Determination of unknown source term and boundary flux in a time-fractional inverse diffusion problem

Document Type : Original Paper


1 1 Department of Mathematics, Khuzestan Science and Research Branch, Islamic Azad University, Ahvaz, Iran, 2 Department of Mathematics, Ahvaz Branch, Islamic Azad University, Ahvaz, Iran.

2 School of Mathematics, Iran University of Science and Technology, Tehran, Iran

3 Department of Mathematics, Payame Noor University, Tehran, Iran


In this work, we consider an inverse fractional parabolic problem that has many applications in different fields. Simultaneously determination of a source term and a boundary flux function in a time-fractional order parabolic equation is investigated using a mollified space marching method. Superposition and Duhamel's principles, the uniqueness of solution with respect to an over determination condition is proved. The stability and convergence of the numerical method are proved and two numerical test problems are conducted to illustrate the ability and accuracy of the numerical method.


Main Subjects

[1] J. Cheng, J. Nakagawa, M. Yamamoto and T. Yamazaki, Uniqueness in an inverse problem for a one dimensional fractional diffusion equation, Inv. Prob. 25 (2009) 115002.
[2] M.D. Echeverry and C.E. Mejía , A two dimensional discrete mollification operator and the numerical solution of an inverse source problem, AXIOMS 7(4), 89 (2018).
[3] A. Fahim, M.A. Fariborzi Araghi, J. Rashidinia and M. Jalalvand, Numerical solution of Volterra partial integro-differential equations based on Sinc-collocation method, Advances in Difference Equations 362 (2017) 1–21. DOI 10.1186/s13662-017-1416 7.
[4] M. Garshasbi and H. Dastour, Estimation of unknown boundary functions in an inverse heat conduction problem using a mollified marching scheme, Numer. Algor. 68 (2015) 769–790.
[5] M. Garshasbi and H. Dastour, A mollified marching solution of an inverse ablation-type moving boundary problem, Comput. Appl. Math. 35 ( 2016) 60–73.
[6] M. Garshasbi, P. Reihani and H. Dastour, A stable numerical solution of a class of semi-linear Cauchy problems, J. Adv. Res. Dyn. Cont. Sys. 4 (2012) 56–67.
[7] R. Yan, M. Han, Q. Ma and X. Ding, A spectral collocation method for nonlinear fractional initial value problems with a variable-order fractional derivative, Comput. Appl. Math. 38, 66 (2019) 1–25.
[8] M. Jalalvand, B. Jazbi and M.R. Mokhtarzadeh, A finite difference method for the smooth solution of linear Volterra integral equations, Int. J. Non-linear Anal. Appl. 4(2) (2013) 1–10.
[9] Y. Lin and C. Xu, Finite difference-spectral approximation for the time-fractional diffusion equation, J. Comp. Phys. 225 (2007) 1533-1552.
[10] Y. Luchko, Maximum principle and its application for the time-fractional diffusion equations, Fract. Calcul. and Appl. Anal., 14 (2011) 110-124.
[11] A. Kumar, A moving boundary problem with space-fractional diffusion logistic population model and density-dependent dispersal rate, Appl. Math. Model. 88 (2020) 51-65.
[12] C.E. Mejía and A. Piedrahita, Solution of a time fractional inverse advection-dispersion problem by discrete mollification, Rev. Colomb. Mat. 51 (2017) 83–102.
[13] C.E. Mejía and A. Piedrahita, finite difference approximation of a two dimensional time fractional advection-dispersion problem, arXiv (2018), arXiv:1807.07393v1.
[14] R. Metzler and J. Klafter, The random walks guide to anomalous diffusion: a fractional dynamics approach, Physics Reports, Volume 339, Issue 1, (2000) 1-77.
[15] D.A. Murio, Mollification and Space Marching, in: K.Woodbury (Ed.), Inverse Engineering Handbook, CRC Press, Boca Raton, Florida, (2002).
[16] D.A. Murio, On the stable numerical evaluation of Caputo fractional derivatives, Comput. Math. Appl. 51 (2006) 1539-1550.
[17] R.R. Nigmatullin, The realization of the generalized transfer equation in a medium with fractal geometry, Phys. Stat. Sol. B 133 (1986) 25-30.
[18] M.A. Zaky, An accurate spectral collocation method for nonlinear systems of fractional differential equations and related integral equations with nonsmooth solutions, Appl. Numer. Math. 154 (2020) 205-222.
[19] C. Milici, G. Drăgănescu and J. T. Machado, Introduction to fractional differential equations, Springer, (2019).
[20] K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, preprint UTMS (2010) 2010–2014.
[21] Z.Z. Sun and X.N. Wu, A full discrete difference scheme for a diffusion-wave system, Appl. Numer. Math. 56 (2006) 193–209.
[22] T. Wei and Z.Q. Zhang, Reconstruction of a time-dependent source term in a time-fractional diffusion equation, Eng Anal Bound Elem 37 (2013), 23–31.
[23] Y. Zhang and X. Xu, Inverse source problem for a fractional diffusion equation, Inv. Prob. 27 (2011) 035010.
[24] L. Zhou and H.M. Selim, Application of the fractional advection-dispersion equations in porous media, Soil Sci. Soc. Am. J. 64 (4) (2003) 1079–1084.