Integrals and derivatives of fractional order based on Laplace type integral transformations with applications

Integrals and derivatives of fractional order theory is being developed. An analogue of the operational calculus is constructed for a differential operator with piecewise constant coefficients. Various constructions of the generalized Laplace transform are proposed. The transformation operators establish a connection between the Mellin – Laplace integral transformations and the generalized Laplace integral transformation. Isomorphism between the space of originals and the space of generalized originals is found. Mellin – Laplace type inversion formulas are established. Theorems on the differentiation of the generalized original and others are proved. A definition of a generalized convolution is given and a formula for its calculation is established, a connection between the generalized and classical convolution is indicated. On the basis of the concept of generalized convolution, a definition of a generalized integral and a generalized fractional derivative is given. Relations between generalized fractional integrals and Riemann-Liouville integrals of fractional order are established. For a model equation of heat conduction with a piecewise constant coefficient, the problem of calculating the heat flow transfer is solved. The heat flow is expressed as a generalized time derivative of the order of 1/2 of the measured temperature dependence at the boundary.