Haar Wavelet Approach for Numerical Solution of Ordinary, Partial and Fractional Differential Equations With Delay

Abstract

In this thesis, the main emphasis is on collocation technique using Haar wavelet. A new method based on Haar wavelet collocation is being formu- lated 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 equa- tions and system of these equations. For delay partial differential equations two methods are considered: the first one is a hybrid method of finite differ- ence scheme and one-dimensional Haar wavelet collocation method while in the second method two-dimensional Haar wavelet collocation method is ap- plied, and a comparative study is performed between the two methods. We also extend the method developed for delay differential equations to solve nu- merically 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 imple- mentations and testing of all methods are performed in MATLAB software. Several numerical experiments are conducted to verify the accuracy, ef- ficiency 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 re- sults 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