An inverse source Cauchy-weighted time-fractional diffusion problem

Abstract

In the present paper, we are concerned with an inverse source Cauchy weighted problem involving a one-dimensional diffusion equation with a time-fractional Riemann-Liouville derivative with $0<\alpha <1$. We start with results on the existence and regularity of the weak solution of the direct problem. Then, we investigate the invertibility of the input-output mapping defined by the additional over-determination integral data in order to the determination of the unknown time-dependent source coefficient.

Keywords

• inverse source problem
• time-fractional diffusion equation
• Riemann-Liouville fractional derivative
• Cauchy-weighted problem
• input-output mapping

References

1. [1] J. Cheng, J. Nakagawa, M. Yamamoto and T. Yamazaki, Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation, Inv. Problems, 25, 2009.
2. [2] A. Demir, F. Kanca and E. Ozbilge,Numerical solution and distinguishability in time fractional parabolic equation, Bound. Value Probl. 142, 2015.
3. [3] R. Faizi and R. Atmania, An inverse source problem of a semilinear time-fractional reaction-diffusion equation, Applicable Analysis, 102 (11), 2939-2959, 2022.
4. [4] R. Gorenflo, F. Mainardi, D. Moretti and P. Paradisi,Time fractional diffusion: a discrete random walk approach, Nonlinear Dynamics, 29, 129-143, 2002.
5. [5] A. Hasanov, A. Demir and A. Erdem,Monotonicity of input–output mappings in inverse coefficient and source problems for parabolic equations, J. Math. Anal. Appl. 335, 1434-1451, 2007.
6. [6] N. Heymans and I. Podlubny,Physical nterpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivative, Rheologica Acta, 45, 765-771, 2006.
7. [7] R. Hilfer,Application of fractional in physics, World scientific publishing company, Singapore, 2000.
8. [8] B. Kaltenbacher and W. Rundell, On an inverse potential problem for a fractional reaction–diffusion equation, Inv. Problems, 35, 2019.

9. [9] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo,Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.
10. [10] A. A. Kilbas, J. J. Trujillo and A. A. Voroshilov,Cauchy type problem for diffusionwave equations with the Riemann-Liouville derivative, Fract. Cal. and Appl. Anal. 8 (4), 403-430, 2005.
11. [11] Z. Li, Y. Liu and M. Yamamoto,Initial-boundary value problems for multi-term timefractional diffusion equations with positive constant coefficients, Appl. Math. and Comput. 257, 381-397, 2015.
12. [12] Yu. Luchko,Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation, Comput. Math. Appl. 59, 1766-1772, 2010.
13. [13] A. Sa’idu and H. Koyunbakan,Inverse fractional Sturm-Liouville problem with eigenparameter in the boundary conditions, Math. Meth. in the Appl. Sc. 45 (17), 11003- 11012, 2022.
14. [14] K. Sakamoto and M. Yamamoto, Inverse source problem with a final overdetermination for a fractional diffusion equation, Math. Contr. and related fields, 1 (4), 509-518, 2011.
15. [15] S. G. Samko, A. A. Kilbas and D. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, 1993.
16. [16] L. Settara and R. Atmania,An inverse coefficient-source problem for a time-fractional diffusion equation, Int. J. of Appl. Math. and Stat. 57 (3), 68-78, 2018.
17. [17] S. Umarov, On fractional Duhamels principle and its applications, J. D. Equations, 252, 5217-5234, 2012.
18. [18] S. Wang, M. Zhang and X. Li, Radial anomalous diffusion in an annulus, Physica A, 390, 3397-3403, 2011.