)D methods, the fourth-order ODEs need to be transformed into a firstorder system and hence the dimension will increase four times.

Conclusion
We have constructed the two pairs of embedded RKFD methods for directly solving special fourth-order ODEs of the form  (V) () = (, ()) using variable step size codes in this paper.These methods are denoted as RKFD5(4) and RKFD6(5), respectively.Based on the methods, the variables step size codes are developed and used to solve special fourth-order ODEs.Numerical results show that the two new embedded RKFD pairs are more effective as compared with the existing embedded Runge-Kutta pairs of the same order in the literature.From the numerical results we conclude that the two new embedded RKFD methods are computationally more efficient in solving special fourthorder ODEs and outperformed the well-known embedded Runge-Kutta methods.

Figure 4 :
Figure 4: The efficiency curves for Problem 2.

Figure 5 :
Figure 5: The efficiency curves for Problem 3.

Figure 6 :
Figure 6: The efficiency curves for Problem 4.

Table 1 :
The comparison results of the total number of successful steps and failure steps between RKFD6(5), RKFD5(4), RK6(5)V, and RK5(4)D methods for all problems.