| Summary: | Following the rise of fractional-order models and consequently, their fractional
governing equations, new analytical and numerical methods have been developed and
studied extensively. However, amidst the multitude of integral transform methods, there
have been no research done on the application of the continuous wavelet transform to
analytically solve fractional differential equations. To fill this gap, the present dissertation
derives an analytical method based on this integral transform by applying it to the
riemann-liouville integral and derivative, and the caputo derivative. With the help
of theorems and techniques in calculus and fractional calculus, important results in
functional analysis and properties of the continuous wavelet transform, it is found that
poisson wavelet transform of order n = 1 is able to yield meaningful results, suitable
for solving fractional-order equations. To demonstrate this, the scheme is applied to
solve two fractional differential equations, defined based on each of the aforementioned
fractional derivatives, wherein the exact solutions were successfully obtained. On the
other hand, much of the numerical analysis of fractional equations have been focused
on that of differential equations and thus, algorithms for fractional integral equations
have been found wanting. Motivated by the prowess of the legendre wavelet in solving
fractional differential equations numerically, this thesis strives to construct a numerical
scheme based on this wavelet for fractional integral equations.
|
|---|
