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.
