Identification of the time-dependent source term in a multi-term time-fractional diffusion equation

  • Y. S. Li
  • L. L. Sun
  • Z. Q. Zhang
  • T. WeiEmail author
Original Paper


The multi-term time-fractional diffusion equation is a useful tool in the modeling of complex systems. This paper aims to identifying a time-dependent source term in a multi-term time-fractional diffusion equation from the boundary Cauchy data. The regularity of the weak solution for the direct problem with homogeneous Neumann boundary condition is proved. We provide the uniqueness and a stability estimate for the inverse time-dependent source problem. On the other hand, the inverse time-dependent source term is formulated into a variational problem by the Tikhonov regularization, with the help of sensitivity problem and adjoint problem we use a conjugate gradient method to find the approximate time-dependent source term. Numerical experiments for five examples in one-dimensional and two-dimensional cases show that our proposed method is effective and stable.


Inverse source problem Multi-term time-fractional diffusion equation Conjugate gradient method 


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  1. 1.
    Adams, E.E., Gelhar, L.W.: Field-study of dispersion in a heterogeneous aquifer.2. spatial moments analysis. Water Resour. Res. 28(12), 3293–3307 (1992)CrossRefGoogle Scholar
  2. 2.
    Berkowitz, B., Scher, H., Silliman, S.E.: Anomalous transport in laboratory-scale, heterogeneous porous media. Water Resour. Res. 36, 149–158 (2000)CrossRefGoogle Scholar
  3. 3.
    Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York (2011)zbMATHGoogle Scholar
  4. 4.
    Caputo, M.: Mean fractional-order-derivatives differential equations and filters. Ann. Univ. Ferrara. 41, 73–84 (1995)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Chechkin, A.V., Gorenflo, R., Sokolov, I.M., Gonchar, V.Y.: Distributed order time fractional diffusion equation. Fract. Calc. Appl. Anal. 6(3), 259–279 (2003)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Coimbra, C.F.M.: Mechanics with variable-order differential operators. Ann. Phys. 12(11–12), 692–703 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Courant, R., Hilbert, D.: Methods of Mathematical Physics, vol. I. Interscience Publishers, Inc., New York (1953)zbMATHGoogle Scholar
  8. 8.
    Daftardar-Gejji, V., Bhalekar, S.: Boundary value problems for multi-term fractional differential equations. J. Math. Anal. Appl. 345(2), 754–765 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dehghan, M., Safarpoor, M., Abbaszadeh, M.: Two high-order numerical algorithms for solving the multi-term time fractional diffusion-wave equations. J. Comput. Appl. Math. 290, 174–195 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems, Mathematics and Its Applications, vol. 375. Kluwer Academic Publishers Group, Dordrecht (1996)CrossRefzbMATHGoogle Scholar
  11. 11.
    Giona, M., Cerbelli, S., Roman, H.E.: Fractional diffusion equation and relaxation in complex viscoelastic materials. Physica A 191(1-4), 449–453 (1992)CrossRefGoogle Scholar
  12. 12.
    Hanke, M., Hansen, P.C.: Regularization methods for large-scale problems. Survey Math. Indust. 3(4), 253–315 (1993)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Hasanov, A., DuChateau, P., Pektaş, B.: An adjoint problem approach and coarse-fine mesh method for identification of the diffusion coefficient in a linear parabolic equation. J. Inverse Ill-Posed Probl. 14(5), 435–463 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hatano, Y., Hatano, N.: Dispersive transport of ions in column experiments: an explanation of long-tailed profiles. Water Resour. Res. 34(5), 1027–1033 (1998)CrossRefGoogle Scholar
  15. 15.
    Henry, B.I., Langlands, T.A.M., Wearne, S.L.: Fractional cable models for spiny neuronal dendrites. Phys. Rev. Lett. 100, 128103 (2008)CrossRefGoogle Scholar
  16. 16.
    Jiang, H., Liu, F., Turner, I., Burrage, K.: Analytical solutions for the multi-term time-fractional diffusion-wave/diffusion equations in a finite domain. Comput. Math. Appl. 64(10), 3377–3388 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Jin, B., Lazarov, R., Liu, Y., Zhou, Z.: The Galerkin finite element method for a multi-term time-fractional diffusion equation. J. Comput. Phys. 281, 825–843 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kilbas, A.A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations, vol. 204. Elsevier, Amsterdam (2006)zbMATHGoogle Scholar
  19. 19.
    Li, G.S., Sun, C.L., Jia, X., Du, D.H.: Numerical solution to the multi-term time fractional diffusion equation in a finite domain. Numer. Math. Theory Methods Appl. 9(3), 337–357 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Li, Z., Imanuvilov, O.Y., Yamamoto, M.: Uniqueness in inverse boundary value problems for fractional diffusion equations. Inverse Prob. 32(1), 015004 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Li, Z., Liu, Y, Yamamoto, M.: Initial-boundary value problems for multi-term time-fractional diffusion equations with positive constant coefficients. Appl. Math. Comput. 257, 381–397 (2015)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Li, Z., Yamamoto, M.: Uniqueness for inverse problems of determining orders of multi-term time-fractional derivatives of diffusion equation. Appl. Anal. 94(3), 570–579 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Liu, F., Meerschaert, M.M., McGough, R.J., Zhuang, P., Liu, Q.: Numerical methods for solving the multi-term time-fractional wave-diffusion equation. Fract. Calc. Appl. Anal. 16(1), 9–25 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Liu, Y.: Strong maximum principle for multi-term time-fractional diffusion equations and its application to an inverse source problem. Comput. Math. Appl. 73(1), 96–108 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Lorenzo, C.F., Hartley, T.T.: Variable order and distributed order fractional operators. Nonlinear Dyn. 29(1-4), 57–98 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Luchko, Y.: Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation. J. Math. Anal. Appl. 374(2), 538–548 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Metzler, R., Klafter, J.: Boundary value problems for fractional diffusion equations. Physica A: Statistical Mechanics and its Applications 278, 107–125 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Metzler, R., Klafter, J.: Subdiffusive transport close to thermal equilibrium: from the Langevin equation to fractional diffusion. Phys. Rev. E 61, 6308–6311 (2000)CrossRefGoogle Scholar
  30. 30.
    Morozov, V.A., Nashed, Z., Aries, A.B.: Methods for Solving Incorrectly Posed Problems. Springer, New York (1984)CrossRefGoogle Scholar
  31. 31.
    Naber, M.: Distributed order fractional subdiffusion. Fractals 12, 23–32 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Thamban Nair, M.: Linear Operator Equations: Approximation and Regularization. World Scientific, Singapore (2009)CrossRefzbMATHGoogle Scholar
  33. 33.
    Ou, Y.H., Hasanov, A., Liu, Z.H.: Inverse coefficient problems for nonlinear parabolic differential equations. Acta Math. Sin. (Engl. Ser.) 24(10), 1617–1624 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Sokolov, I.M., Chechkin, A.V., Klafter, J.: Distributed-order fractional kinetics. Acta Phys. Polon. B 35, 1323–1341 (2004)Google Scholar
  35. 35.
    Sokolov, I.M., Klafter, J.: From diffusion to anomalous diffusion: a century after Einstein’s Brownian motion. Chaos 15(2), 1–7 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Taylor, M.E.: Partial differential equations I. Basic Theory, Second Edition, Applied Mathematical Sciences, vol. 115. Springer, New York (2011)Google Scholar
  37. 37.
    Wei, L.L.: Stability and convergence of a fully discrete local discontinuous Galerkin method for multi-term time fractional diffusion equations. Numer. Algorithms 76(3), 695–707 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Wei, T., Li, X.L., Li, Y.S.: An inverse time-dependent source problem for a time-fractional diffusion equation. Inverse Prob. 32, 085003 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Ye, H., Gao, J.M., Ding, Y.S.: A generalized Gronwall inequality and its application to a fractional differential equation. J. Math. Anal. Appl. 328(2), 1075–1081 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Yuste, S.B., Acedo, L., Lindenberg, K.: Reaction front in an A + B C reaction-subdiffusion process. Phys. Rev. E 69, 036126 (2004)CrossRefGoogle Scholar
  41. 41.
    Yuste, S.B., Lindenberg, K.: Subdiffusion-limited reactions. Chem. Phys. 284, 169–180 (2002)CrossRefGoogle Scholar
  42. 42.
    Zhao, Y.M., Zhang, Y.D., Liu, F., Turner, I., Tang, Y.F., Anh, V.: Convergence and superconvergence of a fully-discrete scheme for multi-term time fractional diffusion equations. Comput. Math. Appl. 73(6), 1087–1099 (2017)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Zheng, M., Liu, F., Anh, V., Turner, I.: A high-order spectral method for the multi-term time-fractional diffusion equations. Appl. Math. Model. 40(7-8), 4970–4985 (2016)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.School of Mathematics and StatisticsLanzhou UniversityGansuPeople’s Republic of China
  2. 2.School of Cyber SecurityGansu Institute of Political Science and LawLanzhouPeople’s Republic of China

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