Abstract
A linear multistep method for the direct solution of initial value problems of ordinary differential equations was presented in this article. Collocation approximation method was adopted in the derivation of the scheme and then the scheme was applied as simultaneous integrator to special third order initial value problem of ordinary differential equations. The new block method possessed the desirable feature of Runge-Kutta method of being self-starting and eliminated the use of predictors. The 3-step block method is P-stable, consistent and more accurate than the existing one. Experimental results confirmed the superiority of the new scheme over the existing method.
Highlights
The mathematical formulation of physical phenomena in science and engineering often leads to initial value problems of the form: y''' = f(x,y),y(a) = y0y'(a)η0,y''(a) = η1 (1)only a limited number of analytical methods are available for solving (1) directly without reducing to a first order system of initial value problems
Awoyemi[1] derived a p-stable linear multistep method for general third order initial value problems of ordinary differential equations which is to be used in form of predictor-corrector forms and like most linear multistep methods, they require starting values from Runge-Kutta methods or any other onestep methods
The block methods in Fatunla[3] are in form of discrete and are proposed for non-stiff special second order ordinary differential equations in form of a predictor- corrector integration process
Summary
Awoyemi[1] derived a p-stable linear multistep method for general third order initial value problems of ordinary differential equations which is to be used in form of predictor-corrector forms and like most linear multistep methods, they require starting values from Runge-Kutta methods or any other onestep methods. Like other linear multistep methods are usually applied to the initial value problems as a single formula but they are not self-starting; and they advance the numerical integration of the ordinary differential equations in one-step at a time, which leads to overlapping of the piecewise polynomials solution model. This study, propose a block multistep method for the direct solution of third order initial value problems of ordinary differential equations
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