Discretization of the Caputo time-fractional advection-diffusion problems with certain wavelet basis function

Authors

  • D. C. Iweobodo
    Department of Mathematics, Dennis Osadebay University, Asaba, Delta State, Nigeria
  • G. C. Abanum
    Department of Mathematics, Dennis Osadebay University, Asaba, Delta State, Nigeria
  • O. Ogoegbulem
    Department of Mathematics, Delta State University, Abraka, Delta State, Nigeria
  • N. I. Ochonogor
    Department of Mathematics, Dennis Osadebay University, Asaba, Delta State, Nigeria
  • I. N. Njoseh
    Department of Mathematics, Delta State University, Abraka, Delta State, Nigeria

Keywords:

Caputo fractional derivatives, Time-fractional advection-diffusion equations (TFADE), Iweobodo-Mamadu-Njoseh wavelet (IMNW), Time discretization, Space discretization

Abstract

Considering the new wavelet-based Galerkin finite element technique constructed with Iweobodo-Mamadu-Njoseh wavelet (IMNW) as the basis function in seeking the numerical solution of time-fractional advection-diffusion equations (TFADE), the TFADE must be simplified to enhance an application of a numerical technique. Thus in this research work, we considered an implementation of the time and space discretization of the TFADE with the
use of IMNW basis function. In order to successfully achieve our result, our methodology inculcated the Caputo fractional derivatives, time fractional advection-diffusion equations (TFADE), Wavelet, IMNW, and Galerkin finite element method. After a successful implementation of the time discretization, an implicit form of TFADE was obtained, followed by the implementation of the space discretization which generated the variational formulation of the equation for easy implementation of the scheme. The illustrated numerical solution from using the new technique provided a resulting numerical evidence which aligns with the exact solution.

Dimensions

[1] M. Kumar & S. Pandit, “Wavelet transform and wavelet based numerical methods: an introduction”, Department of Mathematics, Motila Nehru National Institute of Technology, Alahabad-2011004 (U. P.) India 9 (2011) 1749. https://www.researchgate.net/publication/285635387_Wavelet_transform_and_wavelet_based_numerical_methods_An_introduction

[2] D. C. Iweobodo, E. J. Mamadu & I. N. Njoseh, “Daubechies wavelet based galerkin method of solving partial differential equations”, Caliphate Journal of Science and Technology (CaJoST) 3 (2705) 62. https://cajost.com.ng/index.php/files/article/view/66.

[3] A. K. Gupta & R. S. Saha, “Wavelet methods for solving fractional order differential equations. Mathematical Problems in Engineering”, Wiley 2014 (2014) 1. https://doi.org/10.1155/2014/140453.

[4] K. Issa, R. A. Bello & U. J. Abubakar, “Approximate Analytical Solution of Fractional-order Generalized Integro-differential Equations Via Fractional Derivative of Shifted Vieta-Lucas P olynomial”, Journal of Nigerian Society of Physical Sciences 6 (2024) 1821. https://doi.org/10.46481/jnsps.2024.1821.

[5] M. Basim, A. Ahmadian, Z. B. Ibrahim & S. Salahshour, “Solving fractional variable-order differential equations of the non-singular derivative using jacobi operational matrix”, Journal of Nigerian Society of Physical Sciences 5 (2023)1221. https://doi.org/10.46481/jnsps.2023.1221.

[6] S. C. Shiralashetti, L. M. Angadi & S. Kumbinarasiaiah, “Wavelet based galerkin method for the numerical solution of one dimensional partial differential equations”, International Research Journal of Engineering and Technology 6 (2019) 2889. https://www.researchgate.net/publication/334762351_Wavelet-based_Galerkin_Method_for_the_Numerical_Solution_of_One_Dimensional_Partial_Differential_Equations.

[7] D. C. Iweobodo, I. N. Njoseh & J. S. Apanapudor, “A new wavele-based galerkin method of weighted residual function for the numerical solution of one dimensional differential equations”, Mathematics and Statistics 11 (2023) 910. https://doi.org/10.13189/ms.2023.110605.

[8] D. C. Iweobodo, I. N. Njoseh & J. S. Apanapudor, “An overview of Iweobodo-Mamadu-Njoseh Wavelet (IMNW) and its steps in solving time fractional advection-diffusion problems”, Asian Research Journal of Mathematics 20 (2024) 59. https://doi.org/10.9734/arjom/2024/v20i3791.

[9] S. E. Fadugba, “Solution of fractional order equations in the domain of the mellin transform”, Journal of the Nigerian Society of Physical Sciences 1 (2019) 138. https://doi.org/10.46481/jnsps.2019.31.

[10] I. Podlubny, “An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications”, in Fractional differential equations, United Kingdom, London, 1998, pp. 318–335. https://www.elsevier.com/books/fractional-differential-equations/podlubny/978-0-12-558840-9

[11] U. Saeed & M. U. Rehman, “Hermite wavelet method for fractional delay differential equations”, Journal of Difference Equations 2014 (2014) 1. https://doi.org/10.1155/2014/359093

[12] A. Rayal, M. Anand, K. Chauhan & P. Bisht, “An overview of Mamadu-Njoseh wavelet and its properties for numerical computations”, Uttaranchal Journal of Applied and Life Sciences 4 (2023) 1. https://www.researchgate.net/publication/386546762_An_Overview_of_Mamadu-Njoseh_Wavelet_and_its_Properties_for_Numerical_Computations.

[13] A. Rayal, P. Bisht, S. Giri, P. A. Patel & M. Prajapati, “Dynamical analysis and numerical treatment of pond pollution model endowned with Caputo fractional derivative using effective wavelets technique”, International Journal of Dynamics and Control 12 (2024) 4218. https://doi.org/10.1007/s40435-024-01494-5.

[14] E. J. Mamadu & I. N. Njoseh, “Numerical solutions of volterra equations using galerkin method with certain orthogonal polynomials”, Journal of Applied Mathematics and Physics 4 (2016) 376. http://dx.doi.org/10.4236/jamp.2016.42044.

[15] M. F. N. Moshen, “Some details of the Galerkin finite element method”, Applied Mathematical Modelling 6 (1982) 168. https://doi.org/10.1016/0307-904X(82)90005-1.

[16] P. G. Cialet, “Introduction to the finite element method”, in The finite elements method for elliptic problems, North-Holland, Netherlands, 1977, pp. 36–109. https://doi.org/10.4236/am.2015.612185.

[17] E. J. Mamadu, I. N. Njoseh & H. I. Ojarikre, “Space discretization of time fractional telegraph equation with mamadu-njoseh basis functions”, Applied Mathematics 13 (2022) 760. https://doi.org/10.4236/am.2022.139048.

[18] J. Chen, F. Liu & K. Burrage, “Finite difference method and a Fourier analysis for the fractional reaction-subdiffusion equation”, Appl. Math. and Comput 198 (2008) 134. https://doi.org/10.1016/j.amc.2007.09.020.

[19] S. Doley, A. V. Kumar & L. Jino, “Upwind scheme of caputo time fractional advection diffusion equation”, Advances and Applications in Mathematical Sciences 21 (2022) 1239. https://www.researchgate.net/publication/359106403_UPWIND_SCHEME_OF_CAPUTO_TIME_FRACTIONAL_ADVECTION_DIFFUSION_EQUATION

Published

2025-08-01

How to Cite

Discretization of the Caputo time-fractional advection-diffusion problems with certain wavelet basis function. (2025). Journal of the Nigerian Society of Physical Sciences, 7(3), 2405. https://doi.org/10.46481/jnsps.2025.2405

Issue

Volume 7, Issue 3, August 2025

Section

Mathematics & Statistics
Copyright & Licensing

Copyright (c) 2025 D. C. Iweobodo, G. C. Abanum, O. Ogoegbulem , N. I. Ochonogor, I. N. Njoseh

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite

Discretization of the Caputo time-fractional advection-diffusion problems with certain wavelet basis function. (2025). Journal of the Nigerian Society of Physical Sciences, 7(3), 2405. https://doi.org/10.46481/jnsps.2025.2405

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