# Problem with nonlocal two-point condition in time for a homogeneous partial differential equation of infinite order with respect to space variables

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We specify a class of unique solvability of a problem with nonlocal boundary condition for a homogeneous partial differential equation of the first order with respect to time and of infinite order with respect to space variables with constant complex coefficients. In the class of quasipolynomials of special form, we indicate formulas for the construction of a solution of the problem that require a finite number of differentiation operations of analytically given functions. For the case where there exists a nonunique solution of the problem, we present an algorithm of the construction of its partial solution.

## Keywords

Partial Differential Equation Space Variable Partial Solution Unique Solvability Infinite Order
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## References

- 1.A. V. Bitsadze and A. A. Samarskii, “On some simplest generalizations of linear elliptic boundary-value problems,”
*Dokl. Akad. Nauk SSSR*,**185**, No. 4, 739–740 (1969).MathSciNetGoogle Scholar - 2.V. M. Borok, “Correctly solvable boundary-value problems in an infinite layer for systems of linear partial differential equations,”
*Izv. Akad. Nauk SSSR. Ser. Mat.*,**35**, No. 1, 185–203 (1971).zbMATHMathSciNetGoogle Scholar - 3.V. M. Borok and M. A. Perel’man, “On classes of uniqueness of solutions of a boundary-value problem in an infinite layer,”
*Izv. Vyssh. Uchebn. Zaved. Mat.*, No. 8, 29–34 (1973).MathSciNetGoogle Scholar - 4.A. A. Dezin,
*General Problems of the Theory of Boundary-Value Problems*[in Russian], Nauka, Moscow (1980).zbMATHGoogle Scholar - 5.P. I. Kalenyuk, I. V. Kohut, and Z. M. Nytrebych, “Differential-Symbol method of solution of a nonlocal boundary-value problem for a heterogeneous partial differential equation,”
*Visn. Lviv. Univ. Ser. Mekh.-Mat.*, No. 62, 60–66 (2003).zbMATHGoogle Scholar - 6.P. I. Kalenyuk, I. V. Kohut, and Z. M. Nytrebych, “Differential-Symbol method of solution of a nonlocal boundary-value problem for an heterogeneous system of partial differential equations,”
*Mat. Met. Fiz.-Mekh. Polya*,**46**, No. 3, 25–31 (2003).zbMATHGoogle Scholar - 7.P. I. Kalenyuk, I. V. Kohut, and Z. M. Nytrebych, “Differential-symbol method of solution of a nonlocal boundary-value problem for a partial differential equation,”
*Mat. Met. Fiz.-Mekh. Polya*,**45**, No. 2, 7–15 (2002).zbMATHGoogle Scholar - 8.P. I. Kalenyuk, I. V. Kohut, and Z. M. Nytrebych, “On the kernel of a problem with a nonlocal two-point condition for a partial differential equation,”
*Mat. Visn. NTSh*,**4**, 116–128 (2007).zbMATHGoogle Scholar - 9.P. I. Kalenyuk and Z. M. Nytrebych,
*Generalized Scheme of Separation of Variables. Differential-Symbol Method*[in Ukrainian], “L’vivs’ka Politekhnika” National University, Lviv (2002).Google Scholar - 10.A. Kh. Mamyan, “General boundary-value problems in a layer,”
*Dokl. Akad. Nauk SSSR*,**267**, No. 2, 292–296 (1982).MathSciNetGoogle Scholar - 11.M. I. Matiichuk, “Nonlocal parabolic boundary-value problem,”
*Ukr. Mat. Zh*.,**48**, No. 3, 362–367 (1996);:**English translation***Ukr. Math. J.,***48**, No. 3, 405–411 (1996).CrossRefMathSciNetGoogle Scholar - 12.A. M. Nakhushev, “On local boundary-value problems with a shift and their relation with loaded equations,”
*Differents. Uravn.*,**21**, No. 1, 92–101 (1985).MathSciNetGoogle Scholar - 13.B. Yo. Ptashnyk,
*Incorrect Boundary-Value Problems for Partial Differential Equations*[in Russian], Naukova Dumka, Kiev (1984).Google Scholar - 14.B. Yo. Ptashnyk, V. S. Il’kiv, I. Ya. Kmit’, and V. M. Polishchuk,
*Nonlocal Boundary-Value Problems for Partial Differential Equations*[in Russian], Naukova Dumka, Kiev (2002).Google Scholar - 15.A. A. Samarskii, “On some problems of the theory of differential equations,”
*Differents. Uravn.*,**16**, No. 11, 1925–1935 (1980).MathSciNetGoogle Scholar - 16.A. L. Skubachevskii, “Nonlocal boundary-value problems with a shift,”
*Mat. Zametki*,**38**, No. 4, 587–598 (1985).MathSciNetGoogle Scholar - 17.L. V. Fardigola, “Well-posed problems in a layer with differential operators in a boundary condition,”
*Ukr. Mat. Zh.*,**44**, No. 8, 1083–1090 (1992);:**English translation***Ukr. Math. J.,***44**, No. 8, 983–989 (1992).zbMATHCrossRefMathSciNetGoogle Scholar - 18.L. V. Fardigola, “Nonlocal two-point boundary-value problems in a layer with a differential operators in the boundary condition,”
*Ukr. Mat. Zh.*,**47**, No. 8, 1122–1128 (1995);:**English translation***Ukr. Math. J.,***47**, No. 8, 1283–1289 (1995).zbMATHCrossRefMathSciNetGoogle Scholar - 19.P. Kalenyuk, I. Kohut, and Z. Nytrebych, “Differential-symbol method of solving the nonlocal boundary value problem in the class of non-uniqueness of its solution,”
*Mat. Stud.*,**20**, No. 1, 53–60 (2003).zbMATHMathSciNetGoogle Scholar

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