Abstract
We consider a fractional diffusion equation with variable space-dependent order of the derivative in a bounded multidimensional domain. The initial data are homogeneous and the right-hand side and its time derivative satisfy some monotonicity conditions. Addressing the inverse problem with final overdetermination, we establish the uniqueness of a solution as well as some necessary and sufficient solvability conditions in terms of a certain constructive operator \( A \). Moreover, we give a simple sufficient solvability condition for the inverse problem. The arguments rely on the Birkhoff–Tarski theorem.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Kian Y., Soccorsi E., and Yamamoto M., “On time-fractional diffusion equations with space-dependent variable order,” Ann. Henri Poincare, vol. 19, no. 12, 3855–3881 (2018).
Van Bockstal K., “Existence of a unique weak solution to a non-autonomous time-fractional diffusion equation with space-dependent variable order,” Adv. Difference Equ. (2021) (Article 314, 43 pp.).
Gripenberg G., Londen S.-O., and Staffans O., Volterra Integral and Functional Equations, Cambridge University, Cambridge (1990).
Zacher R., “Weak solutions of abstract evolutionary integro-differential equations in Hilbert spaces,” Funkc. Ekv., vol. 52, no. 1, 1–18 (2009).
Wittbold P., Wolejko P., and Zacher R., Bounded Weak Solutions of Time-Fractional Porous Medium Type and More General Nonlinear and Degenerate Evolutionary Integro-Differential Equations [Preprint] (2008) (arXiv: 2008.10919v1).
Artyushin A.N., “Fractional integral inequalities and their applications to degenerate differential equations with the Caputo fractional derivative,” Sib. Math. J., vol. 61, no. 2, 208–221 (2020).
Orlovsky D.G., “Parameter determination in a differential equation of fractional order with Riemann–Liouville fractional derivative in a Hilbert space,” J. Sib. Fed. Univ. Math. Phys., vol. 8, no. 1, 55–63 (2015).
Sakamoto K. and Yamamoto M., “Inverse source problem with a final overdetermination for a fractional diffusion equation,” Math. Control Related Fields, vol. 1, no. 4, 509–518 (2011).
Van Bockstal K., “Uniqueness for inverse source problems of determining a space dependent source in time-fractional equations with non-smooth solutions,” Fractal Fract., vol. 5, no. 4 (2021) (Article 169, 11 pp.).
Kinash N. and Janno J., “Inverse problems for a generalized subdiffusion equation with final overdetermination,” Math. Model. Anal., vol. 24, no. 2, 236–262 (2019).
Kinash N. and Janno J., “Inverse problem to identify a space-dependent diffusivity coefficient in a generalized subdiffusion equation from final data,” Proc. Est. Acad. Sci., vol. 71, no. 1, 3–15 (2022).
Janno J. and Kinash N., “Reconstruction of an order of derivative and a source term in a fractional diffusion equation from final measurements,” Inverse Problems, vol. 34, no. 2 (2018) (Article 025007, 19 pp.).
Alimov S. and Ashurov R., “Inverse problem of determining an order of the Riemann–Liouville time-fractional derivative,” Progr. Fract. Differ. Appl., vol. 8, no. 4, 467–474 (2022).
Prilepko A.I. and Kostin A.B., “Inverse problems of the determination of the coefficient in parabolic equations. I,” Sib. Math. J., vol. 33, no. 3, 489–496 (1992).
Prilepko A.I., Orlovsky D.G., and Vasin I.A., Methods for Solving Inverse Problems in Mathematical Physics, Marcel Dekker, New York (2000).
Lyusternik L.A. and Sobolev V.I., A Concise Course in Functional Analysis, Higher School Publishing House, Moscow (1982) [Russian].
Zacher R., “Boundedness of weak solutions to evolutionary partial integro-differential equations with discontinuous coefficients,” J. Math. Anal. Appl., vol. 348, no. 1, 137–149 (2008).
Nakhushev A.M., Fractional Calculus and Its Applications, Fizmatgiz, Moscow (2003) [Russian].
Zacher R., “A weak Harnack inequality for fractional evolution equations with discontinuous coefficients,” Ann. Sc. Norm. Super. Pisa Cl. Sci., vol. 12, no. 5, 903–940 (2013).
Meyer-Nieberg P., Banach Lattices, Springer, Berlin (1991).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Sibirskii Matematicheskii Zhurnal, 2023, Vol. 64, No. 4, pp. 675–686. https://doi.org/10.33048/smzh.2023.64.402
Rights and permissions
About this article
Cite this article
Artyushin, A.N. An Inverse Problem of Recovering the Variable Order of the Derivative in a Fractional Diffusion Equation. Sib Math J 64, 796–806 (2023). https://doi.org/10.1134/S003744662304002X
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S003744662304002X