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
Initial Value Solvers for Direct Solution of Fourth Order Ordinary Differential Equations in a Block from Using Chebyshev Polynomial as Basis Function (Published)
The numerical computation of fourth order ordinary differential equations cannot be gloss over easily due to its significant and importance. There have been glowing needs to find an appropriate numerical method that will handle effectively fourth order ordinary differential equations without resolving such an equation to a system of first order ordinary differential equations. To this end, this presentation focuses on direct numerical computation to fourth order ordinary differential equations without resolving such equations to a system of first order ordinary differential equations. The method is not predictor – corrector one due to its limitation in the level of accuracy. The method is order wise christened “Block Method” which is a self-starting method. In order to achieve this objective, Chebyshev polynomial is hereby used as basis function.
Keywords: Chebyshev, consistency, continuous, corrector, multistep, predictor, zero – stability