Solution of Fractional Differential Equation in Terms of Distribution Theory

Abstract

The initial-value problem of a fractional differential equation is studied, assuming that the initial values are given as values of the function and its integer-order derivatives. The problem of the equation with constant coefficients is investigated in terms of distribution theory. The description is given in the style of operational calculus.

Solution of Differential Equations with the Aid of an Analytic Continuation of Laplace Transform

“…Yosida [1] [2] studied the Equation (1.1) for 1 σ = with ( ) 0 f t = , by using Mikusiński's operational calculus [5]. In [3] [4], operational calculus in terms of distribution theory is used, which was developed for the initial-value problem of fractional differential equation with constant coefficients in our preceding papers [6] [7]. In [3], the derivative is the ordinary Riemann-Liouville fractional derivative, so that the fractional derivative of a function ( ) u t exists only when ( ) u t is locally integrable on 0 >  , and the integral ( ) In [4], we adopted this analytic continuation of Riemann-Liouville fractional derivative, and the following condition, in place of Condition 1.…”

Solution of Inhomogeneous Differential Equations with Polynomial Coefficients in Terms of the Green’s Function

“…We consider the space D R [11,12]. A regular distribution in D R is such a distribution that it corresponds to a function which is locally integrable on R and has a support bounded on the left.…”

“…We now discuss the solution of inhomogeneous equations in terms of the Green's function and distribution theory. In [11,12], the solution of inhomogeneous differential equation with constant coefficients is discussed in terms of the Green's function and distribution theory. In Section 6, we discuss it in terms of the Green's function and the AC-Laplace transform, where we obtain the solution which is not obtained with the aid of the usual Laplace transform.…”

Mollification of fractional derivatives using rapidly decaying harmonic wavelet

“…In a recent paper [8], the present authors studied fractional differential equations in terms of distribution theory. In that paper, the fractional integral (1.1) and the fractional derivative (1.2) are defined for the regular right-sided distributions, which are locally integrable in R and have a support bounded on the left.…”