A REGULARIZATION METHOD FOR BACKWARD PROBLEMS OF SINGULARLY PERTURBED PARABOLIC AND FRACTIONAL DIFFUSION EQUATIONS

Jin Wen; Yun-Long Liu; Xue-Juan Ren; Donal O'Regan

doi:10.11948/20240424

2025 Volume 15 Issue 4

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2025 > 15(4): 2212-2237

Next Article Previous Article

Jin Wen, Yun-Long Liu, Xue-Juan Ren, Donal O'Regan. A REGULARIZATION METHOD FOR BACKWARD PROBLEMS OF SINGULARLY PERTURBED PARABOLIC AND FRACTIONAL DIFFUSION EQUATIONS[J]. Journal of Applied Analysis & Computation, 2025, 15(4): 2212-2237. doi: 10.11948/20240424

Citation:

Jin Wen, Yun-Long Liu, Xue-Juan Ren, Donal O'Regan. A REGULARIZATION METHOD FOR BACKWARD PROBLEMS OF SINGULARLY PERTURBED PARABOLIC AND FRACTIONAL DIFFUSION EQUATIONS[J]. Journal of Applied Analysis & Computation, 2025, 15(4): 2212-2237. doi: 10.11948/20240424

A REGULARIZATION METHOD FOR BACKWARD PROBLEMS OF SINGULARLY PERTURBED PARABOLIC AND FRACTIONAL DIFFUSION EQUATIONS

1.
Department of Mathematics, Northwest Normal University, Lanzhou 730070, China
2.
College of Information Engineering, Tarim University, No. 1487 Tarim Avenue East, Alar 843300, Xinjiang, China
3.
School of Mathematical and Statistical Sciences, University of Galway, Ireland

Author Bio: Email: liuyunlong0103@163.com(Y.-L. Liu); Email: renxuejuan1010@163.com(X.-J. Ren); Email: donal.oregan@universityofgalway.ie(D. O'Regan)
Corresponding author: Email: wenj@nwnu.edu.cn/wenjin0421@163.com(J. Wen)

Abstract

Abstract

In this paper, backward problems of singularly perturbed parabolic and fractional diffusion equations are studied from the additional temperature data at fixed time $ t=T$. We analyze the ill-posedness of these two inverse problems, and apply the quasi-reversibility regularization method to solve these problems. Then we obtain the convergence rates of logarithmic and Hölder types for the backward problems. Finally, several one- and two-dimensional numerical examples are given to verify the effectiveness and feasibility of the proposed method.
- Singularly perturbed equations /
- backward problems /
- quasi-reversibility method /
- convergence rates
MSC: 35R25, 65M30

FullText(HTML)

References

[1]	A. Y. Bomba and O. A. Fursachik, Inverse singularly perturbed problems of convection-diffusion type in quadrangular curvilinear domains, Mat. Metodi Fiz.-Mekh. Polya, 2009, 52(3), 59–66. Google Scholar
[2]	X. Cai, D. L. Cai, R. Q. Wu and K. H. Xie, High accuracy non-equidistant method for singular perturbation reaction-diffusion problem, Applied Mathematics and Mechanics, 2009, 30, 175–182. doi: 10.1007/s10483-009-0205-8 CrossRef Google Scholar
[3]	P. Carter, A stabilizing effect of advection on planar interfaces in singularly perturbed reaction-diffusion equations, SIAM J. Appl. Math., 2024, 84(3), 1227–1253. doi: 10.1137/23M1610872 CrossRef Google Scholar
[4]	D. Chaikovskii, A. Liubavin and Y. Zhang, Asymptotic expansion regularization for inverse source problems in two-dimensional singularly perturbed nonlinear parabolic PDEs, CSIAM Trans. Appl. Math., 2023, 4(4), 721–757. doi: 10.4208/csiam-am.SO-2022-0017 CrossRef Google Scholar
[5]	D. Chaikovskii and Y. Zhang, Convergence analysis for forward and inverse problems in singularly perturbed time-dependent reaction-advection-diffusion equations, J. Comput. Phys., 2022, 470, Paper No. 111609, 32. Google Scholar
[6]	J. Dardé, Iterated quasi-reversibility method applied to elliptic and parabolic data completion problems, Inverse Probl. Imaging, 2016, 10(2), 379–407. doi: 10.3934/ipi.2016005 CrossRef Google Scholar
[7]	A. M. Denisov, Approximate solution of an inverse problem for a singularly perturbed integro-differential heat equation, Comput. Math. Math. Phys., 2023, 63(5), 837–844. doi: 10.1134/S0965542523050081 CrossRef Google Scholar
[8]	A. M. Denisov, Approximation of the solution of an inverse problem for a singularly perturbed system of partial differential equations, Differ. Equ., 2023, 59(6), 762–768. doi: 10.1134/S0012266123060058 CrossRef Google Scholar
[9]	J. R. Dorroh and X. P. Ru, The application of the method of quasi-reversibility to the sideways heat equation, J. Math. Anal. Appl., 1999, 236(2), 503–519. doi: 10.1006/jmaa.1999.6462 CrossRef Google Scholar
[10]	N. V. Duc, N. V. Thang and N. T. Thành, The quasi-reversibility method for an inverse source problem for time-space fractional parabolic equations, J. Differential Equations, 2023, 344, 102–130. doi: 10.1016/j.jde.2022.10.029 CrossRef Google Scholar
[11]	L. Eldén, Approximations for a Cauchy problem for the heat equation, Inverse Problems, 1987, 3, 263–273. doi: 10.1088/0266-5611/3/2/009 CrossRef Google Scholar
[12]	S. Haber, Boundary Value Problem for a Singularly Perturbed Differential Equation, ProQuest LLC, Ann Arbor, MI, Thesis (Ph. D.)–Massachusetts Institute of Technology, 1955. Google Scholar
[13]	S. Haber and N. Levinson, A boundary value problem for a singularly perturbed differential equation, Proc. Amer. Math. Soc., 1955, 6, 866–872. doi: 10.1090/S0002-9939-1955-0074634-3 CrossRef Google Scholar
[14]	S. I. Kabanikhin, Inverse and Ill-posed Problems: Theory and Applications, Walter de Gruyter GmbH & Co. KG, Berlin, 2012, vol. 55, xvi+459. Google Scholar
[15]	S. Karasuljić and H. Ljevaković, On construction of a global numerical solution for a semilinear singularly-perturbed reaction diffusion boundary value problem, Mat. Bilten, 2020, 44(2), 131–148. Google Scholar
[16]	A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, in: North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, 2006, vol. 204, p. xvi+523. Google Scholar
[17]	N. Kumar, S. Toprakseven, N. Singh Yadav and J. Y. Yuan, A Crank-Nicolson WG-FEM for unsteady 2D convection-diffusion equation with nonlinear reaction term on layer adapted mesh, Appl. Numer. Math., 2024, 201, 322–346. doi: 10.1016/j.apnum.2024.03.013 CrossRef Google Scholar
[18]	R. Lattès and J.-L. Lions, The Method of Quasi-Reversibility. Applications to Partial Differential Equations, Modern Analytic and Computational Methods in Science and Mathematics, American Elsevier Publishing Co., Inc., New York, 1969. Google Scholar
[19]	T. T. Le and L. H. Nguyen, T.-P. Nguyen and W. Powell, The quasi-reversibility method to numerically solve an inverse source problem for hyperbolic equations, J. Sci. Comput., 2021, 87(3), Paper No. 90, 23. Google Scholar
[20]	Y. S. Li and T. Wei, An inverse time-dependent source problem for a time-space fractional diffusion equation, Appl. Math. Comput., 2018, 336, 257–271. Google Scholar
[21]	T. Linss, Maximum-norm error analysis of a non-monotone FEM for a singularly perturbed reaction-diffusion problem, BIT, 2007, 47(2), 379–391. doi: 10.1007/s10543-007-0118-z CrossRef Google Scholar
[22]	G. Q. Liu and Y. C. Su, A uniformly convergent difference scheme for the singular perturbation problem of a high order elliptic differential equation, Applied Mathematics and Mechanics, 1996, 17(5), 413-421. doi: 10.1007/BF00131089 CrossRef Google Scholar
[23]	D. V. Lukyanenko, A. A. Borzunov and M. A. Shishlenin, Solving coefficient inverse problems for nonlinear singularly perturbed equations of the reaction-diffusion-advection type with data on the position of a reaction front, Commun. Nonlinear Sci. Numer. Simul., 2021, 99, Paper No. 105824, 10. Google Scholar
[24]	D. V. Lukyanenko, V. B. Grigorev, V. T. Volkov and M. A. Shishlenin, Solving of the coefficient inverse problem for a nonlinear singularly perturbed two-dimensional reaction-diffusion equation with the location of moving front data, Comput. Math. Appl., 2019, 77(5), 1245–1254. doi: 10.1016/j.camwa.2018.11.005 CrossRef Google Scholar
[25]	D. V. Lukyanenko, M. A. Shishlenin and V. T. Volkov, Solving of the coefficient inverse problems for a nonlinear singularly perturbed reaction-diffusion-advection equation with the final time data, Commun. Nonlinear Sci. Numer. Simul., 2018, 54, 233–247. doi: 10.1016/j.cnsns.2017.06.002 CrossRef Google Scholar
[26]	D. V. Lukyanenko, M. A. Shishlenin and V. T. Volkov, Asymptotic analysis of solving an inverse boundary value problem for a nonlinear singularly perturbed time-periodic reaction-diffusion-advection equation, J. Inverse Ill-Posed Probl., 2019, 27(5), 745–758. doi: 10.1515/jiip-2017-0074 CrossRef Google Scholar
[27]	D. V. Lukyanenko, V. T. Volkov, N. N. Nefedov, L. Recke and K. Schneider, Analytic-numerical approach to solving singularly perturbed parabolic equations with the use of dynamic adapted meshes, Model. Anal. Inf. Sist., 2016, 23(3), 334–341. doi: 10.18255/1818-1015-2016-3-334-341 CrossRef Google Scholar
[28]	Z. Mao and L. B. Liu, A moving grid algorithm for a strongly coupled system of singularly perturbed convection-diffusion problems, Math. Appl. (Wuhan), 2018, 31(3), 653–660. Google Scholar
[29]	M. A. Mohye, J. B. Munyakazi and T. G. Dinka, A second order numerical method for two-parameter singularly perturbed time-delay parabolic problems, J. Math. Model., 2023, 11(4), 745–766. Google Scholar
[30]	I. Podlubny, Fractional Differential Equations, An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and Some of their Applications, in: Mathematics in Science and Engineering, Academic Press, Inc., San Diego, CA, 1999, vol. 198, p. xxiv+340. Google Scholar
[31]	Z. Qian, C.-L. Fu and R. Shi, A modified method for a backward heat conduction problem, Appl. Math. Comput., 2007, 185(1), 564–573. Google Scholar
[32]	A. Seal, S. Natesan and S. Toprakseven, A dimensional-splitting weak Galerkin finite element method for 2d time-fractional diffusion equation, J. Sci. Comput., 2024, 98(3), Paper No. 56, 26. Google Scholar
[33]	G. I. Shishkin and L. P. Shishkina, A stable standard difference scheme for a singularly perturbed convection-diffusion equation under computer perturbations, Tr. Inst. Mat. Mekh., 2014, 20(1), 322–333. Google Scholar
[34]	C. M. Song, H. Li and J. N. Li, Initial boundary value problem for the singularly perturbed Boussinesq-type equation, Discrete Contin. Dyn. Syst., 2013, 709–717. Google Scholar
[35]	S. Toprakseven, A weak Galerkin finite element method for time fractional reaction-diffusion-convection problems with variable coefficients, Appl. Numer. Math., 2021, 168, 1–12. doi: 10.1016/j.apnum.2021.05.021 CrossRef Google Scholar
[36]	S. Toprakseven and S. Dinibutun, A weak Galerkin finite element method for parabolic singularly perturbed convection-diffusion equations on layer-adapted meshes, Electron. Res. Arch., 2024, 32(8), 5033–5066. doi: 10.3934/era.2024232 CrossRef Google Scholar
[37]	S. Toprakseven and N. Srinivasan, An efficient weak Galerkin FEM for third-order singularly perturbed convection-diffusion differential equations on layer-adapted meshes, Appl. Numer. Math., 2024, 204, 130–146. doi: 10.1016/j.apnum.2024.06.009 CrossRef Google Scholar
[38]	R. Vulanović and L. Teofanov, A modification of the Shishkin discretization mesh for one-dimensional reaction diffusion problems, Applied Mathematics and Computation, 2013, 220, 104–116. doi: 10.1016/j.amc.2013.05.055 CrossRef Google Scholar
[39]	J. G. Wang and T. Wei, An iterative method for backward time-fractional diffusion problem, Numer. Methods Partial Differential Equations, 2014, 30(6), 2029–2041. doi: 10.1002/num.21887 CrossRef Google Scholar
[40]	J. G. Wang and T. Wei, Quasi-reversibility method to identify a space-dependent source for the time-fractional diffusion equation, Appl. Math. Model., 2015, 39(20), 6139–6149. doi: 10.1016/j.apm.2015.01.019 CrossRef Google Scholar
[41]	J. Wen, L. M. Huang and Z. X. Liu, A modified quasi-reversibility method for inverse source problem of Poisson equation, Inverse Probl. Sci. Eng., 2021, 29(12), 2098–2109. doi: 10.1080/17415977.2021.1902516 CrossRef Google Scholar
[42]	J. Wen, Z. Y. Li and Y. P. Wang, Solving the backward problem for time-fractional wave equations by the quasi-reversibility regularization method, Adv. Comput. Math., 2023, 49(6), Paper No. 80, 26. Google Scholar
[43]	J. Wen, Y. L. Liu and D. O'Regan, An efficient iterative quasi-reversibility method for the inverse source problem of time-fractional diffusion equations, Numerical Heat Transfer Part B-Fundamentals, 2024. Google Scholar
[44]	J. Wen, X. J. Ren and S. J. Wang, Simultaneous determination of source term and initial value in the heat conduction problem by modified quasi-reversibility regularization method, Numerical Heat Transfer Part B-Fundamentals, 2022, 82(3–4), 112–127. Google Scholar
[45]	J. Wen, X. J. Ren and S. J. Wang, Simultaneous determination of source term and the initial value in the space-fractional diffusion problem by a novel modified quasi-reversibility regularization method, Physica Scripta, 2023, 98(2), 025201. doi: 10.1088/1402-4896/acaa68 CrossRef Google Scholar
[46]	J. Wen, C. W. Yue, Z. X. Liu and D. O'Regan, A fractional Landweber iteration method for simultaneous inversion in a time-fractional diffusion equation, J. Appl. Anal. Comput., 2023, 13(6), 3374–3402. Google Scholar
[47]	F. Yang and C. L. Fu, Two regularization methods to identify time-dependent heat source through an internal measurement of temperature, Math. Comput. Modelling, 2011, 53(5–6), 793–804. Google Scholar
[48]	F. Yang and C. L. Fu, A mollification regularization method for the inverse spatial-dependent heat source problem, J. Comput. Appl. Math., 2014, 255, 555–567. doi: 10.1016/j.cam.2013.06.012 CrossRef Google Scholar
[49]	Z. Q. Zhang and T. Wei, Identifying an unknown source in time-fractional diffusion equation by a truncation method, Applied Mathematics and Computation, 2013, 219(11), 5972–5983. doi: 10.1016/j.amc.2012.12.024 CrossRef Google Scholar