Using a LDG method for solving an inverse source problem of the time-fractional diffusion equation

Document Type : Research Article

Authors

Isfahan University of Technology

Abstract

In this paper, we apply a local discontinuous Galerkin (LDG) method to solve some fractional inverse problems. In fact, we determine a timedependent source term in an inverse problem of the time-fractional diffusion equation. The method is based on a finite difference scheme in time and a LDG method in space. A numerical stability theorem as well as an error estimate is provided. Finally, some numerical examples are tested to confirm theoretical results and to illustrate effectiveness of the method. It must be pointed out that proposed method generates stable and accurate numerical approximations without using any regularization methods which are necessary for other numerical methods for solving such ill-posed inverse problems.

Keywords


1. Cheng, J., Nakagawa, J., Yamamoto, M. and Yamazaki, T. Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation, Inverse Probl, 25 (11) (2009), 115002.
2. Cockburn, B., Kanschat, G., Perugia, I. and Schotzau, D. Superconvergence of the local discontinuous Galerkin method for elliptic problems on cartesian grids, SIAM J. Numer. Anal, 39 (2001), 264-285.
3. Cockburn, B. and Shu, C.W. The Runge-Kutta discontinuous Galerkin method for conservation laws, V: multidimensional systems, J. Comput. Phys, 141 (1998), 199-224.
4. Cockburn, B. and Shu, C.W. The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal, 35 (1998), 2440-2463.
5. Deng, W.H. Finite element method for the space and time fractional Fokker-Planck equation, SIAM J. Numer. Anal, 47 (2008), 204-226.
6. Deng, W.H. and Hesthaven, J.S. Local discontinuous Galerkin methods for fractional diffusion equations, Math. Modelling Numer. Anal, 47 (2013),1845-1864.
7. Dou, F.F. and Hon, Y.C. Kernel-based approximation for Cauchy problem of the time-fractional diffusion equation, Eng. Anal. Boundary Elem, 36(2012), 1344-1352.
8. Dou, F.F. and Hon, Y.C. Numerical computation for backward time fractional diffusion equation, Eng. Anal. Boundary Elem, 40 (2014), 138-146.
9. Jiang, Y.J. and Ma, J.T. High-order finite element methods for time fractional partial differential equations, J. Comput. Appl. Math, 235 (2011), 3285-3290.
10. Jin, B.T. and Rundell, W. An inverse problem for a one-dimensional time-fractional diffusion problem, Inverse Probl, 28 (7) (2012), 075010.
11. Li, C.Z. and Chen, Y. Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion, Comput. Math. Appl, 62 (2011), 855-875.
12. Lin, Y.M. and Xu, C.J. Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys, 225 (2007), 1533-1552.
13. Liu, J.J. and Yamamoto, M. A backward problem for the time-fractional diffusion equation, Appl. Anal, 89 (2010), 1769-1788.
14. Liu, Y., Zhang, M., Li, H., Li, J. High-order local discontinuous Galerkin method combined with WSGD-approximation for a fractional subdiffusion equation, Comput. Math. Appl, 73 (2017), 1298-1314.
15. Metzler, R., Gl¨ockle, W.G. and Nonnenmacher, T.F. Fractional model equation for anomalous diffusion, Physica A, 211 (1994), 13-24.
16. Mohammadi, M., Mokhtari, R. and Panahipour, H. Solving two parabolic inverse problems with a nonlocal boundary condition in the reproducing kernel space, Appl. Comput. Math, 13 (2014), 91-106.
17. Mohammadi, M., Mokhtari, R. and Toutian, F. Solving an inverse prob lem for a parabolic equation with a nonlocal boundary condition in the reproducing kernel space, Iranian J. Numer. Anal. Optimization, 4 (2014),57-76.
18. Murio, D.A. Stable numerical solution of a fractional-diffusion inverse heat conduction problem, Comput. Math. Appl, 53 (2007), 1492-1501.
19. Murio, D.A. Time fractional IHCP with Caputo fractional derivatives, Comput. Math. Appl, 56 (2008), 2371-2381.
20. Murio, D.A. Stable numerical evaluation of Gr¨unwald-Letnikov fractional derivatives applied to a fractional IHCP, Inverse Probl. Sci. Eng, 17 (2009), 229-243.
21. Pourgholi, R. , Esfahani, A., Abtahi, M. A numerical solution of a two dimensional IHCP, J. Appl. Math. Comput, 41 (2013), 61-79.
22. Qian, Z. Optimal modified method for a fractional-diffusion inverse heat conduction problem, Inverse Probl. Sci. Eng, 18 (2010), 521-533.
23. Rashedi, K., Adibi, H., Dehghan. M. Determination of space-timedependent heat source in a parabolic inverse problem via the Ritz-Galerkin technique, Inverse Probl. Sci. Eng, 22 (2014), 1077-1108.
24. Sakamoto, K. and Yamamoto, M. Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl, 382 (2011), 426-447.
25. Tuan, V.K. Inverse problem for fractional diffusion equation, Fractional Calculus Appl. Anal, 14 (2011), 31-55.
26. Wang, T., Wang, Y.M. A modified compact ADI method and its extrapolation for two-dimensional fractional subdiffusion equations, J. Appl. Math. Comput, 52 (2016), 439-476.
27. Wei, T. and Wang, J.G. A modified quasi-boundary value method for an inverse source problem of the time-fractional diffusion equation, Appl. Numer. Math, 78 (2014), 95-111.
28. Wei, T. and Zhang, Z.Q. Reconstruction of a time-dependent source term in a time-fractional diffusion equation, Eng. Anal. Boundary Elem, 37 (2013), 23-31.
29. Wei, T., Zhang, Z.Q. Stable numerical solution to a Cauchy problem for a time fractional diffusion equation, Eng. Anal. Boundary Elem, 40 (2014), 128-137.
30. Xu, Q. and Hesthaven, J.S. Discontinuous Galerkin method for fractional convection-diffusion equations, To appear in SIAM J. Numer. Anal 31. Xu, Y. and Shu, C.W. Local Discontinuous Galerkin method for the Camassa-Holm equation, 46 (2008), 1998-2021.
32. Yeganeh, S., Mokhtari, R., Hesthaven, J.S. Space-dependent source determination in a time-fractional diffusion equation using a local discontinuous Galerkin method, BIT Numer Math, DOI 10.1007/s10543-017-0648-y.
33. Zhang, Y. and Xu, X. Inverse source problem for a fractional diffusion equation, Inverse Probl, 27 (3) (2011), 035010.
34. Zheng, G.H. and Wei, T. Spectral regularization method for a Cauchy problem of the time fractional advection-dispersion equation, J. Comput. Appl. Math, 233 (2010), 2631-2640.
35. Zheng, G.H. and Wei, T. A new regularization method for Cauchy problem of the fractional diffusion equation, Adv. Comput. Math, 36 (2012), 377-398.