Generalization of the Laplace transform for solving differential equations with piecewise constant coefficients

The article develops the theory of integral transforms in order to obtain operational calculus for the study of transient events. An analogue of the Laplace transform is introduced, which can be applied to expressions with a piecewise constant factor before the differentiation operator. Concepts such as original function, the Laplace transform, convolution are defined. Theorems on differentiation of the original, on differentiation of the Laplace transform and others have been proved. A generalized convolution definition is given and a formula for calculation such convolution is proved. Based on the concept of convolution, a fractional integral is defined. The transmutation operators method is the main tool in the theory of generalized operational calculus. The generalized Laplace integral transforms introduced in the article and the classical Laplace integral transforms are connected with its help. The solution to the heat problem with piecewise constant coefficients for the semi-infinite rod is found.