THE ORDER AND ERROR CONSTANT OF A RUNGE-KUTTA TYPE METHOD FOR THE NUMERICAL SOLUTION OF INITIAL VALUE PROBLEM

Raihanatu Muhammad

doi:10.33003/fjs-2020-0402-256

Abstract

Implicit Runge- Kutta methods are used for solving stiff problems which mostly arise in real life situations. Analysis of the order, error constant, consistency and convergence will help in determining an effective Runge- Kutta Method (RKM) to use. Due to the loss of linearity in Runge –Kutta Methods and the fact that the general Runge –Kutta Method makes no mention of the differential equation makes it impossible to define the order of the method independently of the differential equation. In this paper, we examine in simpler details how to obtain the order, error constant, consistency and convergence of a Runge -Kutta Type method (RKTM) when the step number .

Highlights

Mathematical modeling of many engineering and physical forward manner than multistep methods (Muhammad, R, Y.A system leads to non-linear ordinary and partial differential Yahaya, A.S Abdulkarim, 2016)
The initial value problem for first order Ordinary Differential Equation is defined by y ′ = f(x, y) y(x0 ) = y0 x ∈ [a, b]
The initial value problem (IVP) for first order Ordinary Differential Equation is defined by y ′ = f(x, y) y(x0 ) = y x ∈ [a, b]

Summary

INTRODUCTION

Mathematical modeling of many engineering and physical forward manner than multistep methods (Muhammad, R, Y.A system leads to non-linear ordinary and partial differential Yahaya, A.S Abdulkarim, 2016). The initial value problem for first order Ordinary Differential Equation is defined by y ′ = f(x, y) y(x0 ) = y0 x ∈ [a, b]. On the current stage and later stages, the method is called an Implicit method This method is more suitable for solving stiff problems due to its high order of accuracy which makes it more superior to the explicit method (Yahaya and Adegboye, 2011). The first and second order Ordinary Differential Equation (ODE) methods are said to be of order p if p is the largest integer for which y(x + h) − y(x) − hφ(x, y(x), h) = 0(hp+1 ). The first and second order Ordinary Differential Equation (ODE) methods are said to be consistent if φ(x, y(x), 0) ≡ f(x, y(x)).

METHODOLOGY

RESULTS

DISCUSSION

CONCLUSION