Generalized double Laplace transform and its application for partial differential equations solving
The transmutation operator method is used to construct the generalized double Laplace transform. In the article, the apparatus of the generalized double Laplace transform is created, the differentiation with a piecewise constant factor is considered. By using the transmutation operator method, the calculation of the generalized double Laplace transform is reduced to the calculation of the classical Laplace transform. Theorems on the general properties of the double Laplace transform are proved: on the differentiation of the function; about shifts; a convolution of two functions is defined, its properties are studied, and a convolution theorem is proved. In the article the applications of the generalized double Laplace transform for solving partial differential equations with piecewise constant coefficients it is discussed. The Cauchy problem for the wave equation with piecewise constant coefficients is solved.