We specify a class of quasipolynomials as a class of unique solvability of the problem with inhomogeneous integral time 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 variables with constant coefficients. In this class, the solution of the problem is represented in the form of action of a differential expression whose symbol is the right-hand side of the integral condition upon a meromorphic function of parameters with subsequent setting these parameters equal to zero. In a wider class of quasipolynomials, namely, in the class of existence of a nonunique solution of the problem, we propose a formula for the construction of its partial solutions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
N. I. Ionkin, “Solution of a boundary-value problem in the theory of heat conduction with nonlocal boundary conditions,” Differents. Uravn., 13, No. 2, 294–304 (1977).
V. S. Il’kiv and T. V. Maherovs’ka, “Problem with integral conditions for a partial differential equation of the second order,” Visn. Nats. Univ. “L’viv. Politekhn.,” Ser. Fiz.-Mat. Nauk., No. 625, 12–19 (2008).
P. I. Kalenyuk, I. V. Kohut, and Z. M. Nytrebych, “Problem with integral condition for a partial differential equation of the first order with respect to time,” Mat. Met. Fiz.-Mekh. Polya, 53, No. 4, 7–16 (2010); English translation: J. Math. Sci., 181, No. 3, 293–304 (2012).
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).
P. I. Kalenyuk, Z. M. Nytrebych, and I. V. Kohut, “On the kernel of a problem with integral condition for a partial differential equation of infinite order,” Visn. Nats. Univ. “L’viv. Politekhn.,” Ser. Fiz.-Mat. Nauky, No. 625, 5–11 (2008).
O. M. Medvid’ and M. M. Symotyuk, “Integral problem for linear partial differential equations,” Mat. Stud., 28, No. 2, 115–140 (2007).
A. M. Nakhushev, Equations of Mathematical Biology [in Russian], Vysshaya Shkola, Moscow (1995).
8 A. Yu. Popov and I. V. Tikhonov, “Exponential classes of solvability in the problem of heat conduction with a nonlocal condition for the time means,” Mat. Sb.., 196, No. 9, 71–102 (2005).
L. S. Pul’kina, “A nonlocal problem with integral conditions for a hyperbolic equation,” Differents. Uravn., 40, No. 7, 887–892. (2004); English translation: Different. Equat., 40, No. 7, 947–953 (2004).
L. V. Fardigola, “Test for the well-posedness of a boundary problem with integral condition in a layer,” 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).
J. R. Cannon, “The solution of the heat equation. Subject to the specification of energy,” Quart. Appl. Math., 21, 155–160 (1963).
D.-Q. Dai and Y. Huang, “On a mixed problem with integral conditions for one-dimensional parabolic equation,” Appl. Anal., 85, No. 9, 1113–1121 (2006).
M. Denche and A. Memou, “Boundary value problem with integral conditions for a linear third-order equation,” J. Appl. Math., No. 11, 553–567 (2003).
M. Ivanchov, Inverse Problems for Equations of Parabolic Type [in Ukrainian], VNTL Publishers, Lviv (2003).
Y. T. Mehraliev and E. I. Azizbekov, “A nonlocal boundary-value problem with integral conditions for second-order hyperbolic equations,” J. Qual. Meas. Anal., 7, No. 1, 27–40 (2011).
S. Mesloub and A. Bouziani, “Mixed problem with integral conditions for a certain class of hyperbolic equations,” J. Appl. Math., 1, No. 3, 107–116 (2001).
H. Molaei, “Optimal control and problem with integral boundary conditions,” Int. J. Contemp. Math. Sci., 6, No. 48, 2385–2390 (2011).
Author information
Authors and Affiliations
Additional information
Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 55, No. 4, pp. 7–15, October–December, 2012.
Rights and permissions
About this article
Cite this article
Kalenyuk, P.І., Kohut, І.V., Nytrebych, Z.М. et al. Problem With Inhomogeneous Integral Time Condition for a Partial Differential Equation of the First Order With Respect to Time and of Infinite Order With Respect to the Space Variables. J Math Sci 198, 1–12 (2014). https://doi.org/10.1007/s10958-014-1768-4
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-014-1768-4