Nothing Special   »   [go: up one dir, main page]
Skip to main content

Advertisement

Springer Nature Link
Log in

The Quasi-Boundary Value Method for Identifying the Initial Value of the Space-Time Fractional Diffusion Equation

  • Published:
Acta Mathematica Scientia Aims and scope Submit manuscript

Abstract

In this article, we consider to solve the inverse initial value problem for an inhomogeneous space-time fractional diffusion equation. This problem is ill-posed and the quasi-boundary value method is proposed to deal with this inverse problem and obtain the series expression of the regularized solution for the inverse initial value problem. We prove the error estimates between the regularization solution and the exact solution by using an a priori regularization parameter and an a posteriori regularization parameter choice rule. Some numerical results in one-dimensional case and two-dimensional case show that our method is efficient and stable.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Explore related subjects

Discover the latest articles and news from researchers in related subjects, suggested using machine learning.

References

  1. Metzler R, Klafter J. Boundary value problems for fractional diffusion equations. Physica A, 2010, 278: 107–125

    MathSciNet  Google Scholar 

  2. Yuste S B, Lindenberg K. Subdiffusion-limited reactions. Chem Phys, 2002, 284: 169–180

    Google Scholar 

  3. Magin R, Feng X, Baleanu D. Solving the fractional order Bloch equation. Concept Magn Reson A, 2009, 34: 16–23

    Google Scholar 

  4. Chen W, Ye L J, Sun H G. Fractional diffusion equations by the kansa method. Comput Math Appl, 2010, 59: 1014–1620

    MathSciNet  MATH  Google Scholar 

  5. Kirane M, Malik A S, Al-Gwaiz M A. An inverse source problem for a two dimensional time fractional diffusion equation with nonlocal boundary condition. Math Methods Appl Sci, 2013, 36: 1056–1069

    MathSciNet  MATH  Google Scholar 

  6. Lesnic D, Hussein S O, Johansson B T. Inverse space-dependent force problems for the wave equation. J Comput Appl Math, 2016, 306: 10–39

    MathSciNet  MATH  Google Scholar 

  7. Lopushansky A, Lopushanska H. Inverse source cauchy problem for a time fractional diffusion-wave equation with distributions. Electron J Differ Equ, 2017, 182: 1–14

    MathSciNet  Google Scholar 

  8. Sakamoto K, Yamamoto M. Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems. J Math Anal Appl, 2011, 382: 426–447

    MathSciNet  MATH  Google Scholar 

  9. Šišková K, SlodiČka M. Recognition of a time-dependent source in a time-fractional wave equation. Appl Numer Math, 2017, 121: 1–17

    MathSciNet  MATH  Google Scholar 

  10. Tuan N H, Le D L, Ngu| V T. Regularization of an inverse source problem for a time fractional diffusion equation. Appl Math Model, 2016, 40: 8244–8264

    MathSciNet  MATH  Google Scholar 

  11. Wei T, Sun L L, Li Y S. Uniqueness for an inverse space-dependent source term in a multi-dimensional time-fractional diffusion equation. Appl Math Lett, 2016, 61: 108–113

    MathSciNet  MATH  Google Scholar 

  12. Wei T, Wang J G. A modified quasi-boundary value method for an inverse source problem of the time- fractional diffusion equation. Appl Numer Math, 2014, 78: 95–111

    MathSciNet  MATH  Google Scholar 

  13. Yang F, Fu C L. The quasi-reversibility regularization method for identifying the unknown source for time-fractional diffusion equation. Appl Math Model, 2015, 39: 1500–1512

    MathSciNet  MATH  Google Scholar 

  14. Yang F, Fu C L, Li X X. A mollification regularization method for unknown source in time-fractional diffusion equation. Int J Comput Math, 2014, 91: 1516–1534

    MathSciNet  MATH  Google Scholar 

  15. Yang F, Fu C L, Li X X. The inverse source problem for time fractional diffusion equation: stability analysis and regularization. Inverse Probl Sci Eng, 2015, 23: 969–996

    MathSciNet  MATH  Google Scholar 

  16. Yang F, Fu C L, Li X X. Identifying an unknown source in space-fractional diffusion equation. Acta Math Sci, 2014, 34: 1012–1024

    MathSciNet  MATH  Google Scholar 

  17. Zhang Z Q, Wei T. Identifying an unknown source in time-fractional diffusion equation by a truncation method. Appl Math Comput, 2013, 219: 5972–5983

    MathSciNet  MATH  Google Scholar 

  18. Cheng H, Fu C L. An iteration regularization for a time-fractional inverse diffusion problem. Appl Math Model, 2012, 36: 5642–5649

    MathSciNet  MATH  Google Scholar 

  19. Yang M, Liu J J. Solving a final value fractional diffusion problem by boundary condition regulafization. Appl Numer Math, 2013, 66: 45–58

    MathSciNet  Google Scholar 

  20. Zhang Z Q, Wei T. An optimal regularization method for space-fractional backward diffusion problem. Math Comput Simulat, 2013, 92: 14–27

    MathSciNet  Google Scholar 

  21. Ren C X, Xu X, Lu S. Regularization by projection for a backward problem of the time-fractional diffusion equation. J Inverse Ill-Pose Probl, 2014, 22: 2011–0021

    MathSciNet  Google Scholar 

  22. Yang F, Fan P, Li X X, Ma X Y. Fourier truncation regularization method for a time-fractional backward diffusion problem with a nonlinear source. Mathematics, 2019, 7: 865

    Google Scholar 

  23. Yang F, Pu Q, Li X X, Li D G. The truncation regularization method for identifying the initial value on non-homogeneous time-fractional diffusion-wave equations. Mathematics, 2019, 7: 1007

    Google Scholar 

  24. Wei Z L, Li Q D, Che J L. Initial value problems for fractional differential equations involving Riemann- Liouville sequential fractional derivative. J Math Anal Appl, 2010, 367: 260–272

    MathSciNet  MATH  Google Scholar 

  25. Wang J G,Wei T, Zhou Y B. Tikhonov regularization method for a backward problem for the time-fractional diffusion. Appl Math Model, 2013, 37: 8518–8523

    MathSciNet  MATH  Google Scholar 

  26. Yang F, Sun Y R, Li X X, Huang C Y. The quasi-boundary value method for identifying the initial value of heat equation on a columnar symmetric domain. Numerical Algorithms 2019, 82(2): 623–639

    MathSciNet  MATH  Google Scholar 

  27. Yang F, Zhang Y, Li X X. Landweber iterative method for identifying the initial value problem of the time- space fractional diffusion-wave equation. Numerical Algorithms, DOI: doi.org/10.1007/s11075-019-00734-6

  28. Yang F, Wang N, Li X X. A quasi-boundary regularization method for identifying the initial value of time-fractional diffusion equation on spherically symmetric domain. J Inverse Ill-posed Probl, 2019, 27(5): 609–621

    MathSciNet  MATH  Google Scholar 

  29. Yang F, Fan P, Li X X. Fourier truncation regularization method for a three-dimensional Cauchy problem of the modified helmholtz equation with perturbed wave number. Mathematics, 2019, 7: 705

    Google Scholar 

  30. Yang F, Fu C L, Li X X. A modified tikhonov regularization method for the cauchy problem of Laplace equation. Acta Mathematica Scientia, 2015, 35B(6): 1339–1348

    MathSciNet  MATH  Google Scholar 

  31. Wei T, Zhang Z Q. Stable numerical solution to a Cauchy problem for a time fractional diffusion eqution. Eng Anal Bound Elem, 2014, 40: 128–137

    MathSciNet  MATH  Google Scholar 

  32. Zheng G H, Wei T. A new regularization method for a Cauchy problem of the time fractional diffusion equation. Adv Comput Math, 2012, 36: 377–398

    MathSciNet  MATH  Google Scholar 

  33. Zhang H W. Modified quasi-boundary value method for a Cauchy problem of semi-linear elliptic equation. Int J Comput Math, 2012, 89: 1689–1703

    MathSciNet  MATH  Google Scholar 

  34. Zheng G H, Wei T. Two regularization methods for solving a Riesz-Feller space-fractional backward diffusion problem. Inverse Probl, 2010, 26: 115017

    MathSciNet  MATH  Google Scholar 

  35. Li G S, Yao D, Jiang H Y, Jia X Z. Numerical inversion of a time-dependent reaction coefficient in a soil-column infiltrating experiment. Comp Model Eng Sci, 2011, 74: 83–107

    MathSciNet  MATH  Google Scholar 

  36. Jin B T, Rundell W. A tutorial on inverse problems for anomalous diffusion processes. Inverse Probl, 2015, 31: 35–43

    MathSciNet  MATH  Google Scholar 

  37. Tatar S, Ulusoy S. An inverse source problem for a one-dimensional space-time fractional diffusion equation. Appl Anal, 2015, 94: 2233–2244

    MathSciNet  MATH  Google Scholar 

  38. Tuan N H, Long L D. Fourier truncation method for an inverse source problem for space-time fractional diffusion equation. Electron J Differ Equ, 2017, 2017: 1–16

    MathSciNet  MATH  Google Scholar 

  39. Podlubny I. Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. San Diego: Academic Press Inc, 1999

    MATH  Google Scholar 

  40. Kilbas A A, Srivastava H M, Trujillo J J. Theory and application of fractional differential equations (North- Holland Mathematic Studies). Elsevier Science}, 2006, 204: 2453–2461

  41. Dang D T, Quan P H, Tuan N H. A quasi-boundary value method for regularizing nonlinear ill-posed problems. Electron J Differ Equ, 2009, 109: 397–415

    MathSciNet  MATH  Google Scholar 

  42. Feng X L, Eldén L, Fu C L. A quasi-boundary-value method for the Cauchy problem for elliptic equations with nonhomgenous Neumann data. J Inverse Ill-posed Probl, 2010, 18: 617–645

    MathSciNet  MATH  Google Scholar 

  43. Feng X L, Ning W T, Qian Z. A quasi-boundary-value method for a Cauchy problem of an elliptic equation in multiple dimensions. Inverse Probl Sci Eng, 2014, 22: 1045–1061

    MathSciNet  MATH  Google Scholar 

  44. Kirane M, Malik S A. Determination of an unknown source term and the temperature distribution for the linear heat equation involving fractional derivative in time. Inverse Probl Sci Eng, 2011, 19: 409–423

    MathSciNet  MATH  Google Scholar 

  45. Wei T, Wang J G. A modified quasi-boundary value method for the backward time-fractional diffusion problem. ESAIM-Math Model Num, 2014, 48: 603–621

    MathSciNet  MATH  Google Scholar 

  46. Podlubny I. Fractional differential equations. San Diego: Academic Press Inc, 1998

    MATH  Google Scholar 

  47. Luchko Y. Initial-boundary-value probelms for the one dimensional time-fractional diffusion equation. Fractional Calculus Appl Anal, 2012, 15: 141–160

    MathSciNet  MATH  Google Scholar 

  48. Podlubny I, Kaccenak M. Mittag-Leffler function. The matlabroutine. http://www.mathworks.com/matlabcentral/fileexchange, 2006

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Lanzhou University of Technology, Lanzhou, 730050, China

    Fan Yang  (杨帆), Yan Zhang  (张燕), Xiao Liu  (刘霄) & Xiaoxiao Li  (李晓晓)

Authors
  1. Fan Yang  (杨帆)
    View author publications

    Search author on:PubMed Google Scholar

  2. Yan Zhang  (张燕)
    View author publications

    Search author on:PubMed Google Scholar

  3. Xiao Liu  (刘霄)
    View author publications

    Search author on:PubMed Google Scholar

  4. Xiaoxiao Li  (李晓晓)
    View author publications

    Search author on:PubMed Google Scholar

Corresponding authors

Correspondence to Fan Yang  (杨帆), Yan Zhang  (张燕), Xiao Liu  (刘霄) or Xiaoxiao Li  (李晓晓).

Additional information

The project is supported by the National Natural Science Foundation of China (11561045, 11961044) and the Doctor Fund of Lan Zhou University of Technology.

Electronic supplementary material

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yang, F., Zhang, Y., Liu, X. et al. The Quasi-Boundary Value Method for Identifying the Initial Value of the Space-Time Fractional Diffusion Equation. Acta Math Sci 40, 641–658 (2020). https://doi.org/10.1007/s10473-020-0304-5

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10473-020-0304-5

Key words

2010 MR Subject Classification