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Two regularization methods for identifying the source term of Caputo–Hadamard type time fractional diffusion-wave equation

  • Fan Yang EMAIL logo , Ruo-Hong Li , Yin-Xia Gao and Xiao-Xiao Li
Published/Copyright: April 9, 2025

Abstract

In this paper, the inverse problem of source term identification for Caputo–Hadamard type time-fractional diffusion-wave equation is studied. Firstly, we prove that the problem is ill-posed, and give the optimal error bound and the conditional stability results. Secondly, we apply fractional Tikhonov regularization method and fractional Landweber iterative regularization method to solve the problem. Based on the conditional stability results, we give error estimates under the a priori regularization parameter selection rule and the a posteriori regularization parameter selection rule respectively. In addition, we give three numerical examples to prove the validity and feasibility of the selected regularization method. What is novel is that we apply L21 formula, Crank–Nicolson format and the finite difference method to discrete equation.

MSC 2020: 35R25; 47A52; 35R30

Award Identifier / Grant number: 12461083

Funding statement: The project is supported by the National Natural Science Foundation of China (No. 12461083) and the Doctor Fund of Lan Zhou University of Technology.

References

[1] R. Almeida, A Caputo fractional derivative of a function with respect to another function, Commun. Nonlinear Sci. Numer. Simul. 44 (2017), 460–481. 10.1016/j.cnsns.2016.09.006Search in Google Scholar

[2] M. Cai, G. E. Karniadakis and C. Li, Fractional SEIR model and data-driven predictions of COVID-19 dynamics of Omicron variant, Chaos 32 (2022), no. 7, Article ID 071101. 10.1063/5.0099450Search in Google Scholar PubMed

[3] Z.-C. Deng, L. Yang, J.-N. Yu and G.-W. Luo, Identifying the diffusion coefficient by optimization from the final observation, Appl. Math. Comput. 219 (2013), no. 9, 4410–4422. 10.1016/j.amc.2012.10.045Search in Google Scholar

[4] Z.-L. Deng, C.-L. Fu, X.-L. Feng and Y.-X. Zhang, A mollification regularization method for stable analytic continuation, Math. Comput. Simulation 81 (2011), no. 8, 1593–1608. 10.1016/j.matcom.2010.11.011Search in Google Scholar

[5] E. Fan, C. Li and Z. Li, Numerical approaches to Caputo–Hadamard fractional derivatives with applications to long-term integration of fractional differential systems, Commun. Nonlinear Sci. Numer. Simul. 106 (2022), Article ID 106096. 10.1016/j.cnsns.2021.106096Search in Google Scholar

[6] G.-H. Gao, Z.-Z. Sun and H.-W. Zhang, A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications, J. Comput. Phys. 259 (2014), 33–50. 10.1016/j.jcp.2013.11.017Search in Google Scholar

[7] M. Gohar, C. Li and Z. Li, Finite difference methods for Caputo–Hadamard fractional differential equations, Mediterr. J. Math. 17 (2020), no. 6, Paper No. 194. 10.1007/s00009-020-01605-4Search in Google Scholar

[8] M. Gohar, C. Li and C. Yin, On Caputo–Hadamard fractional differential equations, Int. J. Comput. Math. 97 (2020), no. 7, 1459–1483. 10.1080/00207160.2019.1626012Search in Google Scholar

[9] H. J. Haubold, A. M. Mathai and R. K. Saxena, Mittag-Leffler functions and their applications, J. Appl. Math. 2011 (2011), Article ID 298628. 10.1155/2011/298628Search in Google Scholar

[10] S. He, C. Di and L. Yang, The mollification method based on a modified operator to the ill-posed problem for 3D Helmholtz equation with mixed boundary, Appl. Numer. Math. 160 (2021), 422–435. 10.1016/j.apnum.2020.10.012Search in Google Scholar

[11] R. Hilfer, Fractional diffusion based on Riemann–Liouville fractional derivatives, J. Phys. Chem. B 104 (2000), no. 16, 3914–3917. 10.1021/jp9936289Search in Google Scholar

[12] M. I. Ivanchov, The inverse problem of determining the heat source power for a parabolic equation under arbitrary boundary conditions, J. Math. Sci. (N. Y.) 88 (1998), no. 3,432–436. 10.1007/BF02365265Search in Google Scholar

[13] B. A. Jacobs, A new Grünwald–Letnikov derivative derived from a second-order scheme, Abstr. Appl. Anal. 2015 (2015), 1–9. 10.1155/2015/952057Search in Google Scholar

[14] S. Jiang, J. Zhang, Q. Zhang and Z. Zhang, Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations, Commun. Comput. Phys. 21 (2017), no. 3, 650–678. 10.4208/cicp.OA-2016-0136Search in Google Scholar

[15] C. Li, Z. Li and Z. Wang, Mathematical analysis and the local discontinuous Galerkin method for Caputo–Hadamard fractional partial differential equation, J. Sci. Comput. 85 (2020), no. 2, Paper No. 41. 10.1007/s10915-020-01353-3Search in Google Scholar

[16] Y. Q. Liang, F. Yang and X. X. Li, A hybrid regularization method for identifying the source term and the initial value simultaneously for fractional pseudo-parabolic equation with involution, Numer. Algorithms (2024), 10.1007/s11075-024-01944-3. 10.1007/s11075-024-01944-3Search in Google Scholar

[17] K. S. Miller and S. G. Samko, Completely monotonic functions, Integral Transform. Spec. Funct. 12 (2001), no. 4, 389–402. 10.1080/10652460108819360Search in Google Scholar

[18] K. S. Nisar, S. Ahmad and A. Ullah, Mathematical analysis of SIRD model of COVID-19 with Caputo fractional derivative based on real data, Results Phys. 21 (2021), Article ID 103772. 10.1016/j.rinp.2020.103772Search in Google Scholar PubMed PubMed Central

[19] S. Orankitjaroen, The solutions of some Riemann–Liouville fractional integral equations, Fractal Fract. 5 (2021), no. 4, Paper No. 154. 10.3390/fractalfract5040154Search in Google Scholar

[20] M. D. Ortigueira, L. Rodríguez-Germá and J. J. Trujillo, Complex Grünwald–Letnikov, Liouville, Riemann–Liouville, and Caputo derivatives for analytic functions, Commun. Nonlinear Sci. Numer. Simul. 16 (2011), no. 11, 4174–4182. 10.1016/j.cnsns.2011.02.022Search in Google Scholar

[21] C. Ou, D. Cen, S. Vong and Z. Wang, Mathematical analysis and numerical methods for Caputo–Hadamard fractional diffusion-wave equations, Appl. Numer. Math. 177 (2022), 34–57. 10.1016/j.apnum.2022.02.017Search in Google Scholar

[22] I. Podlubny, Fractional Differential Equations, Math. Sci. Eng. 198, Academic Press, San Diego, 1999. Search in Google Scholar

[23] H. Pollard, The completely monotonic character of the Mittag-Leffler function E a ( x ) , Bull. Amer. Math. Soc. 54 (1948), 1115–1116. 10.1090/S0002-9904-1948-09132-7Search in Google Scholar

[24] A. L. Qian, X. T. Xiong and Y. J. Wu, On a quasi-reversibility regularization method for a Cauchy problem of the Helmholtz equation, J. Comput. Appl. Math. 233 (2010), no. 8, 1969–1979. 10.1016/j.cam.2009.09.031Search in Google Scholar

[25] L. Qiao, F. Yang and X. Li, Simultaneous identification of the unknown source term and initial value for the time fractional diffusion equation with local and nonlocal operators, Chaos Solitons Fractals 189 (2024), no. 1, Article ID 115601. 10.1016/j.chaos.2024.115601Search in Google Scholar

[26] K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl. 382 (2011), no. 1, 426–447. 10.1016/j.jmaa.2011.04.058Search in Google Scholar

[27] T. Schröter and U. Tautenhahn, On the optimality of regularization methods for solving linear ill-posed problems, Z. Anal. Anwend. 13 (1994), no. 4, 697–710. 10.4171/zaa/494Search in Google Scholar

[28] A. Shidfar, A. Babaei and A. Molabahrami, Solving the inverse problem of identifying an unknown source term in a parabolic equation, Comput. Math. Appl. 60 (2010), no. 5, 1209–1213. 10.1016/j.camwa.2010.06.002Search in Google Scholar

[29] M. Subramanian and T. Nandha Gopal, Analysis of boundary value problem with multi-point conditions involving Caputo–Hadamard fractional derivative, Proyecciones 39 (2020), no. 6, 1555–1575. 10.22199/issn.0717-6279-2020-06-0093Search in Google Scholar

[30] U. Tautenhahn, Optimal stable solution of Cauchy problems for elliptic equations, Z. Anal. Anwend. 15 (1996), no. 4, 961–984. 10.4171/zaa/740Search in Google Scholar

[31] U. Tautenhahn, Optimal stable approximations for the sideways heat equation, J. Inverse Ill-Posed Probl. 5 (1997), no. 3, 287–307. 10.1515/jiip.1997.5.3.287Search in Google Scholar

[32] U. Tautenhahn, Optimality for ill-posed problems under general source conditions, Numer. Funct. Anal. Optim. 19 (1998), no. 3–4, 377–398. 10.1080/01630569808816834Search in Google Scholar

[33] U. Tautenhahn and R. Gorenflo, On optimal regularization methods for fractional differentiation, Z. Anal. Anwend. 18 (1999), no. 2, 449–467. 10.4171/zaa/892Search in Google Scholar

[34] U. Tautenhahn, U. Hämarik and B. Hofmann, Conditional stability estimates for ill-posed PDE problems by using interpolation, Numer. Funct. Anal. Optim. 34 (2013), no. 12, 1370–1417. 10.1080/01630563.2013.819515Search in Google Scholar

[35] H. Thorsten, Regularization of exponentially ill-posed problems, Numer. Funct. Anal. Optim. 21 (2000), no. 3–4, 439–464. 10.1080/01630560008816965Search in Google Scholar

[36] N. A. Triet, T. T. Binh, N. D. Phuong, D. Baleanu and N. H. Can, Recovering the initial value for a system of nonlocal diffusion equations with random noise on the measurements, Math. Methods Appl. Sci. 44 (2021), no. 6, 5188–5209. 10.1002/mma.7102Search in Google Scholar

[37] G. Vainikko, On the optimality of methods for ill-posed problems, Z. Anal. Anwend. 6 (1987), no. 4, 351–362. 10.4171/zaa/256Search in Google Scholar

[38] J.-G. Wang, Y.-B. Zhou and T. Wei, A posteriori regularization parameter choice rule for the quasi-boundary value method for the backward time-fractional diffusion problem, Appl. Math. Lett. 26 (2013), no. 7, 741–747. 10.1016/j.aml.2013.02.006Search in Google Scholar

[39] T. Wei, X. L. Li and Y. S. Li, An inverse time-dependent source problem for a time-fractional diffusion equation, Inverse Problems 32 (2016), no. 8, Article ID 085003. 10.1088/0266-5611/32/8/085003Search in Google Scholar

[40] Z. Wei, Q. Li and J. Che, Initial value problems for fractional differential equations involving Riemann–Liouville sequential fractional derivative, J. Math. Anal. Appl. 367 (2010), no. 1, 260–272. 10.1016/j.jmaa.2010.01.023Search in Google Scholar

[41] X. T. Xiong and X. M. Xue, A fractional Tikhonov regularization method for identifying a space-dependent source in the time-fractional diffusion equation, Appl. Math. Comput. 349 (2019), 292–303. 10.1016/j.amc.2018.12.063Search in Google Scholar

[42] L.-L. Yan, F. Yang and X.-X. Li, The fractional Tikhonov regularization method for simultaneous inversion of the source term and initial value in a space-fractional Allen–Cahn equation, J. Appl. Anal. Comput. 14 (2024), no. 4, 2257–2282. 10.11948/20230364Search in Google Scholar

[43] F. Yang, Y. Cao and X. Li, Identifying source term and initial value simultaneously for the time-fractional diffusion equation with Caputo-like hyper-Bessel operator, Fract. Calc. Appl. Anal. 27 (2024), no. 5, 2359–2396. 10.1007/s13540-024-00304-1Search in Google Scholar

[44] F. Yang, J.-L. Fu and X.-X. Li, A potential-free field inverse time-fractional Schrödinger problem: Optimal error bound analysis and regularization method, Math. Methods Appl. Sci. 44 (2021), no. 2, 1219–1251. 10.1002/mma.6826Search in Google Scholar

[45] F. Yang, Q. Pu and X.-X. Li, The fractional Tikhonov regularization methods for identifying the initial value problem for a time-fractional diffusion equation, J. Comput. Appl. Math. 380 (2020), Article ID 112998. 10.1016/j.cam.2020.112998Search in Google Scholar

[46] F. Yang, Q. Sun and X. Li, Two regularization methods for identifying the source term problem on the time-fractional diffusion equation with a hyper-Bessel operator, Acta Math. Sci. Ser. B (Engl. Ed.) 42 (2022), no. 4, 1485–1518. 10.1007/s10473-022-0412-5Search in Google Scholar

[47] F. Yang, Q. Wang and X. Li, A fractional Landweber iterative regularization method for stable analytic continuation, AIMS Math. 6 (2021), no. 1, 404–419. 10.3934/math.2021025Search in Google Scholar

[48] F. Yang, H.-H. Wu and X.-X. Li, Three regularization methods for identifying the initial value of homogeneous anomalous secondary diffusion equation, Math. Methods Appl. Sci. 44 (2021), no. 17, 13723–13755. 10.1002/mma.7654Search in Google Scholar

[49] F. Yang, H.-H. Wu and X.-X. Li, Three regularization methods for identifying the initial value of time fractional advection-dispersion equation, Comput. Appl. Math. 41 (2022), no. 1, Paper No. 60. Search in Google Scholar

[50] F. Yang, H.-H. Wu and X.-X. Li, Three regularization methods for identifying the initial value of time fractional advection-dispersion equation, Comput. Appl. Math. 41 (2022), no. 1, Paper No. 60. 10.1007/s40314-022-01762-0Search in Google Scholar

[51] F. Yang, J.-M. Xu and X.-X. Li, Simultaneous inversion of the source term and initial value of the time fractional diffusion equation, Math. Model. Anal. 29 (2024), no. 2, 193–214. 10.3846/mma.2024.18133Search in Google Scholar

[52] F. Yang, L.-L. Yan, H. Liu and X.-X. Li, Two regularization methods for identifying the unknown source of Sobolev equation with fractional Laplacian, J. Appl. Anal. Comput. 15 (2025), no. 1, 198–225. Search in Google Scholar

[53] F. Yang, Y. Zhang and X.-X. Li, Landweber iterative method for an inverse source problem of time-space fractional diffusion-wave equation, Comput. Methods Appl. Math. 24 (2024), no. 1, 265–278. 10.1515/cmam-2022-0240Search in Google Scholar

[54] Z. Yang, X. Zheng and H. Wang, Well-posedness and regularity of Caputo–Hadamard fractional stochastic differential equations, Z. Angew. Math. Phys. 72 (2021), no. 4, Paper No. 141. 10.1007/s00033-021-01566-ySearch in Google Scholar

[55] C.-Y. Zhang, F. Yang and X.-X. Li, Identifying the source term and the initial value simultaneously for Caputo–Hadamard fractional diffusion equation on spherically symmetric domain, Comput. Appl. Math. 43 (2024), no. 4, Paper No. 161. 10.1007/s40314-024-02679-6Search in Google Scholar

[56] S. Zhang, Monotone iterative method for initial value problem involving Riemann–Liouville fractional derivatives, Nonlinear Anal. 71 (2009), no. 5–6, 2087–2093. 10.1016/j.na.2009.01.043Search in Google Scholar

Received: 2024-07-21
Revised: 2025-02-16
Accepted: 2025-02-28
Published Online: 2025-04-09
Published in Print: 2025-06-01

© 2025 Walter de Gruyter GmbH, Berlin/Boston

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