mathematics Review Some Schemata for Applications of the Integral Transforms of Mathematical Physics Yuri Luchko Department of Mathematics, Physics, and Chemistry, Beuth University of Applied Sciences Berlin, Luxemburger Str. 10, 13353 Berlin, Germany;
[email protected] Received: 18 January 2019; Accepted: 5 March 2019; Published: 12 March 2019 Abstract: In this survey article, some schemata for applications of the integral transforms of mathematical physics are presented. First, integral transforms of mathematical physics are defined by using the notions of the inverse transforms and generating operators. The convolutions and generating operators of the integral transforms of mathematical physics are closely connected with the integral, differential, and integro-differential equations that can be solved by means of the corresponding integral transforms. Another important technique for applications of the integral transforms is the Mikusinski-type operational calculi that are also discussed in the article. The general schemata for applications of the integral transforms of mathematical physics are illustrated on an example of the Laplace integral transform. Finally, the Mellin integral transform and its basic properties and applications are briefly discussed. Keywords: integral transforms; Laplace integral transform; transmutation operator; generating operator; integral equations; differential equations; operational calculus of Mikusinski type; Mellin integral transform MSC: 45-02; 33C60; 44A10; 44A15; 44A20; 44A45; 45A05; 45E10; 45J05 1. Introduction In this survey article, we discuss some schemata for applications of the integral transforms of mathematical physics to differential, integral, and integro-differential equations, and in the theory of special functions. The literature devoted to this subject is huge and includes many books and reams of papers.