
Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2018, Volume 22, Number 2, Pages 236–253
(Mi vsgtu1569)




Differential Equations and Mathematical Physics
Construction of Mikusiński operational calculus based on the convolution algebra of distributions. Methods for solving mathematical physics problems
I. L. Kogan^{} ^{} Russian State Agrarian University  Moscow Agricultural Academy after K. A. Timiryazev, Moscow, 127550, Russian Federation
Abstract:
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 secondorder 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 halfline $x\in \mathbb R_{+}$, nonstationary 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 nonstationary 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 halfaxis 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.
Keywords:
calculus of Mikusiński, space of distributions, convolution of distributions, convolution algebra, Laplace transform, Duhamel integral
DOI:
https://doi.org/10.14498/vsgtu1569
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Article
UDC:
517.982.45
MSC: 44A40, 35E20 Received: October 17, 2017 Revised: February 11, 2018 Accepted: March 12, 2018 First online: March 28, 2018
Citation:
I. L. Kogan, “Construction of Mikusiński operational calculus based on the convolution algebra of distributions. Methods for solving mathematical physics problems”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 22:2 (2018), 236–253
Citation in format AMSBIB
\Bibitem{Kog18}
\by I.~L.~Kogan
\paper Construction of Mikusi\'nski operational calculus based on the convolution algebra of distributions. Methods for solving mathematical physics problems
\jour Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.]
\yr 2018
\vol 22
\issue 2
\pages 236253
\mathnet{http://mi.mathnet.ru/vsgtu1569}
\crossref{https://doi.org/10.14498/vsgtu1569}
\elib{http://elibrary.ru/item.asp?id=35467729}
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Cycle of papers
 Construction of Mikusiński operational calculus based on the convolution algebra of distributions. Methods for solving mathematical physics problems
I. L. Kogan Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2018, 22:2, 236–253
 Construction of Mikusinski operational calculus based on the convolution algebra of distributions. Basic provisions
I. L. Kogan Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2012, 2(27), 44–52
 Construction of Mikusinski operational calculus based on the convolution algebra of distributions. The theorems and the beginning of use
I. L. Kogan Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2013, 3(32), 56–68

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