# On the source problem for the diffusion equations with conformable derivative

## Keywords:

Conformable derivative, ill-posed, source function, diffusion equations## Abstract

In this article, we are interested in the problem of finding the source function of the diffusions equations \(\partial_{t}^\alpha - \Delta u = f(x)\) (where \(f\) as the unknown source function and \(\alpha\in (0,1)\)). Furthermore, the fractional derivative \(\alpha\) of \(u\) is defined by the Conformable time derivative. This is an ill-posed problem. So, we use the Reguralized Tikhonov method to construct a regularization solution, and the estimation of convergence is also discussed.## References

A. Abdeljawad, R. P. Agarwal, E. Karapinar and P. S. Kumari: Solutions of the Nonlinear Integral Equation and Fractional Differential Equation Using the Technique of a Fixed Point with a Numerical Experiment in Extended b-Metric Space, Symmetry, 11 (2019), 686.

B. Alqahtani, H. Aydi, E. Karapinar and V. Rakocevic: A Solution for Volterra Fractional Integral Equations by Hybrid Contractions, Mathematics, 7 (2019), 694.

V. T. T. Ha, N. N. Hung and N. D. Phuong: Identifying inverse source for Diffusion equation with Conformable time derivative by Fractional Tikhonov method, Adv. Theory Nonlinear Anal. Appl., 6 (4) (2022), 433–450.

Y. Han, X. Xiong and X. Xue: A fractional Landweber method for solving backward time-fractional diffusion problem, Comput. Math. Appl., 78 (2019), 81–91.

A. Kirsch: An Introduction to the Mathematical Theory of Inverse Problems, Volume 120 of Applied Mathematical Sciences, second edition, Springer, New-York (2011).

A. R. Khalil, A.Yousef and M.Sababheh: A new definition of fractional derivetive , J. Comput. Appl. Math., 264 (2014), 65–70.

L. D. Long, N. H. Luc, Y. Zhou and C. Nguyen: Identification of Source term for the Time-Fractional Diffusion-Wave Equation by Fractional Tikhonov Method, Mathematics, 7 (10) (2019).

N. A. Triet, V. V. Au, L. D. Long, D. Baleanu and N. H. Tuan: Regularization of a terminal value problem for time fractional diffusion equation, Math. Methods Appl. Sci., 43 (6) (2020), 3850–3878.

N. H. Tuan, L. D. Long: Fourier truncation method for an inverse source problem for space-time fractional diffusion equation, Electron. J. Differ. Equ., 122 (2017), 1–16.

N. H. Tuan, L. D. Long and N. V. Thinh: Regularized solution of an inverse source problem for a time fractional diffusion equation, Appl. Math. Model., 40 (2016), 8244–8264.

F. Yang, C. L. Fu: The quasi-reversibility regularization method for identifying the unknown source for time fractional diffusion equation, Appl. Math. Model., 39 (2015), 1500–1512.

F. Yang, Y. P. Ren, X. X. Li: Landweber iterative method for identifying a spacedependent source for the time-fractional diffusion equation, Bound. Value Probl., 2017 (1) (2017), 163.

F. Yang, X. Liu, X. X. Li: Landweber iterative regularization method for identifying the unknown source of the timefractional diffusion equation, Adv. Differ. Equ., 2017 (1) (2017), 388.

## Downloads

## Published

## How to Cite

*Modern Mathematical Methods*,

*2*(2), 55–64. Retrieved from https://modernmathmeth.com/index.php/pub/article/view/24

## Issue

## Section

## License

Copyright (c) 2024 Modern Mathematical Methods

This work is licensed under a Creative Commons Attribution 4.0 International License.