Applications of Mathematics, Vol. 67, No. 5, pp. 573-592, 2022
Application of Rothe's method to a parabolic inverse problem with nonlocal boundary condition
Yong-Hyok Jo, Myong-Hwan Ri
Received February 9, 2021. Published online October 19, 2021.
Abstract: We consider an inverse problem for the determination of a purely time-dependent source in a semilinear parabolic equation with a nonlocal boundary condition. An approximation scheme for the solution together with the well-posedness of the problem with the initial value $u_0\in H^1(\Omega)$ is presented by means of the Rothe time-discretization method. Further approximation scheme via Rothe's method is constructed for the problem when $u_0\in L^2(\Omega)$ and the integral kernel in the nonlocal boundary condition is symmetric.
References: [1] E. I. Azizbayov: The nonlocal inverse problem of the identification of the lowest coefficient and the right-hand side in a second-order parabolic equation with integral conditions. Bound. Value Probl. 2019 (2019), Article ID 11, 19 pages. DOI 10.1186/s13661-019-1126-z | MR 3900856
[2] D. Bahuguna, V. Raghavendra: Application of Rothe's method to nonlinear integro-differential equations in Hilbert spaces. Nonlinear Anal., Theory Methods Appl. 23 (1994), 75-81. DOI 10.1016/0362-546X(94)90252-6 | MR 1288499 | Zbl 0810.34054
[3] V. Buda, R. Chegis, M. Sapagovas: A model of multiple diffusion from a limited source. Differ. Uravn. Primen. 38 (1986), 9-14. (In Russian.) Zbl 0621.76097
[4] S. Carl, V. Lakshmikantham: Generalized quasilinearization method for reaction-diffusion equations under nonlinear and nonlocal flux conditions. J. Math. Anal. Appl. 271 (2002), 182-205. DOI 10.1016/S0022-247X(02)00114-2 | MR 1923755 | Zbl 1010.65041
[5] A. Chaoui, A. Guezane-Lakoud: Solution to an integrodifferential equation with integral condition. Appl. Math. Comput. 266 (2015), 903-908. DOI 10.1016/j.amc.2015.06.004 | MR 3377607 | Zbl 1410.65354
[6] M. R. Cui: Convergence analysis of compact difference schemes for diffusion equation with nonlocal boundary conditions. Appl. Math. Comput. 260 (2015), 227-241. DOI 10.1016/j.amc.2015.03.039 | MR 3343264 | Zbl 1410.65304
[7] D. S. Daoud: Determination of the source parameter in a heat equation with a non-local boundary condition. J. Comput. Appl. Math. 221 (2008), 261-272. DOI 10.1016/j.cam.2007.10.060 | MR 2458768 | Zbl 1152.65096
[8] W. A. Day: A decreasing property of solutions of parabolic equations with applications to thermoelasticity. Q. Appl. Math. 40 (1983), 468-475. DOI 10.1090/qam/693879 | MR 0693879 | Zbl 0514.35038
[9] R. H. De Staelen, M. Slodička: Reconstruction of a convolution kernel in a semilinear parabolic problem based on a global measurement. Nonlinear Anal., Theory Methods Appl., Ser. A 112 (2015), 43-57. DOI 10.1016/j.na.2014.09.002 | MR 3274282 | Zbl 1302.35435
[10] D. Glotov, W. E. Hames, A. J. Meir, S. Ngoma: An integral constrained parabolic problem with applications in thermochronology. Comput. Math. Appl. 71 (2016), 2301-2312. DOI 10.1016/j.camwa.2016.01.017 | MR 3501321 | Zbl 1443.35053
[11] D. Glotov, W. E. Hames, A. J. Meir, S. Ngoma: An inverse diffusion coefficient problem for a parabolic equation with integral constraint. Int. J. Numer. Anal. Model. 15 (2018), 552-563. MR 3789578 | Zbl 1395.35103
[12] J. Kačur: Method of Rothe in Evolution Equations. Teubner Texte zur Mathematik 80. Teubner, Leipzig (1985). MR 0834176 | Zbl 0582.65084
[13] A. I. Kozhanov: On the solvability of a boundary-value problem with a non-local boundary condition for linear parabolic equations. Vestn. Samar. Gos. Tekh. Univ., Ser. Fiz.-Mat. Nauki 30 (2004), 63-69. (In Russian.) DOI 10.14498/vsgtu308
[14] N. Merazga, A. Bouziani: On a time-discretization method for a semilinear heat equation with purely integral conditions in a nonclassical function space. Nonlinear Anal., Theory Methods Appl., Ser. A 66 (2007), 604-623. DOI 10.1016/j.na.2005.12.005 | MR 2274872 | Zbl 1105.35044
[15] J. Nečas: Direct Methods in the Theory of Elliptic Equations. Springer Monographs in Mathematics. Springer, Berlin (2012). DOI 10.1007/978-3-642-10455-8 | MR 3014461 | Zbl 1246.35005
[16] A. I. Prilepko, D. G. Orlovsky, I. A. Vasin: Methods for Solving Inverse Problems in Mathematical Physics. Pure and Applied Mathematics, Marcel Dekker 231. Marcel Dekker, New York (2000). DOI 10.1201/9781482292985 | MR 1748236 | Zbl 0947.35173
[17] K. Rektorys: The Method of Discretization in Time and Partial Differential Equations. Mathematics and Its Applications (East European Series) 4. Reidel Publishing, Dordrecht (1982). MR 0689712 | Zbl 0505.65029
[18] R. E. Showalter: Monotone Operators in Banach Space and Nonlinear Partial Differential Equations. Mathematical Surveys and Monographs 49. American Mathematical Society, Providence (1997). DOI 10.1090/surv/049 | MR 1422252 | Zbl 0870.35004
[19] M. Slodička: Recovery of an unknown flux in parabolic problems with nonstandard boundary conditions: Error estimates. Appl. Math., Praha 48 (2003), 49-66. DOI 10.1023/A:1022954920827 | MR 1954503 | Zbl 1099.65081
[20] M. Slodička: Semilinear parabolic problems with nonlocal Dirichlet boundary conditions. Inverse Probl. Sci. Eng. 19 (2011), 705-716. DOI 10.1080/17415977.2011.579608 | MR 2819541 | Zbl 1239.65059
[21] K. Van Bockstal, R. H. De Staelen, M. Slodička: Identification of a memory kernel in a semilinear integrodifferential parabolic problem with applications in heat conduction with memory. J. Comput. Appl. Math. 289 (2015), 196-207. DOI 10.1016/j.cam.2015.02.019 | MR 3350770 | Zbl 1319.35305
[22] H.-M. Yin: On a class of parabolic equations with nonlocal boundary conditions. J. Math. Anal. Appl. 294 (2004), 712-728. DOI 10.1016/j.jmaa.2004.03.021 | MR 2061353 | Zbl 1060.35057
Affiliations: Yong-Hyok Jo (corresponding author), Department of Applied Mathematics, Kim Chaek University of Technology, Pyongyang, DPR of Korea, e-mail: jyh73120@star-co.net.kp; Myong-Hwan Ri, Institute of Mathematics, State Academy of Sciences, Pyongyang, DPR of Korea, e-mail: math.inst@star-co.net.kp
