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
On the Accuracy of Binomial Model and Monte Carlo Method for Pricing European Options (Published)
We consider the accuracy of two numerical methods for determining the price of an options namely Binomial model and Monte Carlo method. Then we compare the convergence and accuracy of the methods to the analytic Black-Scholes price of European option. Binomial model is very simple but powerful technique that can be used to solve many complex options pricing problem. Monte Carlo method is very flexible in handling high dimensional financial problems. Moreover Binomial model is more accurate and converges faster than Monte Carlo method when pricing European options.
Keywords: Accuracy, Binomial Model, European option, Monte Carlo Method