Boundary value problems for matrix Euler--Poisson--Darboux equation with data on~a~characteristic

A. A. Andreev ; E. A. Maksimova

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 19 (2015) no. 4, pp. 603-612.

Résumé

We consider the system of $n$ partial differential equations in matrix notation (the system of Euler–Poisson–Darboux equations). For the system we formulate the Cauchy–Goursat and Darboux problems for the case when the eigenvalues of the coefficient matrix lie in $(0; 1/2)$. The coefficient matrix is reduced to the Jordan form, which allows to separate the system to the $r$ independent systems, one for each Jordan cell. The coefficient matrix in the obtained systems has the only one eigenvalue in the considered interval. For a system of equations having the only coefficient matrix in form of Jordan cell, which is the diagonal or triangular matrix, we can construct the solution using the properties of matrix functions. We form the Riemann–Hadamard matrices for each of $r$ systems using the Riemann matrix for the considered system, constructed before. That allow to find out the solutions of the Cauchy–Goursat and Darboux problems for each system of matrix partial differential equations. The solutions of the original problems are represented in form of the direct sum of the solutions of systems for Jordan cells. The correctness theorem for the obtained solutions is formulated.

Détail

Keywords: Riemann method, Darboux problem, partial differential equations
Mots-clés : Cauchy–Goursat problem, system of Euler–Poisson–Darboux equations.

Bibliographie

[1] Khairullin R. S., Zadacha Koshi dlia uravneniia Eilera–Puassona–Darbu [The Cauchy problem for the Euler–Poisson–Darboux equation], Kazan Univ., Kazan, 2014, 275 pp. (In Russian)

[2] Andreev A. A., “On a class of systems of differential equations of hyperbolic type”, Differentsial'nye uravneniia [Partial differential equations], v. 16, Ryazan. Gos. Ped. Inst., Ryazan, 1980, 9–14 (In Russian)

[3] Andreev A. A., “On the Riemann method for one system hyperbolic equations with multiple characteristics”, Correct Boundary Value Problems for Nonclassical Equations of Mathematical Physics, IM SOAN SSSR, Novosibirsk, 1981, 13–16 (In Russian)

[4] Elianu I. P., “Recherches sur les systèmes d'équations linéaires aux dérivées partielles du type de Laplace”, Studii şi cercetǎri matematice, Academia Republicii Populate Române, Institutul de Matematica, 4:1–2 (1953), 155–196 | MR | Zbl

[5] Riemann B., “Über die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite”, Abhandlungen der Königlichen Gesellschaft der Wissenschaften in Göttingen, 8 (1860), 43–65 Retrieved from (August 08, 2015) https://eudml.org/doc/135717

[6] Gellerstedt S., Sur un problème aux limites pour une équation linèaire aux dérivées partielles du second ordre de type mixte, Almqvist och Wiksells, Uppsala, 1935, vii+92 pp. | Zbl

[7] Andreev A. A., “The Cauchy-Goursat and Darboux problems for a system of Euler–Poisson–Darboux equations”, Differentsial'nye uravneniia s chastnymi proizvodnymi [Partial differential equations], Kuybyshev. Gos. Ped. Inst., Kuybyshev, 1983, 53–57 (In Russian)

[8] Spitsyn V. L., “About a Hadamard-Riemann method for the one system of second-order hyperbolic type”, Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 1999, no. 7, 19–26 (In Russian) | DOI

[9] Andreev A. A., Maksimova E. A., “The solution of the Cauchy problem for a system of hyperbolic type with singular characteristics”, Proceedings of the Eighth All-Russian Scientific Conference with international participation (15–17 September 2011). Part 3, Matem. Mod. Kraev. Zadachi, Samara State Technical Univ., Samara, 2011, 11–17 (In Russian)

[10] Maksimova E. A., “On Cauchy Problem for system of $n$ Euler–Poisson–Darboux equations in the plane”, Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2012, no. 1(26), 21–30 (In Russian) | DOI

[11] Gantmakher F. R., Teoriia matrits [Theory of matrices], Nauka, Moscow, 1988, 549 pp. (In Russian) | MR | Zbl

[12] Tyrtyshnikov E. E., Matrichnyi analiz i lineinaia algebra [Matrix analysis and linear algebra], Fizmatlit, Moscow, 2007, 476 pp. (In Russian)

[13] Handbook of mathematical functions with formulas, graphs, and mathematical tables, A Wiley-Interscience Publication. Selected Government Publications, eds. M. Abramowitz, I. A. Stegun, John Wiley Sons, Inc, New York, 1984, xiv+1046 pp. | MR | Zbl

[14] Higher transcendental functions, v. I, Bateman Manuscript Project, California Institute of Technology, ed. A. Erdélyi, Robert E. Krieger Publishing Company, Malabar, Florida, 1981, xxvi+302 pp. | Zbl

[15] Bitsadze A. V., Equations of the mixed type, Pergamon Press, Oxford, London, New York, Paris, 1964, xiii+160 pp. | MR | Zbl | Zbl

[16] Lancaster P., Tismenetsky M., The theory of matrices, Computer Science and Applied Mathematics, Academic Press (Harcourt Brace Jovanovich, Publishers), Orlando, 1985, xv+570 pp. | MR | Zbl

[17] Marcus M., Minc H., A survey of matrix theory and matrix inequalities, Allyn and Bacon, Inc, Boston, 1964, xvi+180 pp. | MR | Zbl