Research article Special Issues

An inverse source problem for a pseudoparabolic equation with memory

  • Received: 28 January 2024 Revised: 10 April 2024 Accepted: 11 April 2024 Published: 18 April 2024
  • MSC : 35R30, 35K10, 35A09, 35A01, 35A02

  • This paper is devoted to investigating the well-posedness, as well as performing the numerical analysis, of an inverse source problem for linear pseudoparabolic equations with a memory term. The investigated inverse problem involves determining a right-hand side that depends on the spatial variable under the given observation at a final time along with the solution function. Under suitable assumptions on the problem data, the existence, uniqueness and stability of a strong generalized solution of the studied inverse problem are obtained. In addition, the pseudoparabolic problem is discretized using extended cubic B-spline functions and recast as a nonlinear least-squares minimization of the Tikhonov regularization function. Numerically, this problem is effectively solved using the MATLAB subroutine lsqnonlin. Both exact and noisy data are inverted. Numerical results for a benchmark test example are presented and discussed. Moreover, the von Neumann stability analysis is also discussed.

    Citation: M. J. Huntul, Kh. Khompysh, M. K. Shazyndayeva, M. K. Iqbal. An inverse source problem for a pseudoparabolic equation with memory[J]. AIMS Mathematics, 2024, 9(6): 14186-14212. doi: 10.3934/math.2024689

    Related Papers:

  • This paper is devoted to investigating the well-posedness, as well as performing the numerical analysis, of an inverse source problem for linear pseudoparabolic equations with a memory term. The investigated inverse problem involves determining a right-hand side that depends on the spatial variable under the given observation at a final time along with the solution function. Under suitable assumptions on the problem data, the existence, uniqueness and stability of a strong generalized solution of the studied inverse problem are obtained. In addition, the pseudoparabolic problem is discretized using extended cubic B-spline functions and recast as a nonlinear least-squares minimization of the Tikhonov regularization function. Numerically, this problem is effectively solved using the MATLAB subroutine lsqnonlin. Both exact and noisy data are inverted. Numerical results for a benchmark test example are presented and discussed. Moreover, the von Neumann stability analysis is also discussed.



    加载中


    [1] A. B. Al'shin, M. O. Korpusov, A. G. Siveshnikov, Blow-up in nonlinear Sobolev type equations, De Gruyter Series in Nonlinear Analysis and Applications, 2011. https://doi.org/10.1515/9783110255294
    [2] S. N. Antontsev, J. I. Diaz, S. Shmarev, Energy methods for free boundary problems: Applications to nonlinear PDEs and fluid mechanics, Progress in Nonlinear Differential Equations and their Applications 48, Birkhäuser, 2002. https://doi.org/10.1115/1.1483358
    [3] M. Amin, M. Abbas, D. Baleanu, M. K. Iqbal, M. B. Riaz, Redefined extended cubic B-spline functions for numerical solution of time-fractional telegraph equation, CMES Comp. Model. Eng., 127 (2021), 361–384. https://doi.org/10.32604/cmes.2021.012720 doi: 10.32604/cmes.2021.012720
    [4] M. Amin, M. Abbas, M. K. Iqbal, D. Baleanu, Numerical treatment of time-fractional Klein-Gordon equation using redefined extended cubic B-spline functions, Front. Phys., 8 (2020), 288. https://doi.org/10.3389/fphy.2020.00288 doi: 10.3389/fphy.2020.00288
    [5] S. N. Antontsev, S. E. Aitzhanov, G. R. Ashurova, An inverse problem for the pseudo-parabolic equation with p-Laplacian, EECT, 11 (2022), 399–414. https://doi.org/10.3934/eect.2021005 doi: 10.3934/eect.2021005
    [6] A. Asanov, E. R. Atamanov, Nonclassical and inverse problems for pseudoparabolic equations, De Gruyter, Berlin, 1997. https://doi.org/10.1515/9783110900149
    [7] E. S. Dzektser, Generalization of the equation of motion of ground waters with free surface, Dokl. Akad. Nauk SSSR, 202 (1972), 1031–1033.
    [8] V. E. Fedorov, A. V. Urasaeva, An inverse problem for linear Sobolev type equations, J. Inverse III-Pose. P., 12 (2004), 387–395. https://doi.org/10.1163/1569394042248210 doi: 10.1163/1569394042248210
    [9] M. Gholamian, J. Saberi-Nadjafi, Cubic B-splines collocation method for a class of partial integro-differential equation, Alex. Eng. J., 57 (2018), 2157–2165. https://doi.org/10.1016/j.aej.2017.06.004 doi: 10.1016/j.aej.2017.06.004
    [10] H. A. Hammad, H. U. Rehman, H. Almusawa, Tikhonov regularization terms for accelerating inertial Mann-Like algorithm with applications, Symmetry, 13 (2021), 554. https://doi.org/10.3390/sym13040554 doi: 10.3390/sym13040554
    [11] M. J. Huntul, M. Tamsir, N. Dhiman, Identification of time-dependent potential in a fourth-order pseudo-hyperbolic equation from additional measurement, Math. Method. Appl. Sci., 45 (2022), 5249–5266. https://doi.org/10.1002/mma.8104 doi: 10.1002/mma.8104
    [12] M. J. Huntul, N. Dhiman, M. Tamsir, Reconstructing an unknown potential term in the third-order pseudo-parabolic problem, Comput. Appl. Math., 40 (2021), 140.
    [13] M. J. Huntul, M. Tamsir, N. Dhiman, An inverse problem of identifying the time-dependent potential in a fourth-order pseudo-parabolic equation from additional condition, Numer. Meth. Part. D. E., 39 (2023), 848–865. https://doi.org/10.1002/num.22778 doi: 10.1002/num.22778
    [14] M. J. Huntul, Identifying an unknown heat source term in the third-order pseudo-parabolic equation from nonlocal integral observation, Int. Commun. Heat Mass, 128 (2021), 105550. https://doi.org/10.1016/j.icheatmasstransfer.2021.105550 doi: 10.1016/j.icheatmasstransfer.2021.105550
    [15] M. J. Huntul, Recovering a source term in the higher-order pseudo-parabolic equation via cubic spline functions, Phys. Scr., 97 (2022), 035004. https://doi.org/10.1088/1402-4896/ac54d0 doi: 10.1088/1402-4896/ac54d0
    [16] K. Kenzhebai, An inverse problem of recovering the right hand side of 1D pseudoparabolic equation, JMCS, 111 (2021), 28–37. https://doi.org/10.26577/JMMCS.2021.v111.i3.03 doi: 10.26577/JMMCS.2021.v111.i3.03
    [17] K. Khompysh, Inverse problem for 1D pseudo-parabolic equation, FAIA, 216 (2017), 382–387. https://doi.org/10.1007/978-3-319-67053-9_36 doi: 10.1007/978-3-319-67053-9_36
    [18] K. Khompysh, A. G. Shakir, The inverse problem for determining the right part of the pseudo-parabolic equation, JMCS, 105 (2020), 87–98. https://doi.org/10.26577/JMMCS.2020.v105.i1.08 doi: 10.26577/JMMCS.2020.v105.i1.08
    [19] N. Khalid, M. Abbas, M. K. Iqbal, D. Baleanu, A numerical investigation of Caputo time fractional Allen-Cahn equation using redefined cubic B-spline functions, Adv. Differ. Equ., 2020 (2020), 1–22. https://doi.org/10.1186/s13662-020-02616-x doi: 10.1186/s13662-020-02616-x
    [20] A. Y. Kolesov, E. F. Mishchenko, N. K. Rozov, Asymptotic methods of investigation of periodic solutions of nonlinear hyperbolic equations, T. Mat. I. Imeni V.A.S., 222 (1998), 3–191.
    [21] M. O. Korpusov, A. G. Sveshnikov, Three-dimensional nonlinear evolution equations of pseudoparabolic type in problems of mathematical physics, Z. V. Mat. I Mat. F., 43 (2003), 1835–1869.
    [22] J. L. Lions, Quelques methodes de resolution des problemes aux limites non-liniaires, Paris, Dunod, 1969.
    [23] A. S. Lyubanova, Inverse problem for a pseudoparabolic equation with integral overdetermination conditions, Differ. Equ., 50 (2014), 502–512. https://doi.org/10.1134/S0012266114040089 doi: 10.1134/S0012266114040089
    [24] Mathworks, Documentation optimization toolbox-least squares (model fitting) algorithms, 2020. Available from: www.mathworks.com.
    [25] Y. T. Mehraliyev, G. K. Shafiyeva, Determination of an unknown coefficient in the third order pseudoparabolic equation with non-self-adjoint boundary conditions, J. Appl. Math., 2014 (2014), 1–7. https://doi.org/10.1155/2014/358696 doi: 10.1155/2014/358696
    [26] Y. T. Mehraliyev, A. T. Ramazanova, M. J. Huntul, An inverse boundary value problem for a two-dimensional pseudo-parabolic equation of third order, Results Appl. Math., 14 (2022), 100274. https://doi.org/10.1016/j.rinam.2022.100274 doi: 10.1016/j.rinam.2022.100274
    [27] N. Mshary, Exploration of nonlinear traveling wave phenomena in quintic conformable Benney-Lin equation within a liquid film, AIMS Math., 9 (2024), 11051–11075. https://doi.org/10.3934/math.2024542 doi: 10.3934/math.2024542
    [28] A. P. Oskolkov, Uniqueness and global solvability for boundary value problems for the equations of motion of water solutions of polymers, Zap. Nauchn. Sem. POMI, 38 (1973), 98–136.
    [29] A. I. Prilepko, D. G. Orlovsky, I. A. Vasin, Methods for solving inverse problems in mathematical physics, Marcel Dekker, New York, Basel, 2000.
    [30] P. Rosenau, Evolution and breaking of ion-acoustic waves, Phys. Fluids, 31 (1988), 1317–1319. https://doi.org/10.1063/1.866723 doi: 10.1063/1.866723
    [31] W. Rundell, Determination of an unknown nonhomogeneous term in a linear partial differential equation from overspecified boundary data, Appl. Anal., 10 (1980), 231–242. https://doi.org/10.1080/00036818008839304 doi: 10.1080/00036818008839304
    [32] B. K. Shivamoggi, A symmetric regularized long‐wave equation for shallow water waves, Phys. Fluids, 29 (1986), 890–891. https://doi.org/10.1063/1.865895 doi: 10.1063/1.865895
    [33] R. E. Showalter, T. W. Ting, Pseudoparabolic partial differential equations, SIAM, 1 (1970), 1–26. https://doi.org/10.1137/0501001 doi: 10.1137/0501001
    [34] M. Tamsir, D. Nigam, N. Dhiman, A. Chauhan, A hybrid B-spline collocation technique for the Caputo time fractional nonlinear Burgers' equation, Beni-Suef U. J. Basic, 12 (2023), 95. https://doi.org/10.1186/s43088-023-00434-0 doi: 10.1186/s43088-023-00434-0
    [35] V. G. Zvyagin, M. V. Turbin, The study of initial-boundary value problems for mathematical models of the motion of Kelvin-Voigt fluids, J. Math. Sci., 168 (2012), 157–308. https://doi.org/10.1007/s10958-010-9981-2 doi: 10.1007/s10958-010-9981-2
    [36] H. Zhang, X. Han, X. Yang, Quintic B-spline collocation method for fourth order partial integro-differential equations with a weakly singular kernel, Appl. Math. Comput., 219 (2013), 6565–6575. https://doi.org/10.1016/j.amc.2013.01.012 doi: 10.1016/j.amc.2013.01.012
  • Reader Comments
  • © 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(654) PDF downloads(69) Cited by(0)

Article outline

Figures and Tables

Figures(5)  /  Tables(1)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog