Abstract:
The main objective of this thesis is to develop numerical methods for solving nonlinear fractional
ordinary differential equations, nonlinear fractional partial differential equations, and linear and
nonlinear fractional delay differential equation. Some methods are proposed by utilizing wavelets
operational matrix methods and quasilinearization technique, these methods are used for the solution
of nonlinear fractional differential equations, we call these methods as wavelet quasilinearization
techniques. According to the wavelet quasilinearization techniques, we convert the fractional nonlinear
differential equation to fractional discretize differential equation by using quasilinearization
technique and apply wavelet methods at each iteration of quasilinearization technique to get the
solution.
We established a technique by utilizing both the Haar wavelet and Picard technique for solving
the fractional nonlinear differential equation. While some methods based on the wavelets methods
and method of steps, used for the solution of linear and nonlinear fractional delay differential
equation. These techniques converts the fractional linear or nonlinear delay differential equation
on a given interval to an fractional linear or nonlinear differential equation without delay over that
interval, by using the function defined on previous interval, and then apply the wavelet method
on the obtained fractional differential equation to find the solution on a given interval. The same
procedure provides the solution on next intervals.
We also developed a method, Gegenbauer wavelet operational matrix method, by using Gegenbauer
polynomials. The Gegenbauer wavelet matrix, Gegenbauer wavelet operational matrix of
fractional integration and Gegenbauer wavelet operational matrix of fractional integration for boundary
value problems are derived, constructed and utilized for the solution of fractional differential
equations.
The convergence and supporting analysis of our methods are also investigated. The comparison
analysis of methods with other existing numerical methods is also performed.
