# On the Theory of Nonlocal Problems with Integral Conditions for Systems of Equations of Hyperbolic Type

- 3 Downloads

We consider a nonlocal problem with integral conditions for a system of hyperbolic equations of the second order. By method of introduction of functional parameters, the investigated problem is reduced to an equivalent problem formed by the Goursat problem for a system of hyperbolic equations with parameters and integral relations. The algorithms used to find the approximate solutions of this problem are constructed and their convergence to the exact solution is demonstrated. Sufficient conditions for the unique solvability of the equivalent problem are obtained in terms of the initial data. Moreover, the conditions of unique solvability of the nonlocal problem with integral conditions for system of hyperbolic equations are established in terms of the coefficients of the system and the kernels from the integral conditions.

## Preview

Unable to display preview. Download preview PDF.

## References

- 1.A. V. Bitsadze,
*Some Classes of Partial Differential Equations*[in Russian], Nauka, Moscow (1981).Google Scholar - 2.B. I. Ptashnik,
*Ill-Posed Boundary-Value Problems for Partial Differential Equations*[in Russian], Naukova Dumka, Kiev (1984).Google Scholar - 3.A. M. Nakhushev,
*Problems with Shifts for Partial Differential Equations*[in Russian], Nauka, Moscow (2006).Google Scholar - 4.T. Kiguradze, “Some boundary-value problems for systems of linear partial differential equations of hyperbolic type,”
*Mem. Different. Equat. Math. Phys.*,**1**, 1–144 (1994).MathSciNetzbMATHGoogle Scholar - 5.J. R. Cannon, “The solution of the heat equation subject to the specification of energy,”
*Quart. Appl. Math.*,**21**, No. 2, 155–160 (1963).MathSciNetCrossRefzbMATHGoogle Scholar - 6.A. A. Samarskii, “Some problems of the contemporary theory of differential equations,”
*Differents. Uravn.*,**16**, No. 11, 1221–1228 (1980).Google Scholar - 7.N. I. Yurchuk, “Mixed problem with integral condition for some parabolic equations,”
*Differents. Uravn.*,**22**, No. 12, 2117–2126 (1986).MathSciNetGoogle Scholar - 8.S. V. Zhestkov, “The Goursat problem with integral boundary conditions,”
*Ukr. Mat. Zh.*,**42**, No. 1, 132–135 (1990);**English translation:***Ukr. Math. J.*,**42**, No. 1, 119–122 (1990).Google Scholar - 9.N. D. Golubeva and L. S. Pul’kina, “One nonlocal problem with integral conditions,”
*Mat. Zametki*,**59**, No. 3, 171–174 (1996).zbMATHGoogle Scholar - 10.A. Bouziani, “Solution forte d’un probleme mixte avec conditions non locales pour une classe liquations hyperboliques,”
*Bull. Cl. Sci., Acad. Roy. Belg.*,**43**, 53–70 (1997).Google Scholar - 11.S. Beilin, “Existence of solution for one-dimensional wave equations with nonlocal conditions,”
*Electron. J. Different. Equat.*,**76**, 1–8 (2001).MathSciNetzbMATHGoogle Scholar - 12.L. S. Pul’kina, “Nonlocal problem with integral conditions for a quasilinear hyperbolic equation,”
*Mat. Zametki*,**70**, Issue 1, 88–95 (2001).Google Scholar - 13.A. I. Kozhanov and L. S. Pul’kina, “On the solvability of boundary-value problems with nonlocal integral boundary condition for multidimensional hyperbolic equations,”
*Differents. Uravn.*,**42**, No. 9, 1166–1179 (2006).zbMATHGoogle Scholar - 14.V. P. Tkach and L. V. Urmancheva, “Numerical-analytic method for finding solutions of systems with distributed parameters and integral condition,”
*Nelin. Kolyv.*,**12**, No. 1, 110–119 (2009);**English translation:***Nonlin. Oscillat.*,**12**, No. 1, 113–122 (2009).Google Scholar - 15.V. S. Il’kiv and B. I. Ptashnyk, “Problems for partial differential equations with nonlocal conditions. Metric approach to the problem of small denominators,”
*Ukr. Mat. Zh.*,**58**, No. 12, 1624–1650 (2006);**English translation:***Ukr. Math. J.*,**58**, No. 12, 1847–1875 (2006).Google Scholar - 16.V. S. Il’kiv, Z. M. Nytrebych, and P. Y. Pukach, “Boundary-value problems with integral conditions for a system of Lamé equations in the space of almost periodic functions,”
*Electron. J. Different. Equat.*,**2016**, No. 304, 1–12 (2016).zbMATHGoogle Scholar - 17.A. M. Kuz’ and B. I. Ptashnyk, “Problem with integral conditions in time for a Sobolev-type system of equations with constant coefficients,”
*Ukr. Mat. Zh.*,**69**, No. 4, 530–549 (2017);**English translation:***Ukr. Math. J.*,**69**, No. 4, 621–645 (2017).Google Scholar - 18.A. T. Asanova and D. S. Dzhumabaev, “Unique solvability of the boundary-value problem with data on characteristics for systems of hyperbolic equations,”
*Zh. Vychisl. Mat. Mat. Fiz.*,**42**, No. 11, 1673–1685 (2002).MathSciNetzbMATHGoogle Scholar - 19.A. T. Asanova and D. S. Dzhumabaev, “On the correct solvability of the nonlocal boundary-value problem for systems of hyperbolic equations,”
*Dokl. Ros. Akad. Nauk*,**391**, No. 3, 295–297 (2003).zbMATHGoogle Scholar - 20.A. T. Asanova, “On the nonlocal boundary-value problem for systems of quasilinear hyperbolic equations,”
*Dokl. Ros. Akad. Nauk*,**411**, No. 1, 5–9 (2006).MathSciNetGoogle Scholar - 21.A. T. Asanova, “On the unique solvability of the nonlocal boundary-value problem with the data on intersecting lines for systems of hyperbolic equations,”
*Differents. Uravn.*,**45**, No. 3, 373–381 (2009).MathSciNetzbMATHGoogle Scholar - 22.A. T. Asanova and D. S. Dzhumabaev, “Well-posedness of nonlocal boundary-value problems with integral condition for the system of hyperbolic equations,”
*J. Math. Anal. Appl.*,**402**, No. 1, 167–178 (2013).MathSciNetCrossRefzbMATHGoogle Scholar - 23.A. T. Asanova, “On solvability of nonlinear boundary-value problems with integral condition for the system of hyperbolic equations,”
*Electron. J. Qual. Theory Different. Equat.*,**63**, 1–13 (2015).MathSciNetzbMATHGoogle Scholar - 24.A. T. Asanova, “Periodic solutions of the system of second-order hyperbolic equations in a plane,”
*Mat. Zametki*,**101**, Issue 1, 20–30 (2017).Google Scholar - 25.A. T. Asanova, “Nonlocal problem with integral conditions for systems of hyperbolic equations in a characteristic rectangle,”
*Izv. Vyssh. Uchebn. Zaved., Ser. Mat.*, No. 5, 11–25 (2017).Google Scholar - 26.A. T. Assanova, “Solvability of a nonlocal problem for a hyperbolic equation with integral conditions,”
*Electron. J. Different. Equat.*,**2017**, No. 170, 1–12 (2017).MathSciNetzbMATHGoogle Scholar - 27.A. T. Asanova, “One approach to the solution of a nonlocal problem for systems of hyperbolic equations with integral conditions,”
*Nelin. Kolyv.*,**20**, No. 4, 435–450 (2017).Google Scholar