The Backward Problem for a Nonlinear Riesz-Feller Diffusion Equation

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Abstract

In this paper, we reconstruct the solution u(x,t) of the backward space-fractional diffusion problem with a locally Lipschitzian nonlinear source

This problem is severely ill-posed in the Hadamard sense, hence, a regularization is in order. In the paper, we introduce one spectral regularization method and establish stability error estimates with optimal order under an a priori choice of regularization parameter. Finally, numerical implementations are given to show the effectiveness of the proposed regularization methods.

Keywords

Space-fractional backward diffusion problem Ill-posed problem Regularization Error estimate 

Mathematics Subject Classification (2010)

26A33 47A52 47J06 65M32 

Notes

Acknowledgements

The authors would like to thank the referees for the careful reading, helpful comments and suggestions leading to the improved version of the paper.

References

  1. 1.
    Cazenave, T., Haraux, A.: An Introduction to Semilinear Evolution Equations. The Clarendon Press, Oxford University Press, New York (1998)MATHGoogle Scholar
  2. 2.
    Cheng, H., Fu, C.-L., Zheng, G.-H., Gao, J.: A regularization for a Riesz-Feller space-fractional backward diffusion problem. Inverse Probl. Sci. Eng. 22 (6), 860–872 (2014)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Clark, G.W., Oppenheimer, S.F.: Quasireversibility methods for non-well-posed problems. Electron. J. Differential Equations 1994(8), 1–9 (1994)MathSciNetMATHGoogle Scholar
  4. 4.
    Mainardi, F., Luchko, Y., Pagnini, G.: The fundamental solution of the space-time fractional diffusion equation. Fract. Calc. Appl. Anal. 4(2), 153–192 (2001)MathSciNetMATHGoogle Scholar
  5. 5.
    Liu, F., Zhuang, P., Anh, V., Turner, I., Burrage, K.: Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation. Appl. Math. Comput. 191(1), 12–20 (2007)MathSciNetMATHGoogle Scholar
  6. 6.
    Gorenflo, R., Mainardi, F.: Some recent advances in theory and simulation of fractional diffusion processes. J. Comput. Appl.Math. 229(2), 400–415 (2009)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Metzler, R., Klafter, J.: Boundary value problems for fractional diffusion equations. Phys. A 278(1-2), 107–125 (2000)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)MATHGoogle Scholar
  9. 9.
    Ray, S.S.: A new approach for the application of Adomian decomposition method for the solution of fractional space diffusion equation with insulated ends. Appl. Math. Comput. 202(2), 544–549 (2008)MathSciNetMATHGoogle Scholar
  10. 10.
    Shi, C., Wang, C., Zheng, G., Wei, T.: A new a posteriori parameter choice strategy for the convolution regularization of the space-fractional backward diffusion problem. J. Comput. Appl. Math. 279, 233–248 (2015)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Hao, D.N.: A mollification method for ill-posed problems. Numer. Math. 68(4), 469–506 (1994)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Hai, D.N.D., Tuan, N.H., Long, L.D., Thong, L.G.Q.: Inverse problem for nonlinear backward space-fractional diffusion equation. J. Inverse Ill-Posed Probl. 25 (4), 423–443 (2017)MathSciNetMATHGoogle Scholar
  13. 13.
    Tuan, N.H., Trong, D.D.: Sharp estimates for approximations to a nonlinear backward heat equation. Nonlinear Anal. 73(11), 3479–3488 (2010)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Tuan, N.H., Trong, D.D., Quan, P.H.: A modified integral equation method of the semilinear backward heat problem. Appl. Math. Comput. 217(12), 5177–5185 (2011)MathSciNetMATHGoogle Scholar
  15. 15.
    Tuan, N.H., Hai, D.N.D., Long, L.D., Thinh, N.V., Kirane, M.M.: On a Riesz - Feller space fractional backward diffusion problem with a nonlinear source. J. Comput. Appl. Math. 312, 103–126 (2017)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Yang, Q., Liu, F., Turner, I.: Numerical methods for fractional partial differential equations with Riesz space fractional derivatives. Appl. Math. Model. 34 (1), 200–218 (2010)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Zhang, Z.Q., Wei, T.: An optimal regularization method for space-fractional backward diffusion problem. Math. Comput. Simulation 92, 14–27 (2013)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Zheng, G.H., Wei, T.: Two regularization methods for solving a Riesz-Feller space-fractional backward diffusion problem. Inverse Prob. 26(11), 22 pp (2010)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Zhao, J., Liu, S., Liu, T.: An inverse problem for space-fractional backward diffusion problem. Math. Methods Appl. Sci. 37(8), 1147–1158 (2014)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceHo Chi Minh City National UniversityHo Chi Minh CityVietnam
  2. 2.Faculty of Basic ScienceHo Chi Minh City University of TransportHo Chi Minh CityVietnam

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