An inverse problem for homogeneous time-fractional diffusion problem on the sphere

Document Type : Research Paper


Division of Applied Mathematics, Thu Dau Mot University, Binh Duong Province, Vietnam


 In this paper, we consider an inverse problem for the time-fractional diffusion equation on the sphere where the final data on the sphere are given. The problem is ill-posed in the sense of Hadamard. Hence, the regularization method has to be used for the stable approximate solution. Then the well-posedness of the proposed regularizing problem and convergence property of the regularizing solution to the exact one is proved. Error estimates for this method are provided together with a selection rule for the regularization parameter.


[1] E. Abdolmaleki, H. Saberi Najafi, Analytical Solution for the Time Fractional Newell-Whitehead-Segel Equation
by Using Modified Residual Power Series Method, International Journal of Nonlinear Analysis and Applications
10(Special Issue (Nonlinear Analysis in Engineering and Sciences)) (2019) 155–167.
[2] M. Badr, A. Yazdani, H. Jafari, Stability of a finite volume element method for the time-fractional advectiondiffusion equation, Numerical Methods for Partial Differential Equations 34(5) (2018) 1459–1471.
[3] N. Faraz, Y. Khan, H. Jafari, A. Yildirim, M. Madani, Fractional variational iteration method via modified
Riemann–Liouville derivative, Journal of King Saud University-Science 23(4) (2011) 413–417.
[4] H. Jafari, H. Tajadodi, New method for solving a class of fractional partial differential equations with applications,
Thermal Science 22(Supplement 1) (2018) 277–286.
[5] D. Kumar, J. Singh, D. Baleanu, A new analysis for fractional model of regularized long-wave equation arising in
ion acoustic plasma waves, Mathematical Methods in the Applied Sciences 40(15) (2017) 5642–5653.
[6] L.D. Long, Y. Zhou, T.T. Binh, N. Can, A Mollification Regularization Method for the Inverse Source Problem
for a Time Fractional Diffusion Equation, Mathematics 7(11) (2019) 1048.
[7] N.H. Luc, D. Baleanu, N.H. Can, Reconstructing the right-hand side of a fractional subdiffusion equation from
the final data, Journal of Inequalities and Applications 1 (2020) 1–15.
[8] W. McLean, Regularity of solutions to a time-fractional diffusion equation, ANZIAM J. 52(2) (2010) 123–138.
[9] I. Podlubny, Fractional differential equations, Academic Press, London, 1999.
[10] S.S. Roshan, H. Jafari, D. Baleanu, Solving FDEs with Caputo-Fabrizio derivative by operational matrix based on
Genocchi polynomials, Mathematical Methods in the Applied Sciences 41(18) (2018) 9134–9141.
[11] K. Sakamoto, M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and
applications to some inverse problems, J. Math. Anal. Appl. 382(1) (2011) 426–447
[12] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional integrals and derivatives: Theory and Applications, Gordon
and Breach Science Publishers, Nauka i Tekhnika, Minsk, 1987.
[13] M. Soluki, S. Rasouli, G. Afrouzi, On a class of nonlinear fractional Schr¨odinger-Poisson systems, International
Journal of Nonlinear Analysis and Applications 10(Special Issue (Nonlinear Analysis in Engineering and Sciences))
(2019) 123–132.
[14] Q.T. Le Gia Approximation of parabolic PDEs on spheres using collocation method, Adv. Comput. Math. 22(4)
(2005) 377–397.
[15] Q.T. L. Gia, N.H. Tuan, T. Tran, Solving the backward heat equation on the unit sphere, ANZIAM J. 56 (2016)
[16] D.D. Trong, N.H. Tuan, A nonhomogeneous backward heat problem: Regularization and error estimates, Electronic Journal of Differential Equations (EJDE)[electronic only], 2008(33) (2008) 1–14.
[17] D.D. Trong, N.H. Tuan, P.H. Quan, A quasi-boundary value method for regularizing nonlinear ill-posed problems,
Electronic Journal of Differential Equations (EJDE)[electronic only], 2009(109) (2009) 1–16.
[18] N.H. Tuan, L.D. Long, S. Tatar, Tikhonov regularization method for a backward problem for the inhomogeneous
time-fractional diffusion equation, Appl. Anal. 97(5) (2018) 842–863.
[19] N.H. Tuan, Y. Zhou, N.H. Can, Identifying inverse source for fractional diffusion equation with Riemann–Liouville
derivative, Computational and Applied Mathematics 39(2) (2020) 1–16.
[20] H. Zeidabadi, R. Pourgholi, S. Tabasi, Solving a nonlinear inverse system of Burgers equations, International
Journal of Nonlinear Analysis and Applications 10(1) (2019) 35–54.
Volume 12, Special Issue
December 2021
Pages 653-662
  • Receive Date: 27 May 2020
  • Revise Date: 09 August 2020
  • Accept Date: 20 August 2020