Geodesic
Generalized Laplace transform based on the differentiation operator with piecewise constant coefficients
Čebyševskij sbornik, Tome 22 (2021) no. 5, pp. 172-184

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The theory of operational calculus is developed on the basis of a differential operator with piecewise constant coefficients. A formula for the generalized Laplace transform is proposed. An inversion formula of Mellin-Laplace type is proved. The concept of a generalized original-function and a generalized image is proposed. A theorem on the isomorphism of the spaces of originals and generalized originals is proved. Using transmutation operators, it is established that the generalized Laplace transform of the generalized original coincides with the Laplace transform of the corresponding original-function. Theorems on differentiation and integration of the generalized original, theorems on homogeneity, similarity, exponential scaling, first and second shifting theorems, and others are proved. In terms of the transmutation operator, a connection between the convolution of generalized original-functions and the corresponding convolution of original-functions is established. An algorithm for solving linear differential equations with piecewise constant coefficients is presented. A solution to the heat equation with a piecewise constant coefficient at the time derivative on the real axis is found. A mixed boundary value problem for the heat equation with a piecewise constant coefficient at the time derivative on the real semiaxis is solved.
Keywords: generalized integral Laplace transform, transmutation operator, generalized original-function, Mellin-Laplace inversion formula. 
@article{CHEB_2021_22_5_a10,
     author = {A. I. Nizhnikov and O. E. Yaremko and N. N. Yaremko},
     title = {Generalized {Laplace} transform based on the differentiation operator with piecewise constant coefficients},
     journal = {\v{C}eby\v{s}evskij sbornik},
     pages = {172--184},
     publisher = {mathdoc},
     volume = {22},
     number = {5},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHEB_2021_22_5_a10/}
}
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A. I. Nizhnikov; O. E. Yaremko; N. N. Yaremko. Generalized Laplace transform based on the differentiation operator with piecewise constant coefficients. Čebyševskij sbornik, Tome 22 (2021) no. 5, pp. 172-184. http://geodesic.mathdoc.fr/item/CHEB_2021_22_5_a10/