Open Access Articles Version 1 https://doi.org/10.12345/3j7aqz32

Numerical Solution of Time-Fractional Sub-Diffusion Equation via Laguerre-Galerkin Method

Vol. 1 No. 1 (2026): Volume 1, Issue 1, (January - June) 2026 Pavitra Dhanda1 Prashant Pandey2
  1. K.S. Saket P.G. College, Ayodhya, India , IN
  2. K.S. Saket P.G. College, Ayodhya, India , IN
Published: June 30, 2026

Abstract

This article presents a Laguerre-Galerkin approximation for the numerical solution of time-fractional sub-diffusion equations involving the Caputo fractional derivative. The proposed methodology employs the separation of variables technique to decompose the governing equation into a spatial ordinary differential equation and a fractional temporal equation into spatial and temporal components. The resulting spatial boundary value problem is approximated using Laguerre polynomial basis functions, while the temporal component is represented through the Mittag-Leffler function. Both Neumann and Dirichlet boundary conditions are investigated. For problems with Dirichlet boundary conditions, a shooting technique is utilized to obtain an equivalent
Neumann formulation suitable for the proposed approach. Numerical experiments are carried out for different values of the fractional order and approximation parameters. The obtained numerical solutions are compared with available exact solutions through graphical and error analysis. 
The results demonstrate that the proposed  method provides accurate approximations with good convergence characteristics and serves as an efficient Computationally tool for the study of time-fractional sub-diffusion models. 

Keywords:
Time-fractional Sub-Diffusion Galerkin Method Mittag-Leffler function Laguerre polynomial

References

Atangana, A. (2018). Non validity of the index law for fractional derivatives. Chaos, Solitons & Fractals, 112, 82–88. Atangana, A., & Alqahtani, R. T. (2016). Numerical approximation of the space-time fractional Hunter–Saxton equation and its applications. Chaos, Solitons & Fractals, 90, 1–9. Atangana, A., & Baleanu, D. (2017). New fractional derivatives with nonlocal and nonsingular kernel: Theory and application to heat transfer model. Thermal Science, 20(2), 763–769. Gorenflo, R., Kilbas, A. A., Mainardi, F., & Rogosin, S. V. (2020). Mittag-Leffler functions, related topics and applications (2nd ed.). Springer. Kilbas, A. A., Srivastava, H. M., & Trujillo, J. J. (2006). Theory and applications of fractional differential equations. Elsevier. Mainardi, F. (2010). Fractional calculus and waves in linear viscoelasticity: An introduction to mathematical models. Imperial College Press. Metzler, R., & Klafter, J. (2000). The random walk’s guide to anomalous diffusion: A fractional dynamics approach. Physics Reports, 339(1), 1–77. Pandey, S. (2026). Focus point on modelling complex real-world problems with fractal and new trends of fractional differentiation. Pandey, S., et al. (2021). Quarteroni, A., Sacco, R., & Saleri, F. (2006). Numerical mathematics (2nd ed.). Springer. Reddy, J. N. (2005). An introduction to the finite element method (3rd ed.). McGraw-Hill. Shen, J., Tang, T., & Wang, L.-L. (2011). Spectral methods: Algorithms, analysis and applications. Springer. Singh, J., Kumar, D., & Baleanu, D. (2017). On the analysis of fractional diabetes model with exponential law. Advances in Difference Equations, 2017, Article 239. Szegő, G. (1975a). Orthogonal polynomials (4th ed.). American Mathematical Society. Szegő, G. (1975b). Numerical methods for global minimization. Prentice-Hall.