Email this article Conditions for the existence and uniqueness of the solution of the Goursat problem for a system of equations with dominant partial derivatives
- Authors: Sozontova E.A.1
-
Affiliations:
- Elabuga Institute of Kazan (Volga Region) Federal University
- Issue: Vol 27, No 2 (2023)
- Pages: 375-383
- Section: Short Communications
- URL: https://journal-vniispk.ru/1991-8615/article/view/145907
- DOI: https://doi.org/10.14498/vsgtu2012
- ID: 145907
Cite item
Full Text
Abstract
An n-dimensional system of equations with dominant partial derivatives of the nth order is being studied. The distinguishing feature of the considered system compared to other systems with partial derivatives is the presence of a first term in the equations on the right side of the system, representing a dominant derivative, while all other derivatives appearing in the system equations are obtained from it by discarding at least one differentiation with respect to any of the independent variables. The aim of the study is to find conditions for the unique solvability of the Goursat problem for the considered system. The main problem is reduced to a system of integral equations, the solution of which exists and is unique when the requirements of continuity of the kernels and right sides of this system are satisfied in the corresponding closed parallelepipeds of variable ranges. Conditions under which the main problem is uniquely solvable have been obtained. The final result in terms of the coefficients of the original system is formulated as a theorem.
About the authors
Elena A. Sozontova
Elabuga Institute of Kazan (Volga Region) Federal University
Author for correspondence.
Email: sozontova-elena@rambler.ru
ORCID iD: 0000-0003-4315-0669
SPIN-code: 4568-9733
Scopus Author ID: http://www.scopus.com/authid/detail.url?authorId=55885264300
ResearcherId: http://www.researcherid.com/rid/O-4039-2016
Cand. Phys. & Math. Sci., Associate Professor, Dept. of Mathematics and Applied Computer Science
Russian Federation, 423600, Elabuga, Kazanskaya st., 89References
- Bitsadze A. V. On structural properties of solutions of hyperbolic systems of partial differential equations of the first order, Mat. Model., 1994, vol. 6, no. 6, pp. 22–31 (In Russian).
- Zhegalov V. I. A problem with normal derivatives in boundary conditions for a system of differential equations, Russian Math. (Iz. VUZ), 2008, vol. 52, no. 8, pp. 58–60. DOI: https://doi.org/10.3103/S1066369X08080070.
- Sozontova E. A. Characteristic problems with normal derivatives for hyperbolic systems, Russian Math. (Iz. VUZ), 2013, vol. 57, no. 10, pp. 37–47. EDN: SKVOOR. DOI: https://doi.org/10.3103/S1066369X13100046.
- Mironova L. B. Boundary-value problems with data on characteristics for hyperbolic systems of equations, Lobachevskii J. Math., 2020, vol. 41, no. 3, pp. 400–406. EDN: ZPBRIK. DOI: https://doi.org/10.1134/S1995080220030130.
- Sozontova E. A. The conditions of solvability of the Goursat problem in quadratures for three-dimensional system of first order, Proc. Voronezh State Univ. Ser. Phys., Math., 2017, no. 2, pp. 128–138 (In Russian). EDN: ZAERAX.
- Mironov A. N., Mironova L. B. Riemann–Hadamard method for one system in three-dimensional space, Differ. Equ., 2021, vol. 57, no. 8, pp. 1034–1041. EDN: BWRRVC. DOI: https://doi.org/10.1134/S0012266121080073.
- Bitsadze A. V. Nekotorye klassy uravnenii v chastnykh proizvodnykh [Some Classes of Partial Differential Equations]. Moscow, Nauka, 1981, 448 pp. (In Russian)
- Zhiber A. V., Startsev S. Ya. Integrals, solutions, and existence problems for Laplace transformations of linear hyperbolic systems, Math. Notes, 2003, vol. 74, no. 6, pp. 803–811. EDN: LHRQKD. DOI: https://doi.org/10.1023/B:MATN.0000009016.91968.ed.
- Startsev S. Ya. Cascade method of Laplace integration for linear hyperbolic systems of equations, Math. Notes, 2008, vol. 83, no. 1, pp. 97–106. EDN: LLHRQR. DOI: https://doi.org/10.1134/S0001434608010124.
- Sozontova E. A. On solvability by quadratures conditions for second order hyperbolic systems, Ufa Math. J., 2016, vol. 8, no. 3, pp. 130–135. EDN: YVGUUD. DOI: https://doi.org/10.13108/2016-8-3-130.
- Sozontova E. A. On the solvability conditions of characteristic problems for one hyperbolic type system, Vestn. Syktyvkar. Univ. Ser 1. Mat. Mekh. Inform., 2020, no. 1(34), pp. 22–28 (In Russian). EDN: BVIINY.
- Mihaylova Yu. G. On Cauchy problem for linear hyperbolic systems of equations with zero generalized Laplace invariants, Ufim. Mat. Zh., 2010, vol. 2, no. 2, pp. 20–26 (In Russian). EDN: MVUKRT.
- Mironova L. B. Linear systems of equations with multiple higher partial derivatives, Thesis of Dissertation (Cand. Phys. & Math. Sci.). Kazan, Kazan State Univ., 2005, 140 pp. (In Russian). EDN: NNQRYV.
- Mironova L. B. Application of Riemann method to one system in three-dimensional space, Russian Math. (Iz. VUZ), 2019, vol. 63, no. 6, pp. 42–50. EDN: GGFHED. DOI: https://doi.org/10.3103/S1066369X19060057.
- Dzhokhadze O. M. The Riemann function for higher-order hyperbolic equations and systems with dominated lower-order terms, Differ. Equ., 2003, vol. 39, no. 10, pp. 1440–1453. EDN: VYSOQG. DOI: https://doi.org/10.1023/B:DIEQ.0000017917.55876.38.
- Sozontova E. A. A conditions of solvability of the Goursat problem in quadratures for the two-dimensional system of high order, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2017, vol. 21, no. 1, pp. 94–111 (In Russian). EDN: YPZFUZ. DOI: https://doi.org/10.14498/vsgtu1479.
Supplementary files
