A Wavelet-Based Approach For The Analytical Solutions Of Fractional Differential Equations And The Numerical Solutions Of Fractional Integral Equations - Loong, Marcus Ong Wei (2024)
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.