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
Multiple Perturbed Collocation Tau-Method For Solving High Order Linear And Non-Linear Boundary Value Problems (Published)
This paper is concerned with the numerical solution of high order linear and nonlinear boundary value problems of ordinary differential equations by Multiple Perturbed Collocation Tau Method (MPCTM). In applying this method to the class of problems mentioned above, we assumed a perturbed approximate solution in terms of Chebyshev Polynomial basis function which is substituted into the special class of the problem considered. Thus, resulting into n-folds integration. After evaluation of n-fold integration, the resulting equation is then collocated at equally spaced interior points and the unknown constants in the approximate solution are then obtained by Gaussian elimination method which are then substituted back into the approximate solution. The proposed method is tested on several numerical examples, the approximate solutions is in agreement with the exact solution. The approximate results obtained by the proposed method confirm the convergence of numerical solutions and are compared favorably with the existing methods available in literature
Keywords: Accuracy, Boundary Value Problem, Chebyshev, Errors, Multiple, Perturbed, collocation
Multiple Perturbed Collocation Tau-Method For Solving High Order Linear And Non-Linear Boundary Value Problems (Published)
This paper is concerned with the numerical solution of high order linear and nonlinear boundary value problems of ordinary differential equations by Multiple Perturbed Collocation Tau Method (MPCTM).In applying this method to the class of problems mentioned above, we assumed a perturbed approximate solution in terms of Chebyshev Polynomial basis function which is substituted into the special class of the problem considered. Thus, resulting into n-folds integration. After evaluation of n-fold integration, the resulting equation is then collocated at equally spaced interior points and the unknown constants in the approximate solution are then obtained by Gaussian elimination method which are then substituted back into the approximate solution.The proposed method is tested on several numerical examples, the approximate solutions is in agreement with the exact solution. The approximate results obtained by the proposed method confirm the convergence of numerical solutions and are compared favorably with the existing methods available in literature.
Keywords: Accuracy, Boundary Value Problem, Chebyshev, Errors, Multiple, Perturbed, collocation