# A Direct Method to Approximate Solution of the Space-fractional Diffusion Equation

## DOI:

https://doi.org/10.11113/mjfas.v20n4.3533## Keywords:

Fractional diffusion equation, Ritz method, Caputo derivative, Legendre polynomials.## Abstract

A class of partial differential equations with fractional derivatives in the spatial variables are called space-fractional diffusion equations. They can be applied to simulate anomalous diffusion, in which the classical diffusion equation does not accurately describe how a plume of particles disperses. Analytically solving fractional diffusion equations can be problematic due to the typically complex structures of fractional derivative models. Hence, this study proposes the utilisation of a satisfier function in combination with the Ritz method to effectively address fractional diffusion equations in the Caputo sense. By employing this approach, the equations are transformed into an algebraic system, so facilitating their solution and providing a numerical result. This method can achieve a high level of accuracy in solving the Caputo fractional diffusion equations by utilising only a small number of terms from the shifted Legendre polynomials in two variables. The precision and effectiveness of our approach may be evaluated, as it yielded dependable approximations of the solutions.

## References

Rehman, A. U., Awrejcewicz, J., Riaz, M. B., & Jarad, F. (2022). Mittag-Leffler form solutions of natural convection flow of second grade fluid with exponentially variable temperature and mass diffusion using Prabhakar fractional derivative. Case Studies in Thermal Engineering, 34, 102018.

Manikandan, K., Serikbayev, N., Aravinthan, D., & Hosseini, K. (2024). Solitary wave solutions of the conformable space–time fractional coupled diffusion equation. Partial Differential Equations in Applied Mathematics, 100630.

Hashemi, M. S., Mirzazadeh, M., Bayram, M., & El Din, S. M. (2023). Numerical approximation of the Cauchy non-homogeneous time-fractional diffusion-wave equation with Caputo derivative using shifted Chebyshev polynomials. Alexandria Engineering Journal, 81, 118–129.

Alqhtani, M., Owolabi, K. M., Saad, K. M., & Pindza, E. (2022). Efficient numerical techniques for computing the Riesz fractional-order reaction-diffusion models arising in biology. Chaos, Solitons & Fractals, 161, 112394.

Abidemi, A., Owolabi, K. M., & Pindza, E. (2022). Modelling the transmission dynamics of Lassa fever with nonlinear incidence rate and vertical transmission. Physica A: Statistical Mechanics and Its Applications, 597.

Olayiwola, M. O., Alaje, A. I., & Yunus, A. O. (2024). A Caputo fractional order financial mathematical model analyzing the impact of an adaptive minimum interest rate and maximum investment demand. Results in Control and Optimization, 14, 100349.

Cengizci, S., & Uğur, Natesan, S. (2024). A stabilized finite element formulation with shock-capturing for solving advection-dominated convection–diffusion equations having time-fractional derivatives. Journal of Computational Science, 76, 102214.

Yang, Z., Chen, X., Chen, Y., & Wang, J. (2024). Accurate numerical simulations for fractional diffusion equations using spectral deferred correction methods. Computers & Mathematics with Applications, 153, 123–129.

Xu, Y., Sun, H., Zhang, Y., Sun, H.-W., & Lin, J. (2023). A novel meshless method based on RBF for solving variable-order time fractional advection-diffusion-reaction equations in linear or nonlinear systems. Computers & Mathematics with Applications, 142, 107–120.

Du, H., & Chen, Z. (2022). Adaptive meshless numerical method of solving 2D variable order time fractional mobile-immobile advection-diffusion equations. Computers & Mathematics with Applications, 124, 42–51.

Salim, T. O. (2009). Some properties relating to the generalized Mittag-Leffler function. Advances in Applied Mathematics and Analysis, 4(1), 21–30.

Shi, P. (2017). Fractional derivatives of some functions. In Robust Adaptive Control for Fractional-Order Systems with Disturbance and Saturation (P. Shi, Ed.).

Dattoli, G., Germano, B., Martinelli, M. R., & Ricci, P. E. (2011). A novel theory of Legendre polynomials. Mathematical and Computer Modelling, 54(1–2), 80–87.

Khan, M. A., & Singh, M. P. (2010). A study of two variables Legendre polynomials. Pro Mathematica, 24(47–48), 201–223.

Mamehrashi, K. (2023). Ritz approximate method for solving delay fractional optimal control problems. Journal of Computational and Applied Mathematics, 417, 114606.

Tao, Y., Chen, C., Zhou, J., & Arvin, H. (2023). Principal parametric resonance analysis of a rotating agglomerated nanocomposite beam employing the Chebyshev–Ritz method. Engineering Analysis with Boundary Elements, 150, 400–412.

Sabermahani, S., Ordokhani, Y., & Razzaghi, M. (2023). Ritz-generalized Pell wavelet method: Application for two classes of fractional pantograph problems. Communications in Nonlinear Science and Numerical Simulation, 119, 107138.

Md Nasrudin, F. S., Phang, C., & Kanwal, A. (2023). Fractal-fractional advection–diffusion–reaction equations by Ritz approximation approach. Open Physics, 21(1), 20220221.

Agarwal, P., & El-Sayed, A. (2018). Non-standard finite difference and Chebyshev collocation methods for solving fractional diffusion equations. Physica A: Statistical Mechanics and Its Applications, 500, 40–49.

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Copyright (c) 2024 Farah Suraya Md Nasrudin, Shafaruniza Mahadi, Nurul Nadiya Abu Hassan

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