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Tow-parameter quasi-boundary value method for a backward abstract time-degenerate fractional parabolic problem

  • Khelili Besma , Boussetila Nadjib ORCID logo EMAIL logo and Benrabah Abderafik ORCID logo
Published/Copyright: January 13, 2025

Abstract

In this article, for a time-degenerate fractional parabolic equation, we study an inverse problem for restoration of the initial condition from the information of the final temperature profile. We show that the considered problem is ill-posed in the sense of Hadamard, i.e., small errors in the measurement data may lead to indefinitely large errors in the solutions. This ill-posed problem is regularized using a modified quasi-boundary value method, and some convergence estimates for the regularized solution are obtained using a priori and posteriori parameter choice rules. Finally, several numerical experiments are presented to demonstrate the accuracy and efficiency of the regularization method.

MSC 2020: 35R25; 35R30; 47A52

References

[1] A. Benrabah and N. Boussetila, Modified nonlocal boundary value problem method for an ill-posed problem for the biharmonic equation, Inverse Probl. Sci. Eng. 27 (2019), no. 3, 340–368. 10.1080/17415977.2018.1461859Search in Google Scholar

[2] A. Benrabah, N. Boussetila and F. Rebbani, Regularization method for an ill-posed Cauchy problem for elliptic equations, J. Inverse Ill-Posed Probl. 25 (2017), no. 3, 311–329. 10.1515/jiip-2015-0075Search in Google Scholar

[3] A. Benrabah, N. Boussetila and F. Rebbani, Modified auxiliary boundary conditions method for an ill-posed problem for the homogeneous biharmonic equation, Math. Methods Appl. Sci. 43 (2020), no. 1, 358–383. 10.1002/mma.5888Search in Google Scholar

[4] M. B. Borikhanov and A. G. Smadiyeva, Cauchy problems for the time-fractional degenerate diffusion equations, J. Math. Mech. Comput. Sci. 117 (2023), 10.26577/JMMCS.2023.v117.i1.02. 10.26577/JMMCS.2023.v117.i1.02Search in Google Scholar

[5] L. Boudabsa and T. Simon, Some properties of the Kilbas–Saigo function, Mathematics 9 (2021), no. 3, Paper No. 217. 10.3390/math9030217Search in Google Scholar

[6] M. Briane and J. Casado-Díaz, Increase of mass and nonlocal effects in the homogenization of magneto-elastodynamics problems, Calc. Var. Partial Differential Equations 60 (2021), no. 5, Paper No. 163. 10.1007/s00526-021-02027-0Search in Google Scholar

[7] P. Cannarsa, A. Doubova and M. Yamamoto, Inverse problem of reconstruction of degenerate diffusion coefficient in a parabolic equation, Inverse Problems 37 (2021), no. 12, Article ID 125002. 10.1088/1361-6420/ac274bSearch in Google Scholar

[8] P. Cannarsa, A. Doubova and M. Yamamoto, Reconstruction of degenerate conductivity region for parabolic equations, Inverse Problems 40 (2024), no. 4, Article ID 045033. 10.1088/1361-6420/ad308aSearch in Google Scholar

[9] C. Cheng, L. Chen and J. Li, Global boundedness and the Allee effect in a nonlocal bistable reaction-diffusion equation in population dynamics, Nonlinear Anal. Real World Appl. 60 (2021), Article ID 103309. 10.1016/j.nonrwa.2021.103309Search in Google Scholar

[10] S. E. Chorfi, L. Maniar and M. Yamamoto, The backward problem for time-fractional evolution equations, Appl. Anal. 103 (2024), no. 12, 2194–2212. 10.1080/00036811.2023.2290273Search in Google Scholar

[11] Z.-C. Deng, K. Qian, X.-B. Rao, L. Yang and G.-W. Luo, An inverse problem of identifying the source coefficient in a degenerate heat equation, Inverse Probl. Sci. Eng. 23 (2015), no. 3, 498–517. 10.1080/17415977.2014.922079Search in Google Scholar

[12] N. M. Dien, On mild solutions of the generalized nonlinear fractional pseudo-parabolic equation with a nonlocal condition, Fract. Calc. Appl. Anal. 25 (2022), no. 2, 559–583. 10.1007/s13540-022-00024-4Search in Google Scholar

[13] S. Djemoui, M. S. Meziani and N. Boussetila, The conditional stability and an iterative regularization method for a fractional inverse elliptic problem of Tricomi–Gellerstedt–Keldysh type, Math. Model. Anal. 29 (2024), no. 1, 23–45. 10.3846/mma.2024.16783Search in Google Scholar

[14] X. Feng, W. Ning and Z. Qian, A quasi-boundary-value method for a Cauchy problem of an elliptic equation in multiple dimensions, Inverse Probl. Sci. Eng. 22 (2014), no. 7, 1045–1061. 10.1080/17415977.2013.850683Search in Google Scholar

[15] D. Geng and H. Wang, Normal form formulations of double-Hopf bifurcation for partial functional differential equations with nonlocal effect, J. Differential Equations 309 (2022), 741–785. 10.1016/j.jde.2021.11.046Search in Google Scholar

[16] J. Hadamard, Lectures on Cauchy’s Problem in Linear Partial Differential Equations, Yale Univerity, New Haven, 1923. Search in Google Scholar

[17] D. N. Hào and N. V. Duc, Regularization of backward parabolic equations in Banach spaces, J. Inverse Ill-Posed Probl. 20 (2012), no. 5–6, 745–763. 10.1515/jip-2012-0046Search in Google Scholar

[18] D. N. Hào and N. V. Duc, A non-local boundary value problem method for semi-linear parabolic equations backward in time, Appl. Anal. 94 (2015), no. 3, 446–463. 10.1080/00036811.2014.970537Search in Google Scholar

[19] D. N. Hào, N. V. Duc and D. Lesnic, A non-local boundary value problem method for the Cauchy problem for elliptic equations, Inverse Problems 25 (2009), no. 5, Article ID 055002. 10.1088/0266-5611/25/5/055002Search in Google Scholar

[20] D. N. Hào, N. V. Duc and H. Sahli, A non-local boundary value problem method for parabolic equations backward in time, J. Math. Anal. Appl. 345 (2008), no. 2, 805–815. 10.1016/j.jmaa.2008.04.064Search in Google Scholar

[21] D. N. Hào, J. Liu, N. V. Duc and N. V. Thang, Stability results for backward time-fractional parabolic equations, Inverse Problems 35 (2019), no. 12, Article ID 125006. 10.1088/1361-6420/ab45d3Search in Google Scholar

[22] N. M. Huzyk, P. Y. Pukach and M. I. Vovk, Coefficient inverse problem for the strongly degenerate parabolic equation, Carpathian Math. Publ. 15 (2023), no. 1, 52–65. 10.15330/cmp.15.1.52-65Search in Google Scholar

[23] V. Isakov, Inverse parabolic problems with the final overdetermination, Comm. Pure Appl. Math. 44 (1991), no. 2, 185–209. 10.1002/cpa.3160440203Search in Google Scholar

[24] V. K. Ivanov, I. V. Melnikova and A. I. Filinkov, Differential-Operator Equations and Ill-Posed Problems, Nauka, Moscow, 1994. Search in Google Scholar

[25] B. Jin, Fractional Differential Equations—An Approach via Fractional Derivatives, Appl. Math. Sci. 206, Springer, Cham, 2021. 10.1007/978-3-030-76043-4Search in Google Scholar

[26] B. Jin and Z. Zhou, Numerical Treatment and Analysis of Time-Fractional Evolution Equations, Appl. Math. Sci. 214, Springer, Cham, 2023. 10.1007/978-3-031-21050-1Search in Google Scholar

[27] S. I. Kabanikhin, Inverse and Ill-Posed Problems, Theory and Applications, Inverse Ill-posed Probl. Ser. 55, Walter de Gruyter, Berlin, 2011. 10.1515/9783110224016Search in Google Scholar

[28] B. Kaltenbacher and W. Rundell, Inverse Problems for Fractional Partial Differential Equations, Grad. Stud. Math. 230, American Mathematical Society, Providence, 2023. 10.1090/gsm/230Search in Google Scholar

[29] V. L. Kamynin, Unique solvability of direct and inverse problems for degenerate parabolic equations in the multidimensional case, J. Math. Sci. (N. Y.) 269 (2023), no. 1, 36–52. 10.1007/s10958-023-06253-zSearch in Google Scholar

[30] B. Khelili, N. Boussetila and F. Rebbani, A modified quasi-boundary value method for an abstract ill-posed biparabolic problem, Open Math 15 (2017), 1649–1666. 10.1515/math-2017-0140Search in Google Scholar

[31] A. A. Kilbas and M. Saigo, On solution of integral equation of Abel–Volterra type, Differential Integral Equations 8 (1995), no. 5, 993–1011. 10.57262/die/1369056041Search in Google Scholar

[32] A. A. Kilbas and M. Saĭgo, Fractional integrals and derivatives of functions of Mittag-Leffler type, Dokl. Akad. Nauk Belarusi 39 (1995), no. 4, 22–26, 123. Search in Google Scholar

[33] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Math. Stud. 204, Elsevier Science, Amsterdam, 2006. Search in Google Scholar

[34] M. Kirane, B. Samet and B. T. Torebek, Determination of an unknown source term temperature distribution for the sub-diffusion equation at the initial and final data, Electron. J. Differential Equations 2017 (2017), Paper No. 257. Search in Google Scholar

[35] Y. S. Li, L. L. Sun, Z. Q. Zhang and T. Wei, Identification of the time-dependent source term in a multi-term time-fractional diffusion equation, Numer. Algorithms 82 (2019), no. 4, 1279–1301. 10.1007/s11075-019-00654-5Search in Google Scholar

[36] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity—An Introduction to Mathematical Models, World Scientific, Hackensack, 2022. Search in Google Scholar

[37] I. V. Melnikova and A. Filinkov, Abstract Cauchy Problems: Three Approaches, Chapman & Hall/CRC Monogr. Surveys Pure Appl. Math. 120, Chapman & Hall/CRC, Boca Raton, 2001. 10.1201/9781420035490Search in Google Scholar

[38] M. S. E. Meziani, S. Djemoui and N. Boussetila, Detection of source term in an abstract fractional subdiffusion model by the quasi-boundary value method with a priori and a posteriori estimate, Eurasian J. Math. Comput. Appl. 11 (2023), 98–123. 10.32523/2306-6172-2023-11-1-98-123Search in Google Scholar

[39] A. T. Nguyen, Z. Hammouch, E. Karapinar and N. H. Tuan, On a nonlocal problem for a Caputo time-fractional pseudoparabolic equation, Math. Methods Appl. Sci. 44 (2021), no. 18, 14791–14806. 10.1002/mma.7743Search in Google Scholar

[40] C. P. Niculescu and L.-E. Persson, Convex Functions and Their Applications, 2nd ed., Springer, Cham, 2018. 10.1007/978-3-319-78337-6Search in Google Scholar

[41] C. V. Pao, Reaction diffusion equations with nonlocal boundary and nonlocal initial conditions, J. Math. Anal. Appl. 195 (1995), no. 3, 702–718. 10.1006/jmaa.1995.1384Search in Google Scholar

[42] I. Podlubny, Fractional Differential Equations, Math. Sci. Eng. 198, Academic Press, San Diego, 1998. Search in Google Scholar

[43] A. I. Prilepko, D. G. Orlovsky and I. A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics, Monogr. Textb. Pure Appl. Math. 231, Marcel Dekker, New York, 2000. Search in Google Scholar

[44] W. Rundell, Determination of an unknown nonhomogeneous term in a linear partial differential equation from overspecified boundary data, Appl. Anal. 10 (1980), no. 3, 231–242. 10.1080/00036818008839304Search in Google Scholar

[45] A. A. Samarskii and P. N. Vabishchevich, Numerical Methods for Solving Inverse Problems of Mathematical Physics, Inverse Ill-posed Probl. Ser. 52, Walter de Gruyter, Berlin, 2007. 10.1515/9783110205794Search in Google Scholar

[46] C. Shi, H. Cheng and W. Fan, An iterative generalized quasi-boundary value regularization method for the backward problem of time fractional diffusion-wave equation in a cylinder, Numer. Algorithms 94 (2023), no. 4, 1619–1651. 10.1007/s11075-023-01549-2Search in Google Scholar

[47] A. G. Smadiyeva, Initial-boundary value problem for the time-fractional degenerate diffusion equation, J. Math. Mech. Comput. Sci. 113 (2022), 32–41. 10.26577/JMMCS.2022.v113.i1.04Search in Google Scholar

[48] A. G. Smadiyeva and B. T. Torebek, Decay estimates for the time-fractional evolution equations with time-dependent coefficients, Proc. A. 479 (2023), no. 2276, Article ID 20230103. 10.1098/rspa.2023.0103Search in Google Scholar

[49] X. Sun and R. Yuan, Hopf bifurcation in a diffusive population system with nonlocal delay effect, Nonlinear Anal. 214 (2022), Article ID 112544. 10.1016/j.na.2021.112544Search in Google Scholar

[50] N. H. Tuan, N. A. Triet, N. H. Luc and N. D. Phuong, On a time fractional diffusion with nonlocal in time conditions, Adv. Difference Equ. (2021), Paper No. 204. 10.1186/s13662-021-03365-1Search in Google Scholar

[51] S. Umarov, Introduction to Fractional Pseudo-Differential Equations with Singular Symbols, Dev. Math. 41, Springer, Cham, 2015. 10.1007/978-3-319-20771-1Search in Google Scholar

[52] T. Wei and Y. Luo, A generalized quasi-boundary value method for recovering a source in a fractional diffusion-wave equation, Inverse Problems 38 (2022), no. 4, Paper No. 045001. 10.1088/1361-6420/ac50b9Search in Google Scholar

[53] T. Wei and J.-G. Wang, A modified quasi-boundary value method for the backward time-fractional diffusion problem, ESAIM Math. Model. Numer. Anal. 48 (2014), no. 2, 603–621. 10.1051/m2an/2013107Search in Google Scholar

[54] H. Zhang and T. Wei, An improved non-local boundary value problem method for a cauchy problem of the Laplace equation, Numer. Algorithms 59 (2012), no. 2, 249–269. 10.1007/s11075-011-9487-0Search in Google Scholar

[55] F. Zouyed and F. Rebbani, A modified quasi-boundary value method for an ultraparabolic ill-posed problem, J. Inverse Ill-Posed Probl. 22 (2014), no. 4, 449–466. 10.1515/jip-2012-0069Search in Google Scholar

Received: 2024-04-14
Revised: 2024-09-25
Accepted: 2024-12-06
Published Online: 2025-01-13
Published in Print: 2025-04-01

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