We investigate the problem with inhomogeneous integral condition for a homogeneous partial differential equation of the first order with respect to time and, in the general case, of infinite order with respect to the space variable with constant coefficients. We prove the existence and uniqueness of a solution of the problem in a class of quasipolynomials of the special form. We construct a solution of this problem with the use of the differential-symbol method. In the case of existence of a nonunique solution of the problem, we propose formulas for the construction of a particular solution of the problem.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
I. A. Belavin, S. P. Kapitsa, and S. P. Kurdyumov, “A mathematical model of global demographic processes with regard for a space distribution,” Zh. Vychisl. Mat. Mat. Fiz., 38, No. 6, 885–902 (1998).
V. M. Borok and E. Kengne, “Classification of integral boundary-value problems in a wide strip,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., No. 5, 3–12 (1994).
V. M. Vihak, “Construction of a solution of the heat conduction problem with integral conditions,” Dopov. Nats. Akad. Nauk Ukr., Ser. A, No. 8, 57–60 (1994).
N. D. Golubeva and L. S. Pul’kina, “On one nonlocal problem with integral conditions,” Mat. Zametki, 59, No. 3, 456–458 (1996).
V. S. Il’kiv and T. V. Maherovs’ka, “Problem with integral conditions for an equation with partial derivatives of the second order,” Visn. Nats. Univ. Lviv. Politekh., Ser. Fiz.-Mat. Nauk., No. 625, 12–19 (2008).
N. I. Ionkin, “Solution of one boundary-value problem in the theory of heat conduction with nonlocal boundary conditions,” Differents. Uravn., 13, No. 2, 294–304 (1977).
P. I. Kalenyuk and Z. M. Nytrebych, A Generalized Scheme of Separation of Variables. Differential-Symbol Method [in Ukrainian], “L’vivs’ka Politekhnika” National University, Lviv (2002).
P. I. Kalenyuk, Z. M. Nytrebych, and I. V. Kohut, “On the kernel of the problem with integral condition for an equation with partial derivatives of infinite order,” Visn. Nats. Univ. Lviv. Politekh., Ser. Fiz.-Mat. Nauk., No. 625, 5–11 (2008).
L. I. Kamynin, “On one boundary-value problem of the theory of heat conduction with nonclassical boundary conditions,” Zh. Vychisl. Mat. Mat. Fiz., 4, No. 6, 1006–1024 (1964).
A. I. Kozhanov and L. S. Pul’kina, “Boundary-value problems with integral boundary conditions for multidimensional hyperbolic equations,” Dokl. Ros. Akad. Nauk, 404, No. 5, 589–592 (2005).
O. M. Medvid’ and M. M. Symotyuk, “Integral problem for linear equations with partial derivatives,” Mat. Stud., 28, No. 2, 115–140 (2007).
A. M. Nakhushev, “On one approximate method for the solution of boundary-value problems for differential equations and its application to the dynamics of soil moisture and subterranean waters,” Differents. Uravn., 18, No. 1, 72–81 (1982).
Z. A. Nakhusheva, “On one nonlocal problem for a partial differential equation,” Differents. Uravn., 22, No. 1, 171–174 (1986).
A. Yu. Popov and I. V. Tikhonov, “Exponential classes of uniqueness in problems of heat conduction,” Dokl. Ros. Akad. Nauk, 389, No. 4, 465–467 (2003).
A. Yu. Popov and I. V. Tikhonov, “Exponential classes of solvability in the problem of heat conduction with nonlocal condition of time mean,” Mat. Sb., 196, No. 9, 71–102 (2005).
B. I. Ptashnyk, V. S. Il’kiv, I. Ya. Kmit’, and V. M. Polishchuk, Nonlocal Boundary-Value Problems for Partial Differential Equations [in Ukrainian], Naukova Dumka, Kyiv (2002).
L. S. Pul’kina, “Nonlocal problem with integral conditions for a hyperbolic equation,” Differents. Uravn., 40, No. 7, 887–892 (2004).
A. A. Samarskii, “Some problems of the theory of differential equations,” Differents. Uravn., 16, No. 11, 1925–1935 (1980).
I. V. Tikhonov, “Theorems of uniqueness in linear nonlocal problems for abstract differential equations,” Izv. Ros. Akad. Nauk, Ser. Mat., 67, No. 2, 133–166 (2003).
L. V. Fardigola, “Integral boundary-value problem in a layer,” Mat. Zametki, 53, Issue 6, 122–129 (1993).
L. V. Fardigola, “Test for propriety in a layer of a boundary problem with integral condition,” Ukr. Mat. Zh., 42, No. 11, 1546–1551 (1990); English translation: Ukr. Math. J., 42, No. 11, 1388–1394 (1990).
A. Bouziani, “Initial boundary-value problems for a class of pseudoparabolic equations with integral boundary conditions,” J. Math. Anal. Appl., 291, 371–386 (2004).
A. Bouziani and N.-E. Benouar, “Mixed problem with integral conditions for a third-order parabolic equation,” Kobe J. Math., 15, No. 1, 47–58 (1998).
J. R. Cannon, “The solution of the heat equation. Subject to the specification of energy,” Quart. Appl. Math., 21, 155–160 (1963).
M. Ivanchov, Inverse Problems for Equations of Parabolic Type, VNTL, Lviv (2003).
Author information
Authors and Affiliations
Additional information
Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 53, No. 4, pp. 7–16, October–December, 2010.
Rights and permissions
About this article
Cite this article
Kalenyuk, P.I., Kohut, I.V. & Nytrebych, Z.M. Problem with integral condition for a partial differential equation of the first order with respect to time. J Math Sci 181, 293–304 (2012). https://doi.org/10.1007/s10958-012-0685-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-012-0685-7