Methods for Solving O-Fractional Differential Equations

Abstract

By inspiration of some new developments in -fractional calculus, we develop a sufficient existence condition of -Laplace transform of solution of particular categories of fractional differential equations. We have prove some known properties of integral and derivatives using generalized Laplace transform. The goal of first part of the thesis is to reveal the efficiency of -Laplace transform for solving -fractional ordinary differential equations. In second part of this thesis a numerical method for solving a class of -fractional differential equations involving Caputo derivative with respect to a function is presented. Initial value problem for certain -fractional differential equation is converted into equivalent second kind of Volterra integral equation. A combination of Simpson’s and Trapezoidal rule is used to transform Volterra equation to a system of algebraic equations. The numerical solutions to the original problem are recovered from a solution of an algebraic system. We also give an error estimate for the function approximation and fractional integral approximation. Error bound for numerical approximation of solution is also derived. The numerical method is tested for various specific problems.