An Inverse Problem for A Time-Fractional Heat Equation: De‎termination of A Time-Dependent Coefficient Via The Laplace ‎Homotopy Analysis Method

Authors

  • Ogugua N. Onyejekwe

    Indian River State College, Fort Pierce, Florida, USA

How to Cite

Onyejekwe, O. N. . (2025). An Inverse Problem for A Time-Fractional Heat Equation: De‎termination of A Time-Dependent Coefficient Via The Laplace ‎Homotopy Analysis Method. International Journal of Applied Mathematical Research, 14(2), 36-39. https://doi.org/10.14419/19gt8385

Received date: May 31, 2025

Accepted date: July 18, 2025

Published date: July 30, 2025

DOI:

https://doi.org/10.14419/19gt8385

Keywords:

Fractional Derivatives; Heat Equation; Homotopy Analysis Method; Inverse Problems; Laplace Transforms; Laplace Homotopy Analysis Method.

Abstract

This study addresses an inverse problem for a time-fractional heat equation involving the Caputo derivative, where the objective is to determine an unknown time-dependent thermal coefficient. The governing equation incorporates a fractional order time derivative, a spatially ‎dependent diffusion term, and a known source term. To solve the inverse problem, we employ the Laplace Homotopy Analysis Method ‎‎(LHAM)—a semi-analytical approach that combines the Laplace transform with the Homotopy Analysis Method. This technique allows for ‎the construction of a convergent series solution for both the temperature distribution ‎ ‎ and the unknown coefficient ‎ ‎. By appropriately selecting the initial guess ‎ ‎, the convergence-control parameter h becomes embedded in the solution process, reducing com-‎computational effort. The proposed method is validated through an illustrative example, demonstrating the effectiveness and accuracy of LHAM ‎in solving time-fractional inverse problems with a source term‎.

References

  1. Podlubny, I. (1999). Fractional differential equations. Academic Press.
  2. Kilbas, A. A., Srivastava, H. M., & Trujillo, J. J. (2006). Theory and applications of fractional differential equations. Elsevier.
  3. Magin, R. L. (2006). Fractional calculus in bioengineering. Begell House.
  4. Mainardi, F. (2010). Fractional calculus and waves in linear viscoelasticity. World Scientific. https://doi.org/10.1142/9781848163300.
  5. Diethelm, K. (2010). The analysis of fractional differential equations. Springer. https://doi.org/10.1007/978-3-642-14574-2.
  6. Sun, H., Li, Z., Zhang, Y., & Chen, W. (2017). Fractional and fractal derivative models for transient anomalous diffusion: Model comparison. Cha-os, Solitons & Fractals, 102(1), 346–356. https://doi.org/10.1016/j.chaos.2017.03.060.
  7. Sun, L., Yan, X., & Wei, T. (2019). Identification of time-dependent coefficients in time-fractional diffusion equations. Journal of Computational Physics, 346, 505–517. https://doi.org/10.1016/j.cam.2018.07.029.
  8. Tikhonov, A. N., & Arsenin, V. Y. (1977). Solutions of ill-posed problems. Winston & Sons.
  9. Engl, H. W., Hanke, M., & Neubauer, A. (1996). Regularization of inverse problems. Kluwer Academic. https://doi.org/10.1007/978-94-009-1740-8.
  10. Liu, F., Anh, V., & Turner, I. (2004). Numerical solution of the space fractional Fokker–Planck equation. Journal of Computational and Applied Mathematics, 166(1), 209–219. https://doi.org/10.1016/j.cam.2003.09.028.
  11. Lin, Y., & Xu, C. (2007). Finite difference/spectral approximations for the time-fractional diffusion equation. Journal of Computational Physics, 225(2), 1533–1552. https://doi.org/10.1016/j.jcp.2007.02.020.
  12. Adomian, G. (1994). Solving frontier problems of physics: The decomposition method. Springer. https://doi.org/10.1007/978-94-015-8289-6.
  13. He, J. H. (1999). Homotopy perturbation technique. Computers & Mathematics with Applications, 40(178), 257–262. https://doi.org/10.1016/S0045-7825(99)00018-3.
  14. Liao, S. (2003). Beyond perturbation: Introduction to the homotopy analysis method. Chapman & Hall/CRC.
  15. Gupta, V. G., & Kumar, P. (2012). Application of Laplace transform homotopy analysis method to fractional diffusion equations. Communications in Nonlinear Science and Numerical Simulation, 12(6), 4518–4530.
  16. Ibraheem, Q. W., & Hussein, M. S. (2023). Determination of time-dependent coefficient in time fractional heat equation. Partial Differential Equa-tions in Applied Mathematics, 7, 100492. https://doi.org/10.1016/j.padiff.2023.100492.
  17. Ibraheem, Q. W., & Hussein, M. S. (2024). Determination of timewise-source coefficient in time-fractional reaction–diffusion equation from first order heat moment. Iraqi Journal of Science, 65(3), 1612–1628. https://doi.org/10.24996/ijs.2024.65.3.35.
  18. Li, M., Li, G., Li, Z., & Li, X. (2020). Determination of time-dependent coefficients in time-fractional diffusion equations by variational iteration method. Journal of Mathematics Research, 12(1), Article 74. https://doi.org/10.5539/jmr.v12n1p74.

Downloads

How to Cite

Onyejekwe, O. N. . (2025). An Inverse Problem for A Time-Fractional Heat Equation: De‎termination of A Time-Dependent Coefficient Via The Laplace ‎Homotopy Analysis Method. International Journal of Applied Mathematical Research, 14(2), 36-39. https://doi.org/10.14419/19gt8385

Received date: May 31, 2025

Accepted date: July 18, 2025

Published date: July 30, 2025