Distribution Theory and Transform Analysis

Solution of Fractional Differential Equation in Terms of Distribution Theory

“…t nÀ1 HðtÞ, as seen from (8), (15) or an equation used in the proof of Lemma 5. By Definition 4 and Lemma 9, F À1 W and F À1 for this case are defined as follows.…”

“…Before concluding this section, a remark is given on the notation Pf; See p. 229 of [15]. When a function f ðtÞ belongs to D 0 , but f 0 ðtÞ does not, the notation Pf is introduced such that Pf½ f 0 ðtÞ ¼ D f ðtÞ.…”

“…The initial-value problem of an ordinary differential equation has been treated with the aid of operational calculus [6,7] and of distribution theory [14,15], in addition to the Laplace transform. The purpose of the present paper is to investigate the initial-value problem of a fractional differential equation with constant coefficients, which was studied in [9,11], in the framework of distribution theory.…”

“…Notation used is similar to the original operational calculus due to Heaviside [6]. In the book of Zemanian [15], arguments are given to show how the Laplace transform can be expressed in terms of distribution theory. We present an equivalent but slightly different definition in this case.…”

“…In discussing the initial-value problem of a differential equation, we use right-sided distributions [15]. Right-sided distributions are the functionals in the space of D 0 R , that is conjugate to the space D R of functions which are infinitely differentiable and have a support bounded on the right.…”