A general class of integral transforms and an expression for their convolution formulas

Abstract

A class of integral transforms which satisfies certain assumptions is considered. Several examples, including the Fourier, Laplace and Mellin transforms, are in this class as well as transforms which are yet to be named. The assumptions allow for certain properties to exist for these integral transforms, such as the shifting and convolution properties. Further conditions are presented which allow for the integral transform operator to be injective on a specific class of functions. Applications of the results are provided and analytical solutions to differential equations are obtained. Numerical solutions are then compared to the analytical solutions to demonstrate the usefulness of this class of transforms for solving a particular family of differential equations.

Keywords:

• Integral transforms
• differential equations
• convolution
• shifting property
• inversion formula

AMS CLASSIFICATIONS:

• 35A22
• 44A35
• 34A12

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Here and throughout the article, for a function f defined on $(a,b)$ we let $f(a)=\underset{t\to a^{+}}{lim}f(t)\mathrm{and}f(b)=\underset{t\to b^{-}}{lim}f(t).$