International Journal of Nonlinear Analysis and Applications
An inverse problem for homogeneous time-fractional diffusion problem on the sphere
Document Type : Research Paper
Author
Division of Applied Mathematics, Thu Dau Mot University, Binh Duong Province, Vietnam
Abstract
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.
Keywords
[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)
262–278.
[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.
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)
262–278.
[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 2021Pages 653-662
- Receive Date: 27 May 2020
- Revise Date: 09 August 2020
- Accept Date: 20 August 2020