Abstract
In this paper, a new scheme of Runge-Kutta (RK) type has been developed while evaluating two slope functions per step and maintaining the third order accuracy of the scheme. Local truncation error is obtained with the help of principal term which is obtained via multi variable Taylor series. It has been shown that the convergence order of the scheme is three and its stability polynomial is also derived. Some numerical examples are taken in order to compare the developed scheme with other existing schemes. It is observed that the developed scheme is better than other selected existing schemes and this comparison has been performed on the basis of slope evaluations per integration step, error analysis and computer time consumed by the scheme under consideration. KEY WORDS : Initial value problems, Runge-Kutta Scheme, Autonomous and non-autonomous differential equations, Zero stability. DOI : 10.7176/MTM/9-8-06 Publication date : August 31st 2019
Highlights
It is observed that ordinary differential equations play vital role in real life situations
Various mathematical models are expressed in terms of ordinary differential equations with one or more than one initial conditions such as mass-spring-damper model, RLC series circuit, beam equation, SIR models in epidemiology, simple pendulum, Vander-pol oscillator, kinetic reactions in chemistry, particle’s trajectory, one-dimensional fluid flow equations, and many others
Ordinary differential equations can be solved analytically and numerically but in most of the situations analytical methods fail to acquire the desired solution and this happens in cases when we come across nonlinear terms in the model under consideration
Summary
It is observed that ordinary differential equations play vital role in real life situations. Rabiei, Faranak et al [11] authors proposed a new method of three explicit Improved Runge-Kutta (IRK) type schemes for solving first order ordinary differential equations. These schemes are two step in nature and requiring lower number of stages as compared to the classical Rungekutta method. [17] proposed a new method Improved Runge-Kutta method Nystrom (IRKM) method for solving second order ordinary differential equations This method www.iiste.org require lower number of function evaluation per step and numerical results are gives to illustrate the efficiency of proposed method and compared with existing method
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