Off Grid Initial Value Solver for First Order Ordinary Differential Equations in Block Form Using Chebyshev Polynomial as Basis Function (Published)
The importance of numerical solution to differential equations cannot be overemphasized. It has been observed that the analytic method of solution to some differential equation often became laborious and practicably impossible. In order to circumvent this problem, then the introduction of an approximate solution became inevitable. This paper focuses on the derivation and application of an appropriate continuous linear multistep method in a block form in solving first order ordinary differential equations by collocating at some selected off grid points and interpolating at only one grid point. To achieve this, Chebyshev polynomial is used as basis function. Some basic properties of Multistep methods were critically examined such as order, consistency, zero stability and region of absolute stability and the level of accuracy of the method was equally compared with an existing method and was found out to performs better that the method compared with.
Keywords: Interpolation, chebyshev polynomial., collocation, consistency, multistep
Off Grid Collocation Four Step Initial Value Solver for Second Order Ordinary Differential Equations (Published)
The derivation and application of a four step Block Linear Multistep Method is hereby presented. To achieve this, Chebyshev polynomial was employed as basis function. Chebyshev polynomial was adopted as basis function based on its level of accuracy among other monomials in the interval [-1, 1]. Block method was adopted in this presentation based on its accuracy over the popular Predictor – Corrector method. The method under consideration gives solution at each grid point within the interval of integration. The method was arrived at by interpolating the polynomial equation and collocating the differential equation at some selected points. The order and error constant of the method were investigated likewise the consistency and zero stability which is one of the desirability property of linear multistep method were equally investigated. The method was applied to solve some second order ordinary differential equations and compare its level of accuracy with the analytical solution and equally compare its level of accuracy with some other existing methods.
Keywords: Block Method., Interpolation, chebyshev polynomial., collocation, corrector, predictor