Inverse initial problem for fractional wave equation with the Hadamard fractional derivative
DOI:
https://doi.org/10.70474/ydmksp42Keywords:
Fractional wave equation, inverse initial problem, Hadamard fractional derivative, Mittag-Leffler functionAbstract
This paper investigates an inverse initial problem for a time-fractional wave equation involving the Hadamard fractional derivative. Unlike the more widely studied Caputo and Riemann–Liouville derivatives, the Hadamard derivative is defined via a logarithmic kernel and exhibits distinct analytical features, making it suitable for modeling processes with slow memory decay and multiplicative structures. The study is motivated by recent advances in fractional calculus and the growing interest in fractional models across applied sciences. Building on prior work concerning the extremum principle and solvability of boundary value problems with Hadamard-type operators, we establish sufficient conditions for the unique solvability of the inverse problem. The analysis is carried out in terms of eigenfunction expansions and leverages properties of the two-parameter Mittag–Leffler function. The findings contribute to the theory of inverse problems for fractional wave equations and highlight the role of Hadamard derivatives in capturing complex temporal dynamics in mathematical models.
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References
Kilbas A.A., Srivastava H.M., Trujillo J.J. Theory and Applications of Fractional Differential Equations. North-Holland Math. Studies, 2006.
Abbas S., Benchohra M., Hamidi N., Henderson J. Caputo-Hadamard fractional differential equations in Banach spaces. Fract. Calc. Appl. Anal. 2018;21(4):1027–1045. 10.1515/fca-2018-0056 DOI: https://doi.org/10.1515/fca-2018-0056
Kilbas A.A. Hadamard-type fractional calculus. J. Korean Math. Soc. 2001;38(6):1191–1204.
Kirane M., Torebek B.T. Extremum Principle for the Hadamard Derivatives and Its Application to Nonlinear Fractional Partial Differential Equations. Fract. Calc. Appl. Anal. 2019;22(2):358–378. 10.1515/fca-2019-0022 DOI: https://doi.org/10.1515/fca-2019-0022
Klimek M. Sequential fractional differential equations with Hadamard derivative. Commun. Nonlinear Sci. Numer. Simul. 2011;16(12):4689–4697. 10.1016/j.cnsns.2011.01.018 DOI: https://doi.org/10.1016/j.cnsns.2011.01.018
Makhlouf A.B., Mchiri L. Some results on the study of Caputo–Hadamard fractional stochastic differential equations. Chaos Solitons Fractals. 2022;155:111757. 10.1016/j.chaos.2021.111757 DOI: https://doi.org/10.1016/j.chaos.2021.111757
Almeida R. Caputo–Hadamard fractional derivatives of variable order. Numer. Funct. Anal. Optim. 2017;38(1):1–19.
Wang Z., Cen D., Vong S. A second-order scheme with nonuniform time grids for Caputo–Hadamard fractional sub-diffusion equations. J. Comput. Appl. Math. 2022;414:114448. 10.1016/j.cam.2022.114448 DOI: https://doi.org/10.1016/j.cam.2022.114448
Ou C., Cen D., Vong S., Wang Z. Mathematical analysis and numerical methods for Caputo-Hadamard fractional diffusion-wave equations. Appl. Numer. Math. 2022;177:34–57. 10.1016/j.apnum.2022.02.017 DOI: https://doi.org/10.1016/j.apnum.2022.02.017
Bulavatsky V.M. A problem with terminal integral condition for one nonlinear fractional-differential equation with bi-ordinal Hilfer–Hadamard derivative. Cybern. Syst. Anal. 2025;61(2):212–220. 10.1007/s10559-025-00761-3 DOI: https://doi.org/10.1007/s10559-025-00761-3
Masood K., Messaoudi S., Zaman F.D. Initial inverse problem in heat equation with Bessel operator. Int. J. Heat Mass Transf. 2002;45(14):2959–2965. DOI: https://doi.org/10.1016/S0017-9310(02)00019-4
Al-Musalhi F., Al-Salti N., Kerbal S. Inverse Problems of a Fractional Differential Equation with Bessel Operator. Math. Model. Nat. Phenom. 2017;12(3):105–113. DOI: https://doi.org/10.1051/mmnp/201712310
Karimov E.T., Tokmagambetov N.E., Usmonov D.A. Inverse-initial problem for time-degenerate PDE involving the bi-ordinal Hilfer derivative. Cybern. Syst. Anal. 2024;60(5):799–809. 10.1007/s10559-024-00717-z DOI: https://doi.org/10.1007/s10559-024-00717-z
Serikbaev D., Tokmagambetov N. Determination of initial data in time-fractional wave equation. Fract. Differ. Calc. 2024;14(1):83–99. 10.7153/fdc-2024-14-05 DOI: https://doi.org/10.7153/fdc-2024-14-05
Pskhu A.V. On real roots of a Mittag-Leffler-type function. Mat. Zametki. 2005;77(4):592–599. 10.4213/mzm2520 DOI: https://doi.org/10.4213/mzm2520
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