Two regularization methods for identifying the initial value of Caputo–Hadamard time-fractional diffusion equation
DOI: https://doi.org/10.3846/mma.2026.23999Abstract
In this paper, the inverse problem of identifying the unknown initial value for time fractional diffusion equation with Caputo-Hadamard derivative is considered. This problem is illposed and two regularization methods are used to solve it. Firstly, we prove that this problem is ill-posed. Secondly, the conditional stability result and the optimal error bound are given. Then, the error estimates of the Quasi-boundary regularization method and the fractional Landweber iterative regularization method under a priori and a posteriori regularization parameter selection rules are given respectively. Finally, numerical examples are given to illustrate the effectiveness of two regularization methods.
Keywords:
Caputo-Hadamard derivative, initial value identification, quasi-boundary regularization method, fractional Landweber iterative regularization methodHow to Cite
Share
License
Copyright (c) 2026 The Author(s). Published by Vilnius Gediminas Technical University.
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
B. Ahmad, A. Alsaedi, S.K. Ntouyas and J. Tariboon. Hadamard-type fractional differential equations, inclusions and inequalities. Springer, 2017. https://doi.org/10.1007/978-3-319-52141-1
G.W. Clark and S.F. Oppenheimer. Quasireversibility methods for non-wellposed problems. Electronic Journal of Differential Equations, 1994(8):1–9, 1994.
R. Garra, F. Mainardi and G. Spada. A generalization of the Lomnitz logarithmic creep law via Hadamard fractional calculus. Chaos, Solitons and Fractals, 102:333–338, 2017. https://doi.org/10.1016/j.chaos.2017.03.032
M. Gohar, C. Li and C. Yin. On Caputo-Hadamard fractional differential equations. International Journal of Computer Mathematics, 97(7):1459–1483, 2020. https://doi.org/10.1080/00207160.2019.1626012
R. Gorenflo, A.A. Kilbas, F. Mainardi and S.V. Rogosin. Mittag-Leffler functions, related topics and applications. Springer, Germany, 2020. https://doi.org/10.1007/978-3-662-61550-8
J. Hadamard. Essai sur l’étude des fonctions, données par leur développement de Taylor. Gauthier-Villars, 1892.
D.N. Hào, N.V. Duc and D. Lesnic. Regularization of parabolic equations backward in time by a non-local boundary value problem method. IMA Journal of Applied Mathematics, 75(2):291–315, 2010. https://doi.org/10.1093/imamat/hxp026
F. Jarad, T. Abdeljawad and D. Baleanu. Caputo-type modification of the Hadamard fractional derivatives. Advances in Difference Equations, 2012(1):142, 2012. https://doi.org/10.1186/1687-1847-2012-142
A.A. Kilbas, H.M. Srivastava and J.J. Trujillo. Theory and applications of fractional differential equations. Elsevier, 2006. https://doi.org/10.1016/s0304-0208(06)x8001-5
T.M. Le, Q.H. Pham, T.D. Dang and T.H. Nguyen. A backward parabolic equation with a time-dependent coefficient: regularization and error estimates. Journal of Computational and Applied Mathematics, 237(1):432–441, 2013. https://doi.org/10.1016/j.cam.2012.06.012
C. Li, Z. Li and Z. Wang. Mathematical analysis and the local discontinuous Galerkin method for Caputo-Hadamard fractional partial differential equation. Journal of Scientific Computing, 85(2):41, 2020. https://doi.org/10.1007/s10915-020-01353-3
C. Li and F. Zeng. Numerical methods for fractional calculus. CRC Press, 2015. https://doi.org/10.1201/b18503
S.Y. Li, R.H. Li, F. Yang, X.X. Li and Z.J. Tian. Identification of the random source in the stochastic caputo-hadamard time-fractional diffusion equation. Journal of Applied Analysis and Computation, 16(2):987–1015, 2026. https://doi.org/10.11948/20250065
Y.-Q. Liang, F. Yang and X.-X. Li. Two regularization methods for identifying the initial value of time-fractional telegraph equation. Computational Methods in Applied Mathematics, 25(4):883–905, 2025. https://doi.org/10.1515/cmam-2024-0159
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 pseudoparabolic equation with involution. Numerical Algorithms, 99:2063–2095, 2025. https://doi.org/10.1007/s11075-024-01944-3
J.J. Liu and M. Yamamoto. A backward problem for the timefractional diffusion equation. Applicable Analysis, 89(11):1769–1788, 2010. https://doi.org/10.1080/00036810903479731
S. Liu and L. Feng. Optimal error bound and modified kernel method for a space-fractional backward diffusion problem. Advances in Difference Equations, 2018(1):268, 2018. https://doi.org/10.1186/s13662-018-1728-2
L. Ma. On the kinetics of Hadamard-type fractional differential systems. Fractional Calculus and Applied Analysis, 23(2):553–570, 2020. https://doi.org/10.1515/fca-2020-0027
C. Ou, D. Cen, S. Vong and Z. Wang. Mathematical analysis and numerical methods for Caputo-Hadamard fractional diffusionwave equations. Applied Numerical Mathematics, 177:34–57, 2022. https://doi.org/10.1016/j.apnum.2022.02.017
H. Pollard. The completely monotonic character of the Mittag-Leffler function Ea(−x). Bulletin Of The American Mathematical Society, 54(12):1115–1116, 1948. https://doi.org/10.1090/S0002-9904-1948-09132-7
Z.S. Ruan, S. Zhang and S.C. Xiong. Solving an inverse source problem for a time fractional diffusion equation by a modified quasi-boundary value method. Evolution Equations and Control Theory, 7(4):669–682, 2018. https://doi.org/10.3934/eect.2018032
M.E. Samei, V. Hedayati and S. Rezapour. Existence results for a fraction hybrid differential inclusion with Caputo-Hadamard type fractional derivative. Advances in Difference Equations, 2019(1):163, 2019. https://doi.org/10.1186/s13662-019-2090-8
D.D. Trong, P.H. Quan and N.H. Tuan. A quasi-boundary value method for regularizing nonlinear ill-posed problems. Electronic Journal of Differential Equations, 2009(109):1–16, 2009.
J.-G. Wang, T. Wei and Y.-B. Zhou. Optimal error bound and simplified Tikhonov regularization method for a backward problem for the time-fractional diffusion equation. Journal of computational and applied mathematics, 279:277– 292, 2015. https://doi.org/10.1016/j.cam.2014.11.026
X. Xiong, X. Xue and Z. Qian. A modified iterative regularization method for ill-posed problems. Applied Numerical Mathematics, 122:108–128, 2017. https://doi.org/10.1016/j.apnum.2017.08.004
F. Yang, R.-H. Li, Y.-X. Gao and X.-X. Li. Two regularization methods for identifying the source term of caputo-hadamard type time fractional diffusionwave equation. Journal of Inverse and Ill-Posed Problems, 33(3):369–399, 2025. https://doi.org/10.1515/jiip-2024-0051
F. Yang, S.-Y. Li and X.-X. Li. Inverse random source problem for the stochastic Caputo-Hadamard time-fractional diffusion equation driven by fractional brownian motion. Chaos, Solitons and Fractals, 201:117233, 2025. https://doi.org/10.1016/j.chaos.2025.117233
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. Journal of Applied Analysis and Computation, 15(1):198–225, 2025. https://doi.org/10.11948/20240065
F. Yang, Y. Zhang and X.-X. Li. Landweber iterative method for an inverse source problem of time-space fractional diffusion-wave equation. Computational Methods in Applied Mathematics, 24(1):265–278, 2024. https://doi.org/10.1515/cmam-2022-0240
S. Yang, X. Xiong and Y. Han. A modified fractional Landweber method for a backward problem for the inhomogeneous time-fractional diffusion equation in a cylinder. International Journal of Computer Mathematics, 97(11):2375–2393, 2020. https://doi.org/10.1080/00207160.2020.1803297
View article in other formats
CrossMark check
Published
Issue
Section
Copyright
Copyright (c) 2026 The Author(s). Published by Vilnius Gediminas Technical University.
License
This work is licensed under a Creative Commons Attribution 4.0 International License.