Solution of the boundary-value problem of heat conduction with periodic boundary conditions

F.  Kanca; I.  Baglan

F. Kanca Dept. Comput. Eng., Fenerbahce Univ., Istanbul, Turkey
I. Baglan Kocaeli Univ., Turkey

Keywords: boundary-value problem, periodic boundary conditions

Abstract

UDC 517.9

We investigate the solution of the inverse problem for a linear two-dimensional parabolic equation with periodic boundary and integral overdetermination conditions.
Under certain natural regularity and consistency conditions imposed on the input data, we establish the existence, uniqueness of the solution and its continuous dependence on the data by using the generalized Fourier method.
In addition, an iterative algorithm is constructed for the numerical solution of this problem.

References

Baglan, Irem. Determination of a coefficient in a quasilinear parabolic equation with periodic boundary condition. Inverse Probl. Sci. Eng. 23 (2015), no. 5, 884--900. doi: 10.1080/17415977.2014.947479

Kanca, Fatma. The inverse problem of the heat equation with periodic boundary and integral overdetermination conditions. J. Inequal. Appl. 2013, 2013:108, 9 pp. doi: 10.1186/1029-242X-2013-108

Kanca, Fatma; Baglan, Irem. An inverse coefficient problem for a quasilinear parabolic equation with nonlocal boundary conditions. Bound. Value Probl. 2013, Paper No. 213, 17 pp. doi: 10.1186/1687-2770-2013-213

Hill, G. W. On the part of the motion of the lunar perigee which is a function of the mean motions of the sun and moon. Acta Math. 8 (1886), no. 1, 1--36. doi: 10.1007/BF02417081

Cannon, J. R. The solution of the heat equation subject to the specification of energy. Quart. Appl. Math. 21 1963 155--160. doi: 10.1090/qam/160437

Cannon, J. R. Determination of an unknown heat source from overspecified boundary data. SIAM J. Numer. Anal. 5 (1968), 275--286. doi: 10.1137/0705024

Dou, Fang-Fang; Fu, Chu-Li; Yang, Fan. Identifying an unknown source term in a heat equation. Inverse Probl. Sci. Eng. 17 (2009), no. 7, 901--913. doi: 10.1080/17415970902916870

P. R. Sharma, G. Methi, Solution of two dimensional parabolic equation subject to non-local conditions using homotopy Perturbation method, J. Comput. Sci. Appl., 1, 12 – 16 (2012).

Choi, Y. S.; Chan, Kwong-Yu. A parabolic equation with nonlocal boundary conditions arising from electrochemistry. Nonlinear Anal. 18 (1992), no. 4, 317--331. doi: 10.1016/0362-546X(92)90148-8

Ozbilge, Ebru; Demir, Ali. Inverse problem for a time-fractional parabolic equation. J. Inequal. Appl. 2015, 2015:81, 9 pp. doi: 10.1186/s13660-015-0602-y

Ozbilge, Ebru; Demir, Ali. Identification of unknown coefficient in time fractional parabolic equation with mixed boundary conditions via semigroup approach. Dynam. Systems Appl. 24 (2015), no. 3, 341--348. MR3445817

Dehghan, Mehdi. Finite difference schemes for two-dimensional parabolic inverse problem with temperature overspecification. Int. J. Comput. Math. 75 (2000), no. 3, 339--349. doi: 10.1080/00207160008804989

Dehghan, Mehdi. Implicit locally one-dimensional methods for two-dimensional diffusion with a non-local boundary condition. Math. Comput. Simulation 49 (1999), no. 4-5, 331--349. doi: 10.1016/S0378-4754(99)00056-7

Wang, Shingmin; Lin, Yan Ping. A finite-difference solution to an inverse problem for determining a control function in a parabolic partial differential equation. Inverse Problems 5 (1989), no. 4, 631--640. MR1009043

Mohebbi, Akbar; Abbasi, Masoume. A fourth-order compact difference scheme for the parabolic inverse problem with an overspecification at a point. Inverse Probl. Sci. Eng. 23 (2015), no. 3, 457--478. doi: 10.1080/17415977.2014.922075

R. Zolfaghari, Parameter determination in a parabolic inverse problem in general dimensions, Comput. Methods Different. Equat., 1(1), 55 – 70 (2013).

Mohebbi, Akbar. A numerical algorithm for determination of a control parameter in two-dimensional parabolic inverse problems. Acta Math. Appl. Sin. Engl. Ser. 31 (2015), no. 1, 213--224. doi: 10.1007/s10255-015-0461-9

Ionkin, N. I. Letter to the editors: "The solution of a certain boundary value problem of the theory of heat conduction with a nonclassical boundary condition'' (Differencialʹnye Uravnenija 13 (1977), no. 2, 294–304). (Russian) Differencialʹnye Uravnenija 13 (1977), no. 10, 1903. MR0603292

Īvanchov, M. Ī.; Pabirīvsʹka, N. V. Simultaneous determination of two coefficients in a parabolic equation in the case of nonlocal and integral conditions. (Ukrainian) ; translated from Ukraïn. Mat. Zh. 53 (2001), no. 5, 589--596 Ukrainian Math. J. 53 (2001), no. 5, 674--684 doi: 10.1023/A:1012570031242

G. Ramm, Mathematical and analytical techniques withapplication to engineering, Springer, NewYork (2005).

doi: 10.1007/b100958