On class of integral equations with partial integrals and its applications

L. B. Mironova

Ufa mathematical journal, Tome 11 (2019) no. 3, pp. 61-77

Résumé

We prove the existence and uniqueness of the solution to one class of systems of integral equations with partial integrals. Equations with partial integrals are equations containing an unknown function in the integrands of integrals of different dimension. The feature of the considered class of integral equations is that the equations involve integrals with both variables and constant upper integration limits. We first prove the unique solvability theorem for integral equations in the three-dimensional space. A similar statement is proved for equations with arbitrary many independent variables. Some applications of the obtained result are provided. For a hyperbolic system with dominant derivatives of the second order with three independent variables, we prove the existence and uniqueness of the solution of the main characteristic problem. The main characteristic problem for the system of equations with higher derivatives of the second order can be considered as an analogue of the Goursat problem for a hyperbolic system with no multiple characteristics. The solution of this problem is constructed explicitly in terms of the Riemann matrix. The Riemann matrix is defined as the solution of a system of Volterra integral equations. The problem with boundary conditions on five sides of the characteristic parallelepiped for this system of equations with higher derivatives of the second order is formulated. By reducing the problem to a system of equations with partial integrals and basing on our results, we prove the existence and uniqueness of the solution to this problem.

Détail

Keywords: integral equation with partial integrals, problem with conditions on the characteristics.

Bibliographie

[1] P. P. Zabreiko, A. S. Kalitvin, E. V. Frolova, “On partial integral equations in the space of continuous functions”, Diff. Equat., 38:4 (2002), 567–576 | DOI | MR | Zbl

[2] J. M. Appell, A. S. Kalitvin, P. P. Zabrejko, Partial Integral Operators and Integro-Differential Equations, M. Dekker, New York, 2000, 560 pp. | MR | Zbl

[3] A. S. Kalitvin, Linear operators with partial integrals, TsChKI, Voronezh, 2000, 252 pp. (in Russian)

[4] A. V. Bitsadze, “On structural properties of solutions of hyperbolic systems of partial differential equations of the first order”, Matem. Model., 6:6 (1994), 22–31 (in Russian) | MR | Zbl

[5] T. V. Chekmarev, “Formulas for the solution of Goursat's problem for a linear system of partial differential equations”, Diff. Equat., 18:9 (1983), 1152–1158 | MR | Zbl

[6] I. E. Pleshchinskaya, “On the equivalence of some classes of elliptic and hyperbolic systems of first order and second order partial differential equations”, Diff. Uravn., 23:9 (1987), 1634–1637 | MR

[7] V. I. Zhegalov, “A problem with normal derivatives in the boundary conditions for a system of differential equations”, Russ. Math. (Iz. VUZ), 52:8 (2008), 58–60 | DOI | MR | Zbl

[8] Yu. G. Voronova, “On Cauchy problem for linear hyperbolic system of the equations with zero generalized Laplace invariants”, Ufimskij Matem. Zhurn., 2:2 (2010), 20–26 (in Russian) | Zbl

[9] A. V. Zhiber, O. S. Kostrigina, “Goursat problem for nonlinear hyperbolic systems with integrals of the first and second order”, Ufimskij Matem. Zhurn., 3:3 (2011), 67–79 (in Russian) | MR | Zbl

[10] E. A. Sozontova, “Characteristic problems with normal derivatives for hyperbolic systems”, Russ. Math. (Iz. VUZ), 57:10 (2013), 37–47 | DOI | MR | Zbl

[11] A. A. Andreeva, J. O. Yakovleva, “The Cauchy problem for a system of the hyperbolic differential equations of the $n$th order with the nonmultiple characteristics”, Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz. Mat. Nauki, 21:4 (2017), 752–759 (in Russian) | DOI | Zbl

[12] L. B. Mironova, “On the Riemann method in $R^n$ for a system with multiple characteristics”, Russ. Math. (Iz. VUZ), 50:1 (2006), 32–37 | MR | Zbl

[13] V. I. Zhegalov, L. B. Mironova, “One system of equations with double major partial derivatives”, Russ. Math. (Iz. VUZ), 51:3 (2007), 9–18 | DOI | MR | Zbl

[14] L. B. Mironova, “On characteristic problem for a system with double higher partial derivatives”, Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz. Mat. Nauki, 43 (2006), 31–37 (in Russian) | DOI

[15] A. V. Sevastjanov, On I.N. Vekua method of solving Volterra type integral equation, Preprint VINITI, No 1373-B97, 1997, 9 pp. (in Russian)

[16] F. R. Gantmacher, The theory of matrices, v. 1, 2, AMS Chelsea Publ., Providence, RI, 1998 | MR | MR

[17] A. N. Kolmogorov, S. V. Fomin, Introductory real analysis, Englewood Cliffs, New Jersey, 1970 | MR | MR | Zbl

[18] V. S. Vladimirov, Equations of mathematical physics, Pure Appl. Math., 3, Marcel Dekker, Inc, New York, 1971 | MR | MR | Zbl

[19] V. I. Zhegalov, A. N. Mironov, Differential equations with higher partial derivatives, Kazan Math. Soc., Kazan, 2001 (in Russian)