×
Triet Le Minh; Tu Tran Quoc; Phong Luu Hong

Recovering initial population density of fractional pseudo-parabolic problem associated with a nonlinear reaction. (English) Zbl 07914779

J. Pseudo-Differ. Oper. Appl. 15, No. 3, Paper No. 56, 31 p. (2024).
Summary: In this paper, we consider an inverse problem related to the fractional pseudo-parabolic equation with a nonlinear source term. Our investigation reveals the ill-posedness of the problem according to Hadamard’s definition. We present two improved variations of the optimal filtering method introduced by T. I. Seidman [SIAM J. Numer. Anal. 33, No. 1, 162–170 (1996; Zbl 0851.65066)] to establish some optimal estimates under some an a priori assumptions on the regularity of the exact solution. Finally, the effectiveness of our algorithm is demonstrated through numerical examples.

MSC:

35R30 Inverse problems for PDEs
35K55 Nonlinear parabolic equations
35K70 Ultraparabolic equations, pseudoparabolic equations, etc.
35R11 Fractional partial differential equations
47J06 Nonlinear ill-posed problems
47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)

Keywords:

time-fractional diffusion equation; fractional pseudo-parabolic; nonlinear source term; ill-posedness; truncation method

Citations:

Zbl 0851.65066
Cite Review PDF
Full Text: DOI

References:

[1] Arendt, W.; Ter Elst, AFM; Warma, M., Fractional powers of sectorial operators via the Dirichlet-to-Neumann operator, Comm. Partial Differ. Equ., 43, 1, 1-24, 2018 · doi:10.1080/03605302.2017.1363229
[2] Barenblatt, GI; Zheltov, IP; Kochina, IN, Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks, J. Appl. Math. Mech., 24, 5, 1286-303, 1960 · Zbl 0104.21702 · doi:10.1016/0021-8928(60)90107-6
[3] Beshtokov, MK, Boundary value problems for a pseudo-parabolic equation with the Caputo fractional derivative, Differ. Equ., 55, 7, 884-93, 2019 · Zbl 1426.35221 · doi:10.1134/S0012266119070024
[4] Cao, Y.; Yin, J.; Wang, C., Cauchy problems of semilinear pseudo-parabolic equations, J. Differ. Equ., 246, 4568-90, 2009 · Zbl 1179.35178 · doi:10.1016/j.jde.2009.03.021
[5] Cao, Y.; Liu, C., Initial boundary value problem for a mixed pseudo- parabolic p-Laplacian type equation with logarithmic nonlinearity, Electron. J. Differ. Equ., 2018, 116, 1-19, 2018 · Zbl 1391.35230
[6] Can, NH; Zhou, Y.; Tuan, NH; Thach, TN, Regularized solution approximation of a fractional pseudo-parabolic problem with a nonlinear source term and random data, Chaos, Solitons Fractals, 2020 · Zbl 1489.35163 · doi:10.1016/j.chaos.2020.109847
[7] Chen, H.; Xu, H., Global existence and blow-up in finite time for a class of finitely degenerate semilinear pseudo-parabolic equations, Acta. Math. Sin. Engl. Ser., 35, 7, 1143-62, 2019 · Zbl 1417.35063 · doi:10.1007/s10114-019-8037-x
[8] Chen, H.; Xu, H., Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity, Discrete. Contin. Dyn. Syst., 39, 2, 1185-203, 2019 · Zbl 1404.35233 · doi:10.3934/dcds.2019051
[9] Chen, H.; Tian, S., Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity, J. Differ. Equ., 258, 12, 4424-42, 2015 · Zbl 1370.35190 · doi:10.1016/j.jde.2015.01.038
[10] Dehghan, M., A computational study of the one-dimensional parabolic equation subject to nonclassical boundary specifications, Numer. Methods Partial Diff. Equa., 22, 220-257, 2006 · Zbl 1084.65099 · doi:10.1002/num.20071
[11] Dehghan, M., The one-dimensional heat equation subject to a boundary integral specification, Chaos Solitons Fractals, 32, 2, 661-675, 2007 · Zbl 1139.35352 · doi:10.1016/j.chaos.2005.11.010
[12] Ding, H.; Zhou, J., Global existence and blow-up for a mixed pseudo-parabolic p-Laplacian type equation with logarithmic nonlinearity, J. Math. Anal. Appl., 478, 393-420, 2019 · Zbl 1447.35202 · doi:10.1016/j.jmaa.2019.05.018
[13] Engl, HW; Hanke, M.; Neubauer, A., Regularization of Inverse Problems, 1996, Dordrecht: Kluwer, Dordrecht · Zbl 0859.65054 · doi:10.1007/978-94-009-1740-8
[14] Guo, B.; Pu, X.; Huang, F., Fractional Partial Differential Equations and Their Numerical Solutions, 2015, Hackensack: Publishing Co. Pvt. Ltd., Hackensack · Zbl 1335.35001 · doi:10.1142/9543
[15] Guo, B.; Han, Y.; Xin, J., Existence of the global smooth solution to the period boundary value problem of fractional nonlinear Schrödinger equation, Appl. Math. Comput., 204, 1, 468-477, 2008 · Zbl 1163.35483
[16] Hadamard, J., Lectures on Cauchy’s Problem in Linear Partial Differential Equations, 1953, New York: Dover, New York · JFM 49.0725.04 · doi:10.1063/1.3061337
[17] Hai, DND, Holder-logarithmic type approximation for nonlinear backward parabolic equations connected with a pseudo-differential operator, Commun. Pure Appl. Anal., 21, 5, 1715-1734, 2022 · Zbl 1487.35224 · doi:10.3934/cpaa.2022043
[18] Huafei, D.; Yadong, S.; Xiaoxiao, Z., Global well-posedness for a fourth order pseudo-parabolic equation with memory and source terms, Disc. Contin. Dyn. Syst. Ser. B, 21, 3, 781-801, 2016 · Zbl 1331.35204 · doi:10.3934/dcdsb.2016.21.781
[19] Jin, L.; Li, L.; Fang, S., The global existence and time-decay for the solutions of the fractional pseudo-parabolic equation, Comput. Math. Appl., 73, 10, 2221-32, 2017 · Zbl 1386.35443 · doi:10.1016/j.camwa.2017.03.005
[20] Mainardi, F.; Luchko, Y.; Pagnini, G., The fundamental solution of the space-time fractional diffusion equation, Fract. Cacl. Appl. Anal., 4, 153-192, 2001 · Zbl 1054.35156
[21] Seidman, TI, Optimal filtering for the backward heat equation, SIAM J. Numer. Anal., 33, 162-170, 1996 · Zbl 0851.65066 · doi:10.1137/0733010
[22] Showalter, RE; Ting, TW, Pseudo-parabolic partial differential equations, SIAM J. Math. Anal., 1, 1, 1-26, 1970 · Zbl 0199.42102 · doi:10.1137/0501001
[23] Sun, F.; Liu, L.; Wu, Y., Global existence and finite time blow-up of solutions for the semilinear pseudo-parabolic equation with a memory term, Appl. Anal., 98, 4, 22, 2019 · Zbl 1407.35122 · doi:10.1080/00036811.2017.1400536
[24] Tautenhahn, U.; Schroter, T., On optimal regularization methods for the backward heat equation, Z. Anal. Anwendungen, 15, 475-493, 1996 · Zbl 0848.65044 · doi:10.4171/zaa/711
[25] Tautenhahn, U., Optimality for ill-posed problems under general source conditions, Num. Funct. Anal. Optim., 19, 377-398, 1998 · Zbl 0907.65049 · doi:10.1080/01630569808816834
[26] Thanh, BLT, On the well-posedness of a spectral fractional forward-backward pseudo-parabolic equation, Comput. Math. Appl., 2018 · Zbl 1442.35526 · doi:10.1016/j.camwa.2018.09.035
[27] Xu, R.; Su, J., Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations, J. Funct. Anal., 264, 2732-63, 2013 · Zbl 1279.35065 · doi:10.1016/j.jfa.2013.03.010
[28] Yang, F.; Li, XX; Li, DG; Wang, L., The simplified Tikhonov regularization method for solving a Riesz-Feller space-fractional backward diffusion problem, Math. Comput. Sci., 11, 91-110, 2017 · Zbl 1516.35544 · doi:10.1007/s11786-017-0292-6
[29] Zhao, J.; Liu, S.; Liu, T., An inverse problem for space-fractional backward diffusion problem, Math. Methods Appl. Sci., 37, 1147-1158, 2014 · Zbl 1476.35340 · doi:10.1002/mma.2876
[30] Zhanga, H.; Lub, J.; Hu, Q., Exponential growth of solution of a strongly nonlinear generalized Boussinesq equation, Comput. Math. Appl., 68, 1787-93, 2014 · Zbl 1369.35029 · doi:10.1016/j.camwa.2014.10.012
[31] Zhu, X.; Li, F.; Li, Y., Global solutions and blow up solutions to a class of pseudo- parabolic equations with nonlocal term, Appl. Math. Comput., 329, 38-51, 2018 · Zbl 1427.35148
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.