Abstract
ABSTRACT
 In this paper, formulation of an efficient numerical schemes for the approximation first-order initial value problems (IVPs) of ordinary differential equations (ODE) is presented. The method is a block scheme for some k-step linear multi-step methods (and) using the Hermite Polynomials a basis function. The continuous and discrete linear multi-step methods (LMM) are formulated through the technique of collocation and interpolation. Numerical examples of ODE have been examined and results obtained show that the proposed scheme can be efficient in solving initial value problems of first order ODE.
Highlights
Numerous problems in Sciences and Engineering are modelled using ordinary differential equations (ODEs)
We proposed an efficient numerical scheme to solve numerically first order initial value problems (IVPs)
James et al (2013), proposed a continuous block method for the solution of second order IVPs with constant step size, the method was developed by interpolation and collocation of power series approximate solution to generate a continuous linear multistep method
Summary
Numerous problems in Sciences and Engineering are modelled using ordinary differential equations (ODEs). We proposed an efficient numerical scheme to solve numerically first order IVPs. The proposed method is a block scheme for some k-step linear multistep methods (for k 1, 2 and 3 ) using Hermite polynomial as the basis functions. Many researchers had developed interest on improving the numerical solution of initial value problems (IVPs) of ordinary differential equation. James et al (2013), proposed a continuous block method for the solution of second order IVPs with constant step size, the method was developed by interpolation and collocation of power series approximate solution to generate a continuous linear multistep method. In this paper, Hermite polynomial is used as a basis function to derive some block methods for the solution of first order initial value problem (1). The required numerical scheme is obtained if we collocate equation (19) at x xn 1 and substitute c0 , c1, c2 and as yn 1
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