Abstract
We consider collocation and interpolation of the approximate solution at some selected grid and off grid points to give a system of non- linear equations, solving for the unknown constants using Guassian elimination method and substituting into the approximate solution gives the continuous block method. We investigate the basic properties of the derived method, nu- merical examples show that the method is suitable for solving fourth order initial value problem of ordinary differential equations.
Highlights
This paper considers approximate solution to problems in the form
Reduction of (1) to systems of first order ordinary differential equations has been established in literature not to be effective in terms of cost of evaluation, time of execution and approximation [2, 3, 4, 5]
Different basis function for the solution of higher order ordinary differential equations have been proposed in literature ranging from power series, Langrange polynomial, Newton’s polynomial, Chebychev polynomial, backward differentiation formula, Fourier series to mention few
Summary
(k) where xn is the initial point, yn are the solutions at the initial point, f is continuous within the interval of integration and satisfies the existence and uniqueness theorem given in [1]. Reduction of (1) to systems of first order ordinary differential equations has been established in literature not to be effective in terms of cost of evaluation, time of execution and approximation [2, 3, 4, 5]. Block method has been established in literature to be more effective than the predictor-corrector method in terms of accuracy, time of implementetion and cost of execution [10, 11]. Adopted one step approach which leads to hybrid method circumvents the Dahlquist stability barrier, gives better stability properties and approximation than the multistep method [12, 13]. We combine the properties of hybrid and block method to develop one step block integrator for the solution of (1)
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