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

Abstract

We discuss the solution of Laplace's differential equation and a fractional differential equation of that type, by using analytic continuations of Riemann-Liouville fractional derivative and of Laplace transform. We show that the solutions, which are obtained by using operational calculus in the framework of distribution theory in our preceding papers, are obtained also by the present method.

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

“…In [1,2], we confirm that the solution K 2 (t) is obtained by using an analytic continuation of the Laplace transform (AC-Laplace transform) for all nonzero values of c ∈ C\Z >0 . The complementary solution of the hypergeometric differential equation, corresponding to K 2 (t), is found to be obtained by using the Laplace transform series, where the Laplace transforms of the solutions are expresssed by a series of powers of s −1 multipied by a power of s, which has zero range of convergence.…”

“…We introduce the analytic continuation of the Laplace transform (AC-Laplace transform) of g ν (t), which is expressed by L H [g ν (t)], as in [1,2], such that…”

“…In our recent papers [1,2], we are concerned with the solution of Kummer's and hypergeometric differential equations, where complementary solutions expressed by the confluent hypergeometric series and the hypergeometric series, respectively, are obtained, by using the Laplace transform, its analytic continuation, distribution theory and the fractional calculus.…”

“…In our preceding papers [7,8] stimulated by Yosida's works [9,10] on Laplace's differential equations, of which typical one is Kummer's equation, we sudied the solution of Kummer's equation and a simple fractional differential equation on the basis of fractional calculus and distribution theory. In [1], we discussed it in terms of the AC-Laplace transform. In [4], we applied the arguments in [1] to the solution of the homogeneous hypergeometric equation.…”

“…In [1], we discussed it in terms of the AC-Laplace transform. In [4], we applied the arguments in [1] to the solution of the homogeneous hypergeometric equation. We now discuss the solution of inhomogeneous equations in terms of the Green's function and distribution theory.…”