Abstract:
This dissertation discusses wavelet numerical schemes for solving linear and nonlinear
fractional di erential equations. A numerical method based on the Boubaker wavelet
for the solution of linear and nonlinear fractional di erential equations and fractional
delay di erential equations is presented. Method of steps is used to deal with frac tional delay di erential equations. Furthermore, we have utilized Quasilinearization
technique in conjunction with Boubaker wavelet method to reduce nonlinear fractional
di erential equations to system of linear fractional di erential equations. We have de rived and constructed the Boubaker wavelet and Boubaker wavelet operational matrix
of fractional integration for the solution of Caputo-Hadamard fractional di erential
equation. We have discussed the error analysis of the Boubaker wavelet. We have also
provided numerical simulation of the proposed wavelet method to check the reliability
and applicability of the proposed method.
In the second part of the thesis, we have developed a technique using shifted
Gegenbauer wavelets. We have constructed operational matrix of shifted Gegenbauer
wavelets, fractional order integration and fractional order derivative to solve the ini tial value problem. Furthermore, solution of nonlinear fractional di erential equation
is obtained by the proposed method in conjunction with the Adomian decomposition
method. We have investigated the convergence of the proposed method. We have pre sented numerical examples and graphical results to check the reliability and e ciency
of the proposed method.
Lastly, we have developed the Katugampola Gegenbauer wavelet (KGW) on the
interval [a,(ρ + a
ρ
)
1/ρ), ρ > 0, a ∈ R, for the numerical solutions of Katugampola
fractional di erential equations. We have proved that these wavelets are orthonormal
within the interval [a,(ρ+a
ρ
)
1/ρ). We have constructed the matrix for KGW, the KGW
integration matrix, and the KGW operational matrix of derivative. We have combined
the KGW method with Adomian decomposition method for the solution of nonlinear
