Abstract
We present a study on numerical solutions of nonlinear ordinary differential equations by applying Runge-Kutta-Fehlberg (RKF) method, a well-known adaptive Runge-kutta method. The adaptive Runge-kutta methods use embedded integration formulas which appear in pairs. Typically adaptive methods monitor the truncation error at each integration step and automatically adjust the step size to keep the error within prescribed limit. Numerical solutions to different nonlinear initial value problems (IVPs) attained by RKF method are compared with corresponding classical Runge-Kutta (RK4) approximations in order to investigate the computational superiority of the former. The resulting gain in efficiency is compatible with the theoretical prediction. Moreover, with the aid of a suitable time-stepping scheme, we show that the RKF method invariably requires less number of steps to arrive at the right endpoint of the finite interval where the IVP is being considered.
Highlights
The Runge-Kutta methods are an important family of predictor-corrector methods for approximation of solutions of ordinary differential equations (ODEs) which were developed by German mathematicians duo C
We report a few numerical results concerning the numerical solutions of four test problems as well as one real-world problem where nonlinear ODEs of different degrees are considered
We have presented the progression of both numerical solutions along the grid points to demonstrate the adaptive nature of the step-size control for RKF method for each problem
Summary
The Runge-Kutta methods are an important family of predictor-corrector methods for approximation of solutions of ordinary differential equations (ODEs) which were developed by German mathematicians duo C. Let’s consider an initial value problem (IVP). To numerically approximate the continuously differentiable solution u(t) of the IVP over the time interval t ∈ [a, b], we subdivide the interval into N equal subintervals and select the mesh points t j = a + jh, j = 0, 1, . N and h = (b − a)/N, where h is called a step size
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