Recovering unknown source function for composite fractional relaxation equations
DOI:
https://doi.org/10.66147/lnaa.20253470Keywords:
Inverse source problem, regularization method., Caputo fractional derivative, diffusion-wave equation, ill-posed problemAbstract
A backward problem for composite fractional relaxation equations is considered with the Caputo fractional derivative. Based on a spectral problem, the representation of solutions is established. Next, we show the mildly ill-posedness in the Hadamard sense. After that, we show the regularization solution using two regularization methods : the Landweber regularization method and the iterative method. Then, the convergence rate between the exact solution and the regularized solution is provided, under the a priori parameter choice rule.
References
[1] Mainardi, F. (2010). Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models , World Scientific Publishing. DOI: https://doi.org/10.1142/p614
[2] Havriliak, S., Havriliak, S. J. (1997). Dielectric and Mechanical Relaxation in Materials: Analysis, Interpretation, and Application to Polymers , Hanser/Gardner Publications.
[3] Magin, R. L. (2006). Fractional Calculus in Bioengineering , Begell House Publishers.
[4] Metzler, R., Klafter, J. (2000). The random walk's guide to anomalous diffusion: a fractional dynamics approach , Physics Reports, 339(1), 1-77. DOI: https://doi.org/10.1016/S0370-1573(00)00070-3
[5] Podlubny, I. (1999). Fractional Differential Equations , Academic Press.
[6] Carcione, J. M. (2014). Wave Fields in Real Media: Wave Propagation in Anisotropic, Anelastic, Porous, and Electromagnetic Media , Elsevier.
[7] Li, X., Zhao, Y. (2008). Fractional order signal processing: a new paradigm , Signal Processing, 88(10), 2346-2357. DOI: https://doi.org/10.1016/j.sigpro.2008.03.002
[8] Barsoukov, E., Macdonald, J. R. (Eds.). (2018). Impedance Spectroscopy: Theory, Experiment, and Applications , John Wiley & Sons. DOI: https://doi.org/10.1002/9781119381860
[9] Povstenko, Y. (2015). Fractional Thermoelasticity , Springer. DOI: https://doi.org/10.1007/978-3-319-15335-3
[10] Scalas, E., Gorenflo, R., Mainardi, F. (2000). Fractional calculus and continuous-time finance , Physica A: Statistical Mechanics and its Applications, 284(1-4), 376-384. DOI: https://doi.org/10.1016/S0378-4371(00)00255-7
[11] Mainardi F. Fractional calculus: some basic problems in continuum and statistical mechanics , Fractals and Fractional Calculus in
Continuum Mechanics. Vienna: Springer; 1997:291-348.
[12] Ashyralyev A. Well-posedness of the Basset problem in spaces of smooth functions , Appl Math Lett. 2011;24:1176-1180. DOI: https://doi.org/10.1016/j.aml.2011.02.002
[13] Karczewska A, Lizama C. Solutions to stochastic fractional relaxation equations , Phys Scr T. 2009;136:014030. 7pp. DOI: https://doi.org/10.1088/0031-8949/2009/T136/014030
[14] Lizama C, Prado H. Fractional relaxation equations on Banach spaces , Appl Math Lett. 2010;23:137-142. DOI: https://doi.org/10.1016/j.aml.2009.08.019
[15] Bazhlekova E. Estimates for a general fractional relaxation equation and application to an inverse source problem, Math Methods Appl
Sci. 2018;41(18):9018-9026. doi:10.1002/mma.4868. DOI: https://doi.org/10.1002/mma.4868
[16] Azhar Ali Zafar, Jan Awrejcewicz, Olga Mazur, Muhammad Bilal Riaz. Study of composite fractional relaxation differential equation using fractional operators with and without singular kernels and special functions , Advances in Continuous and Discrete Models. https://doi.org/10.1186/s13662-021-03227-w DOI: https://doi.org/10.1186/s13662-021-03227-w
[17] Fan Z, Dong Q, Li G. Approximate controllability for semilinear composite fractional relaxation equations , Frac Calc Appl Anal.
2016;19(1):267-284
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Copyright (c) 2025 Doan Vuong Nguyen; Nguyen Hoang Tuan (Corresponding Author)
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