Skip to main content
SpringerLink
Log in

Regularization of the Inverse Problem for Time Fractional Pseudo-parabolic Equation with Non-local in Time Conditions

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

This paper is devoted to identifying an unknown source for a time-fractional diffusion equation in a general bounded domain. First, we prove the problem is non-well posed and the stability of the source function. Second, by using the Modified Fractional Landweber method, we present regularization solutions and show the convergence rate between regularization solutions and sought solution are given under a priori and a posteriori choice rules of the regularization parameter, respectively. Finally, we present an illustrative numerical example to test the results of our theory.

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

Similar content being viewed by others

References

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

    Article  Google Scholar 

  2. Biswas, I. H., Anant, K. M., Guy, V.: On the Cauchy problem of a degenerate parabolic-hyperbolic PDE with Lvy noise. Advances in Nonlinear Analysis, 8(1), 809–844 (2019)

    Article  Google Scholar 

  3. Can, N. H., Luc, N. H., Baleanu, D., et al.: Inverse source problem for time fractional diffusion equation with Mittag-Leffler kernel. Advances in Difference Equations, 210, (2020)

  4. Chen, W., Li, C.: Maximum principles for the fractional p-Laplacian and symmetry of solutions. Adv. Math., 335, 735–758 (2018)

    Article  MATH  Google Scholar 

  5. Dokuchaev, N.: On recovering parabolic diffusion from their time-averages. Calculus of Variations and Partial Differential Equations, 58(1), 1–14 (2019)

    Article  MATH  Google Scholar 

  6. Fredrik, A. H., Peter, L.: Regularity of solutions of the parabolic normalized p-Laplace equation. Advances in Nonlinear Analysis, 9(1), 7–15 (2020)

    MATH  Google Scholar 

  7. Han, Y. Z., Xiong, X. T., Xue, X. M.: A fractional Landweber method for solving backward time-fractional diffusion problem. Computers and Mathematics with Applications, 78, 81–91 (2019)

    Article  MATH  Google Scholar 

  8. Hung, N. N., Binh, H. D., Luc, N. H., et al.: Stochastic sub-diffusion equation with conformable derivative driven by standard Brownian motion. Advances in Theory of Nonlinear Analysis and its Applications, 5(3), 287–299 (2021)

    Google Scholar 

  9. Ikehata, R., Suzuki, T.: Stable and unstable sets for evolution equations of parabolic and hyperbolic type. Hiroshima Math. J., 26, 475–491 (1996)

    Article  MATH  Google Scholar 

  10. Karapinar, E., Binh, H. D., Luc, N. H., et al.: On continuity of the fractional derivative of the time-fractional semilinear pseudo-parabolic systems, Advances in Difference Equations, 2021(1), 1–24 (2021)

    Article  MATH  Google Scholar 

  11. Kilbas, A. A., Saigo, M., Saxena, R. K.: Generalized Mittag-Leffler function and generalized fractional calculus operators. Integral Transforms Spec. Funct., 15(1), 31–49 (2004)

    Article  MATH  Google Scholar 

  12. Lian, W., Wang, J., Xu, R.: Global existence and blow up of solutions for pseudo-parabolic equation with singular potential. J. Differential Equations, 269, 4914–4959 (2020)

    Article  MATH  Google Scholar 

  13. Le, D. L., Nguyen, H. L., Zhou, Y.: Identification of source term for the time-fractional diffusion-wave equation by Fractional Tikhonov method. Mathematics, 7(10), 934 (2019)

    Article  Google Scholar 

  14. Le, D. L., Yong, Z., Tran, T. B., et al.: A mollification regularization method for the inverse source problem for a time fractional diffusion equation. Mathematics, 7, 1048 (2019)

    Article  Google Scholar 

  15. Luc, N. H., Huynh, L. N., Baleanu, D., et al.: Identifying the space source term problem for a generalization of the fractional diffusion equation with hyper-Bessel operator. Adv. Difference Equ., 1, 1–23 (2020)

    MATH  Google Scholar 

  16. Liu, Y., Xu, R., Yu, T.: Global existence, nonexistence and asymptotic behavior of solutions for the Cauchy problem of semilinear heat equations. Nonlinear Anal., 68, 3332–3348 (2008)

    Article  MATH  Google Scholar 

  17. Nguyen, H. T., Le, D. L., Nguyen, V. T.: Regularized solution of an inverse source problem for a time fractional diffusion equation. Applied Mathematical Modelling, 40(19–20), 8244–8264 (2016)

    Article  MATH  Google Scholar 

  18. Pavol, Q., Souplet, P.: Superlinear Parabolic Problems, Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts, Birkhäuser Verlag, Basel, 2007

    MATH  Google Scholar 

  19. Phuong, N. D., Luc, N. H., Long, L. D.: Modified quasi boundary value method for inverse source problem of the bi-parabolic equation. Advances in Theory of Nonlinear Analysis and Its Applications, 4(3), 132–142 (2020)

    Google Scholar 

  20. Phuong, N. D.: Note on a Allen-Cahn equation with Caputo-Fabrizio derivative. Results in Nonlinear Analysis, 4(3), 179–185 (2021)

    Google Scholar 

  21. Podlubny, I.: Fractional Diffusion Equation, Mathematics in Science and Engineering, New York, Academic Press, 1999

    MATH  Google Scholar 

  22. Saanouni, T.: Global and non global solutions for a class of coupled parabolic systems. Adv. Nonlinear Anal., 9, 1383–1401 (2020)

    Article  MATH  Google Scholar 

  23. Sakamoto, K., Yamamoto, M.: Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems. Journal of Mathematical Analysis and Applications, 382, 426–447 (2011)

    Article  MATH  Google Scholar 

  24. Sevinik, A. R., Aksoy, U., Karapinar, E., et al.: On the solution of a boundary value problem associated with a fractional differential equation. Mathematical Methods in the Applied Sciences.https://doi.org/10.1002/mma.6652

  25. Sevinik, A. R., Aksoy, U., Karapinar, E., et al.: Uniqueness of solution for higher-order nonlinear fractional differential equations with multi-point and integral boundary conditions. Revista de la Real Academia de Ciencias Exactas, Fsicas y Naturales. Serie A. Matemticas, 115(3), 1–16 (2021)

    MATH  Google Scholar 

  26. Sevinik, A. R., Aksoy, U., Karapinar, E., et al.: On the solutions of fractional differential equations via Geraghty type hybrid contractions. Appl. Comput. Math., 20(2), 313–333 (2021)

    Google Scholar 

  27. Tuan, N. H., Zhou, Y., Long, L. D., et al.: Identifying inverse source for fractional diffusion equation with Riemann-Liouville derivative. Computational and Applied Mathematics, 39(2), 1–16 (2020)

    Article  Google Scholar 

  28. Tuan, N. H., Huynh, L. N., Ngoc, T. B., et al.: On a backward problem for nonlinear fractional diffusion equations. Applied Mathematics Letters, 92, 76–84 (2019)

    Article  MATH  Google Scholar 

  29. Tuan, N. H., Debbouche, A., Ngoc, T. B.: Existence and regularity of final value problems for time fractional wave equations. Comput. Math. Appl., 78(5), 1396–1414 (2019)

    Article  MATH  Google Scholar 

  30. Tuan, N. H., Long, L. D., Thinh, V. N., et al.: On a final value problem for the time-fractional diffusion equation with inhomogeneous source. Inverse Problems in Science and Engineering, 25(9), 1367–1395 (2017)

    Article  MATH  Google Scholar 

  31. Tuan, N. A., O’Regan, D., Baleanu, D., et al.: On time fractional pseudo-parabolic equations with nonlocal integral conditions. Evolution Equations & Control Theory, 11(1), 225 (2022)

    Article  MATH  Google Scholar 

  32. Wang, J. G., Zhou, Y. B., Wei, T.: Two regularization methods to identify a space-dependent source for the time-fractional diffusion equation. Appl. Numer. Math., 68, 39–57 (2013)

    Article  MATH  Google Scholar 

  33. Wang, X., Xu, R.: Global existence and finite time blowup for a nonlocal semilinear pseudo-parabolic equation. Advances in Nonlinear Analysis, 10(1), 261–288 (2021)

    Article  MATH  Google Scholar 

  34. Wei, T., Li, X. L., Li, Y. S.: An inverse time-dependent source problem for a time-fractional diffusion equation. Inverse Problems, 32(8), 085003 (2016)

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  36. Xu, R., Lian, W., Niu, Y.: Global well-posedness of coupled parabolic systems. Sci. China Math., 63, 321–356 (2020)

    Article  MATH  Google Scholar 

  37. Xu, R., Su, J.: Global existence and finite time blow-up for a class of semilinear pseudoparabolic equations. J. Funct. Anal., 264, 2732–2763 (2013)

    Article  MATH  Google Scholar 

  38. Xu, R., Wang, X., Yang, Y.: Blowup and blowup time for a class of semilinear pseudoparabolic equations with high initial energy. Appl. Math. Lett., 83, 176–181 (2018)

    Article  MATH  Google Scholar 

  39. Yang, S., Xiong, X., Han, Y.: A modified fractional Landweber method for a backward problem for the inhomogeneous time-fractional diffusion equation in a cylinder. International Journal of Computer Mathematics, 97(11), 2375–2393 (2020)

    Article  MATH  Google Scholar 

  40. Zhou, Y., Wei, H. J., Ahmad, B., et al.: Existence and regularity results of a backward problem for fractional diffusion equations. Mathematical Methods in the Applied Sciences, 42(18), 6775–6790 (2019)

    Article  MATH  Google Scholar 

Download references

Acknowledgements

We thank the referees for their time and comments. This research is supported by Industrial University of Ho Chi Minh City (IUH) under Grant Number 130/HD-DHCN. The authors Le Dinh Long and Anh Tuan Nguyen are supported by Van Lang University.

Author information

Authors and Affiliations

  1. Faculty of Fundamental Science, Industrial University of Ho Chi Minh City, Ho Chi Minh City, Vietnam

    Nguyen Duc Phuong

  2. Division of Applied Mathematics, Science and Technology Advanced Institute, Van Lang University, Ho Chi Minh City, Vietnam

    Le Dinh Long & Anh Tuan Nguyen

  3. Faculty of Technology, Van Lang University, Ho Chi Minh City, Vietnam

    Le Dinh Long & Anh Tuan Nguyen

  4. Department of Mathematics, Cankaya University, Ankara, Turkey

    Dumitru Baleanu

  5. Lebanese American University, Beirut, Lebanon

    Dumitru Baleanu

  6. Institute of Space Sciences, Magurele-Bucharest, Romania

    Dumitru Baleanu

Authors
  1. Nguyen Duc Phuong
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Le Dinh Long
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Anh Tuan Nguyen
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Dumitru Baleanu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Le Dinh Long.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Phuong, N.D., Long, L.D., Nguyen, A.T. et al. Regularization of the Inverse Problem for Time Fractional Pseudo-parabolic Equation with Non-local in Time Conditions. Acta. Math. Sin.-English Ser. 38, 2199–2219 (2022). https://doi.org/10.1007/s10114-022-1234-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-022-1234-z

Keywords

MR(2010) Subject Classification

Search

Navigation