Filter regularization method for a nonlinear Riesz-Feller space-fractional backward diffusion problem with temporally dependent thermal conductivity
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Dinh Nguyen Duy Hai
Abstract
This paper concerns a backward problem for a nonlinear space-fractional diffusion equation with temporally dependent thermal conductivity. Such a problem is obtained from the classical diffusion equation by replacing the second-order space derivative with a Riesz-Feller derivative of order α ∈ (0, 2), which is usually used to model the anomalous diffusion. We show that the problem is severely ill-posed. Using the Fourier transform and a filter function, we construct a regularized solution from the data given inexactly and explicitly derive the convergence estimate in the case of the local Lipschitz reaction term. Special cases of the regularized solution are also presented. These results extend some earlier works on the space-fractional backward diffusion problem.
Acknowledgements
I would like to thank Prof. Dang Duc Trong for fruitful discussions on backward problems. Also, I wish to thank the Editor and the Reviewers for their most helpful comments on this paper. This work is supported by Vietnam National University-Ho Chi Minh City (VNU-HCM) under Grant number B2021-18-02.
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Articles in the same Issue
- Frontmatter
- Editorial
- FCAA related news, events and books (FCAA–volume 24–4–2021)
- Research Paper
- Three representations of the fractional p-Laplacian: Semigroup, extension and Balakrishnan formulas
- Tutorial paper
- The bouncing ball and the Grünwald-Letnikov definition of fractional derivative
- Research Paper
- Fractional diffusion-wave equations: Hidden regularity for weak solutions
- Censored stable subordinators and fractional derivatives
- Variational methods to the p-Laplacian type nonlinear fractional order impulsive differential equations with Sturm-Liouville boundary-value problem
- Multivariable fractional-order PID tuning by iterative non-smooth static-dynamic H∞ synthesis
- Filter regularization method for a nonlinear Riesz-Feller space-fractional backward diffusion problem with temporally dependent thermal conductivity
- The rate of convergence on fractional power dissipative operator on compact manifolds
- Fractional Langevin type equations for white noise distributions
- Local existence and non-existence for a fractional reaction–diffusion equation in Lebesgue spaces
- Maximum principles and applications for fractional differential equations with operators involving Mittag-Leffler function
- The Crank-Nicolson type compact difference schemes for a loaded time-fractional Hallaire equation
- Robust fractional-order perfect control for non-full rank plants described in the Grünwald-Letnikov IMC framework
- Properties of the set of admissible “state control” pair for a class of fractional semilinear evolution control systems
Articles in the same Issue
- Frontmatter
- Editorial
- FCAA related news, events and books (FCAA–volume 24–4–2021)
- Research Paper
- Three representations of the fractional p-Laplacian: Semigroup, extension and Balakrishnan formulas
- Tutorial paper
- The bouncing ball and the Grünwald-Letnikov definition of fractional derivative
- Research Paper
- Fractional diffusion-wave equations: Hidden regularity for weak solutions
- Censored stable subordinators and fractional derivatives
- Variational methods to the p-Laplacian type nonlinear fractional order impulsive differential equations with Sturm-Liouville boundary-value problem
- Multivariable fractional-order PID tuning by iterative non-smooth static-dynamic H∞ synthesis
- Filter regularization method for a nonlinear Riesz-Feller space-fractional backward diffusion problem with temporally dependent thermal conductivity
- The rate of convergence on fractional power dissipative operator on compact manifolds
- Fractional Langevin type equations for white noise distributions
- Local existence and non-existence for a fractional reaction–diffusion equation in Lebesgue spaces
- Maximum principles and applications for fractional differential equations with operators involving Mittag-Leffler function
- The Crank-Nicolson type compact difference schemes for a loaded time-fractional Hallaire equation
- Robust fractional-order perfect control for non-full rank plants described in the Grünwald-Letnikov IMC framework
- Properties of the set of admissible “state control” pair for a class of fractional semilinear evolution control systems
