Public Defense of a Ph.D Research Scholar at Department of Mathematics

Mr. Rohul Amin, a Ph.D. Research Scholar at the Department of Mathematics, University of Peshawar has  submitted his thesis entitled "Haar Wavelet Approach for Numerical Solution of Ordinary, Partial and Fractional Differential Equations with Delay"  to the University of Peshawar, in partial fulfillment of the requirement for the award of degree of  Doctor of Philosophy (Ph. D) in the discipline of Mathematics.

All those having interest in the said research work, are cordially invited to attend the occasion. The participants would be allowed to raise relevant questions after the presentation by the scholar for further judgment and evaluation of the examiners.

Abstract:

In this thesis, the main emphasis is on collocation technique using Haar wavelet. A new method based on Haar wavelet collocation is being formulated for numerical solution of delay differential equations, delay differential systems, delay partial differential equations and fractional delay differential equations. The numerical method is applied to both linear and nonlinear time invariant delay differential equations, time-varying delay differential equations and system of these equations. For delay partial differential equations two methods are considered: the first one is a hybrid method of finite difference scheme and one-dimensional Haar wavelet collocation method while in the second method two-dimensional Haar wavelet collocation method is applied, and a comparative study is performed between the two methods. We also extend the method developed for delay differential equations to solve numerically fractional delay differential equations using Caputo derivatives and Haar wavelet. Here we consider fractional derivatives in the Caputo sense. Also we designed algorithms for all the new developed methods. The implementations and testing of all methods are performed in MATLAB software.

Several numerical experiments are conducted to verify the accuracy, efficiency and convergence of the proposed method. The proposed method is also compared with some of the existing numerical methods in the literature and is applied to a number of benchmark test problems. The numerical results are also compared with the exact solutions and the performance of the method is demonstrated by calculating the maximum absolute errors, mean square root errors and experimental rates of convergence for different number of collocation points. The numerical results show that the method is simply applicable, accurate, efficient and robust.