Geodesic
Generalization of the Laplace transform for solving differential equations with piecewise constant coefficients
Čebyševskij sbornik, Tome 23 (2022) no. 2, pp. 5-20.

Voir la notice de l'article provenant de la source Math-Net.Ru

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.
Keywords: Generalization of the Laplace transform, inversion formula, transformation operator, fractional integral. 
@article{CHEB_2022_23_2_a0,
     author = {F. S. Avdeev and O. E. Yaremko and N. N. Yaremko},
     title = {Generalization of the {Laplace} transform for solving differential equations with piecewise constant coefficients},
     journal = {\v{C}eby\v{s}evskij sbornik},
     pages = {5--20},
     publisher = {mathdoc},
     volume = {23},
     number = {2},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHEB_2022_23_2_a0/}
}
TY  - JOUR
AU  - F. S. Avdeev
AU  - O. E. Yaremko
AU  - N. N. Yaremko
TI  - Generalization of the Laplace transform for solving differential equations with piecewise constant coefficients
JO  - Čebyševskij sbornik
PY  - 2022
SP  - 5
EP  - 20
VL  - 23
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/CHEB_2022_23_2_a0/
LA  - ru
ID  - CHEB_2022_23_2_a0
ER  - 
%0 Journal Article
%A F. S. Avdeev
%A O. E. Yaremko
%A N. N. Yaremko
%T Generalization of the Laplace transform for solving differential equations with piecewise constant coefficients
%J Čebyševskij sbornik
%D 2022
%P 5-20
%V 23
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/CHEB_2022_23_2_a0/
%G ru
%F CHEB_2022_23_2_a0
F. S. Avdeev; O. E. Yaremko; N. N. Yaremko. Generalization of the Laplace transform for solving differential equations with piecewise constant coefficients. Čebyševskij sbornik, Tome 23 (2022) no. 2, pp. 5-20. http://geodesic.mathdoc.fr/item/CHEB_2022_23_2_a0/

[1] Aghili Arman, “New trends in Laplace type integral transforms with applications”, Boletim da Sociedade Paranaense de Matematica, 35:1 (2017), 173–193 | DOI | MR | Zbl

[2] Baeumer Boris, “On the Inversion of the Convolution and Laplace Transform”, Transactions of the American Mathematical Society, 355:3 (2003), 1201–1212 | DOI | MR | Zbl

[3] Brychkov Yu. A. , Prudnikov A. P. , Shishov V. S., “Operational calculus”, Itogi Nauki i Tekhn. Ser. Mat. Anal., 16, VINITI, M., 1979, 99–148 | MR

[4] Ermolova N.Y., Tirkkonen O., “Laplace Transform of Product of Generalized Marcum Q, Bessel I, and Power Functions With Applications”, IEEE Transactions on Signal Processing, 2014, 2938–2944 | MR | Zbl

[5] Ganzha E.I., “On Laplace and Dini transformations for multidimensional equations with a decomposable principal symbol”, Programming and Computer Software, 38 (2012), 150–155 | DOI | MR | Zbl

[6] Gonzalez-Acuna Rafael G., Gutierrez-Vega Julio C., “Transition integral transform obtained from generalization of the Fourier transform”, Ain Shams Engineering Journal, 10:4 (2019), 841–845 | DOI

[7] Jarad Fahd, Abdeljawad Thabet, “A modified Laplace transform for certain generalized fractional operators”, Results in Nonlinear Analysis, 1:2 (2018), 88–98 | MR

[8] Koepf Wolfram, Kim Insuk, Rathie Arjun K., “On a New Class of Laplace-Type Integrals Involving Generalized Hypergeometric Functions”, Axioms, 8:3 (2019), 87 | DOI | MR | Zbl

[9] Li S., Shemyakova E., Voronov Th., “Differential operators on the superline, Berezinians, and Darboux transformations”, Lett. Math. Phys., 107:9 (2017), 1686–1714 | MR

[10] Matveev V.B., Salle M.A., Darboux transformations and solitons, Springer Series in Nonlinear Dynamics, Springer-Verlag, Berlin, 1991 | DOI | MR | Zbl

[11] Milovanovic G. V., Parmar R. K., Rathie A. K., “A study of generalized summation theorems for the series with an applications to Laplace transforms of convolution type integrals involving Kummer's functions”, Applicable analysis and discrete mathematics, 12:1 (2018), 257–272 | DOI | MR | Zbl

[12] Napalkov V. V., Mullabaeva A. U., “On one class of differential operators and their application”, Proceedings of the Steklov Institute of Mathematics, 288 (2015), 142–155 | DOI | MR

[13] Pinelas Sandra, Xavier G. B. A., Kumar S. U. Vasantha, Meganathan M., “Laplace - Fibonacci transform by the solution of second order generalized difference equation”, Non autonomous Dynamical Systems, 4:1 (2017), 22–30 | DOI | MR | Zbl

[14] Sharma V. D., Thakare M. M., “Introduction of generalized Laplace-fractional Mellin transform”, International journal of engineering sciences research technology, 5 (2016), 667–670

[15] Sharma V. D., Thakare M. M., “Generalized Laplace-Fractional Mellin Transform and Operators”, International Journal of Pure Applied Sciences Technology, 16:1 (2013), 20–25 | MR

[16] Tsarev S.P., “Generalized Laplace Transformations and Integration of Hyperbolic Systems of Linear Partial Differential Equations”, Proc. ISSAC, ed. Labahn G., 2005, 325–331 | DOI | MR | Zbl

[17] Zaikina S. M., “Generalized Integral Laplace Transform and Its Application to Solving Some Integral Equations”, Vestnik Samarskogo Gosudarstvennogo Tehnivceskogo Universiteta. Seria: Fiziko-Matematicheskie Nauki, 2014, no. 1(34), 19–24 | DOI | Zbl

[18] Jeffreys H., Jeffreys B., Methods of Mathematical Physics, 3rd ed., Cambridge Univ. Press, 1956 | MR | Zbl

[19] Sitnik Sergei M., Yaremko Oleg, Yaremko Natalia, “Transmutation Operators and Applications”, Transmutation Operators Boundary Value Problems, Springer Nature, Switzerland, 2020, 447–466 | MR

[20] Yaremko O.E., “Transformation operator and boundary value problems”, Differential Equation, 40:8 (2004), 1149–1160 | DOI | MR | Zbl