Construction of Mikusi\'nski operational calculus based on the convolution algebra of distributions. Methods for solving mathematical physics problems

I. L. Kogan

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 22 (2018) no. 2, pp. 236-253

Résumé

A new justification is given for the Mikusinsky operator calculus entirely based on the convolution algebra of generalized functions $D'_{+}$ and $D'_{-}$, as applied to the solution of linear partial differential equations with constant coefficients in the region $(x;t)\in \mathbb R (\mathbb R_{+})\times \mathbb R_{+}$. The mathematical apparatus used is based on the current state of the theory of generalized functions and its one of the main differences from the theory of Mikusinsky is that the resulting images are analytical functions of a complex variable. This allows us to legitimate the Laplace transform in the algebra $D'_{+} $ $( x\in \mathbb R_{+} )$, and apply the algebra to the region of negative values of the argument with the use of algebra $D'_{-}$. On classical examples of second-order equations of hyperbolic and parabolic type, in the case $x\in \mathbb R$, questions of the definition of fundamental solutions and the Cauchy problem are stated, and on the segment and the half-line $x\in \mathbb R_{+}$, non-stationary problems in the proper sense are considered. We derive general formulas for the Cauchy problem, as well as circuit of fundamental solutions definition by operator method. When considering non-stationary problems we introduce the compact proof of Duhamel theorem and derive the formulas which allow optimizing obtaining of solutions, including problems with discontinuous initial conditions. Examples of using series of convolution operators of generalized functions are given to find the originals. The proposed approach is compared with classical operational calculus based on the Laplace transform, and the theory of Mikusinsky, having the same ratios of the original image on the positive half-axis for normal functions allows us to consider the equations posed on the whole axis, to facilitate the obtaining and presentation of solutions. These examples illustrate the possibilities and give an assessment of the efficiency of the use of operator calculus.

Détail

Keywords: calculus of Mikusiński, space of distributions, Duhamel integral.
Mots-clés : convolution of distributions, convolution algebra, Laplace transform