Abstract
In this paper three numerical methods are discussed to find the approximate solutions of a systems of first order ordinary differential equations. Those are Classical Runge-Kutta method, Modified Euler method and Euler method. For each methods formulas are developed for n systems of ordinary differential equations. The formulas explained by these methods are demonstrated by examples to identify the most accurate numerical methods. By comparing the analytical solution of the dependent variables with the approximate solution, absolute errors are calculated. The resulting value indicates that classical fourth order Runge-Kutta method offers most closet values with the computed analytical values. Finally from the results the classical fourth order is more efficient method to find the approximate solutions of the systems of ordinary differential equations.
Highlights
Many physical phenomenon in sciences and engineering are modeled by using a systems of n first order ordinary differential equations defined by [1, 2, 3] as dy1 dx =f1(x, y1, y2, ..., yn) dy2 dx f2(x, y1, y2, ..., yn) dyn dx fn(x, y1, y2, ..., yn) (1)Where each equation represents the first derivative of each unknown functions as a mapping depending on the independent variable x, and n unknown functions f1, f2, ..., fn and the initial conditions f1(x0), f2(x0), ..., fn(x0) are prescribed.The comparison between a domain decomposition method and Runge Kutta methods for system of ordinary differential equation was analysed by [4]
The main purpose of this paper is to compare the numerical methods by obtaining the approximate solutions of a systems of first order ordinary differential equations
For demonstration purpose consider a systems of two ordinary differential equations, Euler methods results the following y1,i+1 = y1,i + hf1(xi, y1,i, y2,i) y2,i+1 = y2,i + hf2(xi, y1,i, y2,i) at each step we compute the vector of approximate values of the two unknown functions from the corresponding vector at the immediately preceding step
Summary
The comparison between a domain decomposition method and Runge Kutta methods for system of ordinary differential equation was analysed by [4]. An nth order initial value problems could be reduced to a systems of n first order ordinary differential equations discussed by [5, 6]. A second order initial value problem is reduced to two first order systems and solved by fourth order and Butcher’s fifth order Runge Kutta Methods [7]. The main purpose of this paper is to compare the numerical methods by obtaining the approximate solutions of a systems of first order ordinary differential equations. A conclusion is given in the last section
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