Skip to main content
SpringerLink
Log in

A Stable Numerical Approach to Solve a Time-Fractional Inverse Heat Conduction Problem

  • Research Paper
  • Published:
Iranian Journal of Science and Technology, Transactions A: Science Aims and scope Submit manuscript

Abstract

This paper studies a time-fractional inverse heat conduction problem for identifying unknown Robin coefficients in the boundary conditions. This inverse problem is generally ill-posed. Thus, a mollification technique is used to obtain a regularized problem. Then, a finite difference marching method is employed to solve this problem and finally, we get estimations to the unknown coefficients. The stability of the approach is proved and an error analysis of the approximate solution is provided. Three examples are investigated to show feasibility and efficiency of the proposed method.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

References

  • Atangana A, Baleanu D (2016) New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model. Therm Sci 20(2):763–769

    Article  Google Scholar 

  • Baleanu D, Tenreiro Machado JA, Luo ACJ (2012) Fractional dynamics and control. Springer, New York

    Book  Google Scholar 

  • Barkai E, Metzler R, Klafter J (2000) From continuous time random walks to the fractional Fokker-Planck equation. Phys Rev E 61:132–8

    Article  MathSciNet  Google Scholar 

  • Chaves AS (1998) A fractional diffusion equation to describe Levy flights. Phys Lett A 239:13–6

    Article  MathSciNet  Google Scholar 

  • Depollier C, Fellah ZEA, Fellah M (2004) Propagation of transient acoustic waves in layered porous media fractional equations for the scattering operators. Nonlinear Dyn 38:181–190

    Article  MathSciNet  Google Scholar 

  • Garshasbi M, Dastour H (2015) Estimation of unknown boundary functionsin an inverse heat conduction problem using a mollified marching scheme. Numer Algorithms 68(4):769–790

    Article  MathSciNet  Google Scholar 

  • Guo B, Pu X, Huang F (2015) Fractional partial differential equations and their numerical solutions. World scientific, Singapore

    Book  Google Scholar 

  • Jin B, Rundell W (2012) An inverse problem for a one-dimensional time-fractional diffusion problem. Inverse Probl 28:075010

    Article  MathSciNet  Google Scholar 

  • Jin B, Rundell W (2015) A tutorial on inverse problems for anomalous diffusion processes. Inverse Probl 31(3):035003

    Article  MathSciNet  Google Scholar 

  • Li B, Wang J (2003) Anomalous heat conduction and anomalous diffusion in one dimensional systems. Phys Rev Lett 91:044301

    Article  Google Scholar 

  • Liu F, Anh V, Turner I (2004) Numerical solution of the space fractional Fokker-Planck equation. J Comput Appl Math 66:209–219

    Article  MathSciNet  Google Scholar 

  • Liu JJ, Yamamoto M, Yan L (2015) On the uniqueness and reconstruction for an inverse problem of the fractional diffusion process. Appl Numer Math 87:1–19

    Article  MathSciNet  Google Scholar 

  • Metzler R, Klafter J (2000) The random walks guide to anomalous diffusion: a fractional dynamics approach. Phys Rep 339(1):1–77

    Article  MathSciNet  Google Scholar 

  • Murio DA (2002) Mollification and space marching. In: Woodbury K (ed) Inverse engineering handbook. CRC Press, Boca Raton

    Google Scholar 

  • Murio DA (2006) On the stable numerical evaluation of Caputo fractional derivatives. Comput Math Appl 51:1539–1550

    Article  MathSciNet  Google Scholar 

  • Murio DA (2007) Implicit finite difference approximation for time fractional diffusion equations. Comput Math Appl 56:1138–1145

    Article  MathSciNet  Google Scholar 

  • Murio DA (2008) Time fractional IHCP with Caputo fractional derivatives. Comput Math Appl 56:2371–2381

    Article  MathSciNet  Google Scholar 

  • Murio DA, Mejia CE (2008) Generalized time fractional IHCP with Caputo fractional derivatives. J Phys 135:1–8

    MATH  Google Scholar 

  • Nguyen HT, Le DL, Nguyen VT (2016) Regularized solution of an inverse source problem for a time fractional diffusion equation. Appl Math Model 40(19):8244–8264

    Article  MathSciNet  Google Scholar 

  • Oldham KB, Spanier J (1974) The fractional calculus: theory and application of differentiation and integration to arbitrary order. Academic Press, New York

    MATH  Google Scholar 

  • Podlubny I (1999) Fractional differential equations. Academic Press, New York

    MATH  Google Scholar 

  • Raberto M, Scalas E, Mainardi F (2002) Waiting-times and returns in high-frequency financial data: an empirical study. Physica A 314:749–55

    Article  Google Scholar 

  • Sayevand K, Jafari H (2016) On systems of nonlinear equations: some modified iteration formulas by the homotopy perturbation method with accelerated fourth-and fifth-order convergence. Appl Math Model 40(2):1467–1476

    Article  MathSciNet  Google Scholar 

  • Taghavi A, Babaei A, Mohammadpour A (2017) A stable numerical scheme for a time fractional inverse parabolic equation. Inverse Probl Sci Eng 25(10):1474–1491

    Article  MathSciNet  Google Scholar 

  • Wang JG, Wei T, Zhou YB (2015) Optimal error bound and simplified Tikhonov regularization method for a backward problem for the time fractional diffusion equation. J Comput Appl Math 279:277–292

    Article  MathSciNet  Google Scholar 

  • Wei T, Zhang ZQ (2013) Reconstruction of a time-dependent source term in a time-fractional diffusion equation. Eng Anal Bound Elem 37:23–31

    Article  MathSciNet  Google Scholar 

  • Wei T, Zhang ZQ (2014) Stable numerical solution to a Cauchy problem for a time fractional diffusion equation. Eng Anal Bound Elem 40:128–137

    Article  MathSciNet  Google Scholar 

  • Yan L, Yang F (2014) Efficient Kansa-type MFS algorithm for time-fractional inverse diffusion problems. Comput Math Appl 67:1507–1520

    Article  MathSciNet  Google Scholar 

  • Yuste SB, Acedo L (2005) An explicit finite difference method and a new Von Neumann-type stability analysis for fractional diffusion equations. SIAM J Numer Anal 42:1862–1874

    Article  MathSciNet  Google Scholar 

  • Zhang Y, Xu X (2011) Inverse source problem for a fractional diffusion equation. Inverse Probl 27(3):035010

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the anonymous referees for their constructive comments and suggestions.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Mazandaran, P.O. Box: 47416-95447, Babolsar, Iran

    Afshin Babaei & Seddigheh Banihashemi

Authors
  1. Afshin Babaei
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Seddigheh Banihashemi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Afshin Babaei.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Babaei, A., Banihashemi, S. A Stable Numerical Approach to Solve a Time-Fractional Inverse Heat Conduction Problem. Iran J Sci Technol Trans Sci 42, 2225–2236 (2018). https://doi.org/10.1007/s40995-017-0360-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40995-017-0360-4

Keywords

Mathematics Subject Classification

Search

Navigation