A Comparative Study on Classical Fourth Order and Butcher Sixth Order Runge-Kutta Methods with Initial and Boundary Value Problems

Differential equations (DEs) are of great use in modeling different real life problems arising in science and engineering (Arora, 2019). Model equations for-med by using DEs get complicated and several times it becomes quite difficult to find its exact solution (Ahmed and Iqbal, 2020). However, to find the exact solution of a complicated model equation a practice is to simplify the model equation and then find the exact solution of the simplified equation, after then they obtained result is used to approximate the original equation (Islam, 2015).


INTRODUCTION:
Differential equations (DEs) are of great use in modeling different real life problems arising in science and engineering (Arora, 2019). Model equations for-med by using DEs get complicated and several times it becomes quite difficult to find its exact solution (Ahmed and Iqbal, 2020). However, to find the exact solution of a complicated model equation a practice is to simplify the model equation and then find the exact solution of the simplified equation, after then they obtained result is used to approximate the original equation (Islam, 2015).
In this circumstance, the approximate result differs from the real one. To avoid such inconvenience researchers, find their faith in numerical techniques to find out the approximate solution of a complicated model equation. In the recent era as a result of the technological revolution, numerical techniques have become the most desirable technique in searching the solution to such a problem that cannot be solved analytically. There are several numerical methods to solve DEs. However, the Euler method is one of the basic methods for solving initial value problems of ordinary differential equations (ODEs). Problems arising in DEs are of two types based on the condition given at the endpoints namely initial value problems (IVPs) and boundary value problems (BVPs), and it In this article, we have used RK fourth order and RK sixth order method to solve initial and boundary value DEs. Also, we have to use the shooting method to convert the boundary value DE to the initial value DE to employ the RK method. The rest of the paper is organized as follows: Section 2 deals with a brief discussion of the RK method, in section 3 our selected problem is introduced, application of RK fourth order and sixth order method to the selected problem is presented in section 4 and 5, respectively, section 6 consists of result and discussion and conclusion is placed in section 7. Finally, the paper ends with a list of references.

RUNGE-KUTTA METHOD
Runge-Kutta method is a universal method that is widely used for solving ODE. The Runge-Kutta techniques are designed in such a way to achieve greater accuracy in approximation where functional values are required to know at some certain point of the subinterval. A brief discussion of the RK 4 th order and 6 th order method is presented in the adjacent subsections.

A. Runge-Kutta 4 th order method
Runge- Kutta 4 th order (RK4) method is the most powerful method to solve ODE. The classical RK method of 4 th order requires four evaluations per step and it gives more accurate results. A widely used technique of classical RK fourth order method was used in this study whose derivation is also presented here.
The Runge-Kutta method finds approximate value for a given . Only first-order ODEs can be solved by using the Runge-Kutta 4 th order method. The formula used to compute the next value +1 from the previous value given below. The value of are 0, 1, 2, 3, … . . . , ( − 0 )/ ℎ. Here, ℎ is step height and +1 = + ℎ ( + ℎ) − ( ) ≅ 1 + 2 + 3 + 4 1 = ℎ ( , ) 2 = ℎ ( + ℎ, + 1 ) 3 = ℎ ( + ℎ, + 2 ) 4 = ℎ ( + ℎ, + 1 )  1 is the increment based on the slope at the beginning of the interval, using  2 is the increment based on the slope at the midpoint of the interval, using + ℎ 1 2  3 is again the increment based on the slope at the midpoint, using + ℎ 2 2  4 is the increment based on the slope at the end of the interval, using + ℎ 3 To find the coefficients , , , , , , from Runge-Kutta formulas, re-product the Taylor series in term of ℎ. The last formula is not a polynomial approximation.

B. Runge-Kutta 6 th order method
Runge-Kutta Sixth order (RK6) method is better for solving an ODE as well as a partial differential equation (PDE) with an IVP. Though in general, higher-order Runge-Kutta gives a better solution than the lower order ones (Hossain et al., 2017). But it is not true for all purposes and differs from case to case. However, it is a well-known result that the Runge-Kutta method of order needs at least stages to the process. It is true for = 1, 2, 3, 4 but suddenly for order = 5 there exist a process with stages 6 due to (Kutta, 1901) but corrected by (Nyström, 1925). Again, in the study of (Huïa, 1956) there are 8 stages processed for = 6. To mitigate, these varieties and complexity Butcher (Butcher, 1963) provide a common form to all Runge-Kutta method by a matrix and two vectors and it is known as Butcher tableau. This method approximates the solution to an ODE of the form ′ = ( , ) with stages is Where, is the node's running index ( = 0, 1, 2, ⋯ ⋯ ), is the index used to label stages (1 ≤ ≤ ), the nodes and for the consistency the weights are defined as ∑ =1 = 1. And the Butcher tableau is Derivation of RK6 method from Butcher tableau can find in any standard numerical analysis book. Due to the paucity of space, we are not presenting the derivation here. For details of derivation please see (Butcher, 1963

Problem statement
There are two parts of differential equations, IVP and BVP on the basics of the given conditions indicated at the endpoints. In our present work, second order initial and boundary value ODEs are solved by RK4 and RK6 methods. Among many different formulas, one most common form of fourth ordered which derived and Butcher's sixth ordered Runge-Kutta methods are used. The BVP problems are firstly converted to a system of IVP through the medium of shooting method.

A. The numerical solution of IVPs
Consider the following second order IVP of ODE: With the initial condition ( 0 ) = 0 , ′ ( 0 ) = . Now this second order IVP can be written in a system of first order differential equations by considering Then the new problem will be, Where, ( , , ) = With the initial condition ( 0 ) = 0 , ( 0 ) = 0 = .

B. The numerical solution of BVPs
Consider the following second order boundary value ODEs: With the boundary condition ( ) = , ( ) = .
The same formulas of RK4 and RK6 methods that are used in solving IVP are used along with the shooting method to solve this BVP. Mainly, in the Shooting method, BVP has been transferred under consideration by an IVP with unknown parameter as of the form, The parameter is to be chosen in a manner to ensure that denotes the solution to the initial-value problem with = . Thus RK4 and RK6 methods are used for solving the IVPs of Eq. (3) for the repeated value of until we carry out the desired level of tolerance and accuracy and the solution ( ) of the BVP achieved.

Application of Runge -Kutta 4 th order and 6 th order method to the initial value problem
In The actual solution of this equation is = 1 + log + 2(log ) 3 . Table 1 to 3 and Fig 1 to 3 show the computed result and maximum absolute error relative to the actual solution. Where the line curve represents the approximated curve and the circle is for actual results.

Application of Runge -Kutta 4 th order and 6 th order method to the boundary value problem
The following BVP was taken for our study and test the efficiency and convergence of Runge-Kutta fourth order and sixth order methods separately and   4-6 show the computed result and maximum absolute error with actual solution. As earlier the line curved represents the approximated curve and circle for the actual results.

Comparison between the obtained results
The obtained result of our model examples of IVP and BVP is expressed through Table 1-3 and Table 4-6 and graphical representations are in Fig 1-3 and Fig 4-6, respectively. From the tables and figures for both examples and both methods, we observed that if step size leads to deterioration the estimated solution converges faster to the exact solution such that in the limit when step size tends to zero the errors go to zero. Also, we observed from Table 1-3 that in solving IVP the method RK4 gives a more accurate and better result than the method RK6 and it has been clarified with the adjacent graphs Fig 1-3.
We also see from Table 4-6 and Fig 4-6 that in solving BVP the integrator RK4 gives better approximation and converges faster to the actual solution than RK6.

CONCLUSION:
In this paper, we have applied Runge-Kutta fourth order and Butcher's sixth ordered methods to solve IVP and BVP of ODEs and the results are found in good agreement with exact solutions. We assume three different step sizes for each problem to arrive at a more accurate result with competition between the exact and approximated solutions in tables and figures. From the above investigation of these methods, observed that the convergence of approximated solutions to exact solutions increases with decreasing step size in both cases and the convergence rate of RK4 is superior to RK6 in comparison to the exact solutions. Hence it is cleared from this study that to find the more accurate result the lower order RK4 method is appropriate than the higher order RK6 method.

ACKNOWLEDGEMENT:
We would like to express our gratitude to our dear students RoksanaYeasmin, Sumaiya Pervin, Sayantani Biswas for their effortless support in completing this paper. We would also like to thanks our colleagues for their advice and support.

CONFLICTS OF INTEREST:
The authors declare that there is no conflict of interest concerning the publication of this paper.