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
