Order Runge-Kutta with Extended Formulation for Solving Ordinary Differential Equations

The mathematical model has been used to understand many phenomena and natural interactions. Since including many variables and parameters, the complex models are not easy to find analytical solutions. In this paper, we analyze one of the family of Runge–Kutta method with an expansion of evaluation function. We applied the proposed method to solve ordinary differential equations problems and compared it with other well-known Runge-Kutta methods. The computation cost and accuracy for each method have been analyzed.


Introduction
Differential equations are the mathematical tool that can be used to model problems in various fields of sciences such as economics, biology, physics, mathematics, robotic, encryption, circuit (Amelia, 2020;Anggriani 2019;Aldila 2020;Vaidyanathan et al., 2017;Sambas et al., 2020;Sukono et al., 2020;Sambas et al., 2019;Mobayen et al., 2019;). A differential equation is an equation that contains a derivative of a function. Based on the number of independent variables, the differential equations are divided into two, namely ordinary differential equations and partial differential equations. The combination of several differential equations is called a system of differential equations (Atkinson, 2008).
Numerical methods are also capable of solving a large, non-linear, and very complex system of differential equations with known initial conditions. A search using numerical methods produces an approximate value of an analytical solution so that the solution contains an error value. Numerical methods can be used to solve differential equation problems with the aid of a computer as a calculation tool. One of the numerical methods used to obtain the exact value of differential equation problems is Runge Kutta.
Mathematical models are designed with varying difficulties, relying on the characteristics of the problem. Some models require high computational cost for simulation, especially for a complex model. Many numerical methods have been developed by researchers for solving nonlinear differential equations. Numerical methods for solving firstorder IVPs often fall into one of two large categories: linear multistep methods, or Runge-Kutta methods (Griffiths, 2010). A further division can be realized by dividing methods into those that are explicit and those that are implicit.
Many researchers analyzed various Runge-Kutta methods an proposed some modifications. Goeken (2000) developed a class of Runge-Kutta method with higher derivatives approximations for the 3rd and 4th-order method. Wu (2003) proposed a new family of Runge-Kutta formula with reduced evaluations of function. Phohomsiri and Udwadia (2004) proposed the Accelerated Runge-Kutta using two functions evaluation per step. Gadisa (2017) have compared the convergence and stability of the Runge-Kutta method with the high order Taylor method. Other researchers have implemented Runge-Kutta 6th order method to solve the initial value problem and partial differential equations model (Al-Shimmary, 2017;Sun 2017). The implicit form of the Runge-Kutta 6th order function and its application to ordinary differential equations problem has been investigated by Ghawadri (2019) and Huang (2019). In this study, we discuss a new family of Runge-Kutta method, which includes the derivation of its formulation and application in computer programming, as well as obtaining the results of solutions given by each Runge Kutta formulation. Apart from this. each method is then compared in terms of error rate, speed, and computational efficiency. We apply the proposed Runge-Kutta formulation for solving non-linear differential equations problem. We also compared the proposed method with other Runge-Kutta family.

Explicit Runge-Kutta methods
The Runge-Kutta method can lead to a large family of methods that have close structure. The following are some of the basic properties used in constructing the formula for the Runge-Kutta Method.

Extended Runge-Kutta Method
Previous research has been done to improve the accuracy of the Runge-Kutta method. Among others are improving the number of terms in Taylor series expansion. One of the approaches is done by Goeken (2008), with adding a term to the evaluation function using higher derivatives and presented new third and fourth-order numerical methods. Technically, the term ′ is added in . This leads to a new family of Runge-Kutta formulation as given in definition 2.

Definition 2.
The generalization of the Extended Runge-Kutta method is stated by where It can be shown, with = 0 ( = 1,2, … ) in (3) this generalization form reduce to classical Runge-Kutta (1). Moreover if = , then we will get two-step Runge-Kutta methods, which can be called 'derivative-free' extended Runge-Kutta methods.
As in Runge-Kutta, equation (3) represents the main function of Runge-Kutta while and are evaluation function. In this paper, we use the sixth order Extended Runge-Kutta and Butcher table to determine the coefficient value of the formulation.

Formulation of Novel Runge-Kutta Methods
In this section, we formulate proposed sixth-order Runge-Kutta Methods with extended iteration function. First, using definition 2 we have extended sixth-order Runge-Kutta ), = 1,2, … 6.
To get the coefficients in equation (3) can be done by simplifying and with the Taylor series. The results obtained, then give a value to one of the coefficients to get a simple evaluation function. The table Butcher for the formulation is shown in Table 1. Satisfying that ∑ = 1, the values in Table 1. are substituted to equation (4) to obtain the Extended 6th order Runge-Kutta solution.

Numerical Simulation
In this chapter, we will show the application of the extended 6th order Runge-Kutta method for solving examples of dynamic models. We use a simple logistic growth model and Lotka-Volterra equation system on a simple interaction of two populations (prey predatory model). Using the formulation that has been derived, we compute the solution of both models by using the Maple 15 program. The parameters used are hypothetical, for simulation purposes.

Application for solving Logistic Growth Model
Our first example concerns the simple scalar ODE problem where 0 , , > 0 are known constants. This is a common model for population dynamics in ecology where represents the number of individuals, the initial growth rate, is the maximum number of individuals that is allowed by the environment (the so-called carrying capacity of the environment). Using the separable differential equations method with the parameter value given in Table 2 we can simply get the exact solution for Model 1 as follows The graph of the solution (6) is displayed in Figure 1. Along with applying the derived extended 6th order Runge-Kutta Method we compared the result of the solution with those from other well-known Runge-Kutta family, that is Runge-Kutta Fehlberg and Classical 6th order Runge-Kutta. In Figure 1 it is shown that the solution of each method approaches well to the analytical solution. We use 25 numerical points for solving Model 5. The error of Runge-Kutta Fehlberg (RKF), 6th order Runge-Kutta (RK 6), and Extended 6th order Runge-Kutta (XRK6) for some time steps in numerical point is shown in Table 3. We use Root Mean Square Error (RMSE) to summarize the performance result for a method, which follows = √ ∑( −̂) .
(7) Table 4 shows the RMSE for each method when used to solve Model 1. XRK6 method has a smaller error than RKF and RK 6. However, in terms of computing cost, the XRK6 requires the largest amount of memory and the longest time to complete a simulation.

Application for solving Lotka-Volterra Model
One of the simpler population models that describe prey interactions is the Lotka-Volterra, named after research by Alfred J. Lotka (1910) andVito Vorterra (1926). Model consists of a pair of non-linear differential equations, which follows Parameter described the natural growth rate of species (prey). The number of prey is diminished by predation . We used hypothetical parameters for simulation shown in Table 5. With the same procedure as in the previous simulation, we compare the solutions of the Lotka-Volterra model given by each method. Figure 3 shows the dynamic populations of prey and predator produced by each method. The peak points and oscillations of each population can also be captured by method.    Table 6 shows the solution produced by Extended able to approach the solution produced by the other two methods. Agreed with the previous simulation, the computation cost required for this method is the most for solving the Model as shown in Table 7.

Conclussion
The family Runge-Kutta method for solving differential equation system has been analyzed. We derived basic properties of Runge-Kutta and proposed the new family of Runge-Kutta with sixth-order and extended formulation. We use the proposed method to solving the simple dynamic models. By comparing the dynamical population of the model and the computational cost which is obtained by each method, the proposed Runge-Kutta method quite close to the analytical solution. Further the results show the proposed method can be an alternative way for solving ODE problems.