A SEMI-ANALYTICAL SOLUTION TO THE TIME-FRACTIONAL ADVECTION–DIFFUSION EQUATION USING FRACTIONAL REDUCED DIFFERENTIAL TRANSFORM METHOD
DOI:
https://doi.org/10.33003/fjs-2025-0910-3850Keywords:
FPDEs, FRDTM, FVIM, Advection-Diffusion, Fractional orderAbstract
A potent tool for simulating models of physical, biological, and dynamic processes is well described by Fractional Partial Differential Equations (FPDE), which is due to their memory effect and non-local properties. The present study applies the Fractional Reduced Differential Transform Method (FRDTM), a semi-analytical technique, to solve Fractional Partial Differential Equations (FPDEs) relevant to advection-diffusion models. The study modifies an existing integer-order model by introducing fractional derivatives and analyzes the influence of these fractional parameters. The FRDTM is validated through comparison with the Fractional Variational Iteration Method (FVIM), thereby demonstrating its accuracy and computational efficiency. Results show that FRDTM performs well in finding an approximate solution for long-time simulations, especially at fractional order α < 1, and is preferable for high accuracy and low computational error, particularly for Advection-Diffusion problems.
References
Abdallah, M. A. and Ishag, K. A. (2023). Fractional reduced differential transform method for solving mutualism model with fractional diffusion. International Journal of Analysis and Applications, 21, 33.
Akgül, A. and Khoshnaw, S. (2019). Application of fractional derivative on non-linear biochemical reaction models. International Journal of Intelligent Networks, 1, 52–58. https://doi.org/10.1016/j.ijin.2020.05.001
Al-Rabtah, A. and Abuasad, S. (2023). Effective modified fractional reduced differential transform method for solving multi-term time-fractional wave-diffusion equations. Symmetry, 15(9), 1721. https://doi.org/10.3390/sym15091721
Amattouch, M. R., Harfaoui, M. and Hadadi, A. (2023). A finite difference scheme for fractional partial differential equation. Journal of Theoretical and Applied Information Technology, 101(1), 262–266. http://www.jatit.org
Area, I., Batarfi, H., Losada, J., Nieto, J. J., Shammakh, W. and Torres, Á. (2015). On a fractional-order Ebola epidemic model. Advances in Difference Equations, 2015(1), 278. https://doi.org/10.1186/s13662-015-0613-6
Arikoglu, A. and Ozkol, I. (2007). Solution of fractional differential equations by using the differential transform method. Chaos, Solitons & Fractals, 34(5), 1473–1481. https://doi.org/10.1016/j.chaos.2006.04.022
Bansi, C. D. K., Tabi, C. B., Motsumi, T. G. and Mohamadou, A. (2018). Fractional blood flow in oscillatory arteries with thermal radiation and magnetic field effects. Journal of Magnetism and Magnetic Materials. https://doi.org/10.1016/j.jmmm.2018.01.079
Bohaienko, V. and Bulavatsky, V. (2020). Fractional-fractal modeling of filtration-consolidation processes in saline saturated soils. Fractal and Fractional, 4(4), 59. https://doi.org/10.3390/fractalfract4040059
Guo, J., Xu, D. and Qiu, W. (2020). A finite difference scheme for the nonlinear time-fractional partial integro-differential equation. Mathematical Methods in the Applied Sciences, 1–21. https://doi.org/10.1002/mma.6128
Jafari, H., Khalique, C. M. and Nazari, M. (2011). Application of the Laplace decomposition method for solving linear and nonlinear fractional diffusion–wave equations. Applied Mathematics Letters, 24(11), 1799–1805. https://doi.org/10.1016/j.aml.2011.04.037
Kenea, K. S. (2018). Analytic solutions of time-fractional diffusion equations by fractional reduced differential transform method (FRDTM). African Journal of Mathematics and Computer Science Research, 11(2), 14–34. https://doi.org/10.5897/AJMCSR2017.0716
Keskin, Y., & Oturang, G. (2010). Application of reduced differential transformation method for solving gas dynamics equation. International Journal of Contemporary Mathematical Sciences, 5(22), 1091–1096.
Maitama, S. and Abdullahi, I. (2016). A new analytical method for solving linear and nonlinear fractional partial differential equations. Progress in Fractional Differentiation & Applications, 2(4), Article 2. https://doi.org/10.18576/pfda/020402
Meerschaert, M. M. and Tadjeran, C. (2004). Finite difference approximations for fractional advection-dispersion flow equations. Journal of Computational and Applied Mathematics, 172(1), 65–77. http://www.sciencedirect.com/science/article/pii/S0377042704000986
Meerschaert, M. M. and Tadjeran, C. (2006). Finite difference approximations for two-sided space-fractional partial differential equations. Applied Numerical Mathematics, 56(1), 80–90. https://doi.org/10.1016/j.apnum.2005.02.008
Momani, S. and Odibat, Z. (2006). Analytical approach to linear fractional partial differential equations arising in fluid mechanics. Physics Letters A, 355(4-5), 271-279. https://doi.org/10.1016/j.physleta.2006.02.048
Moosavi Noori, S. R. and Taghizadeh, N. (2020). Study of convergence of reduced differential transform method for different classes of differential equations. International Journal of Differential Equations, 2021, 6696414. https://doi.org/10.1155/2021/6696414
Muhamad, D. J., Asep, K. S., Endang, R. and Saputra, J. (2021). Application of fractional differential equation in economic growth model: A systematic review approach. AIMS Mathematics, 6(9), 10266–10280. https://doi.org/10.3934/math.2021594
Patel, H., Patel, T. and Pandit, D. (2023). An efficient technique for solving fractional-order diffusion equations arising in oil pollution. Journal of Ocean Engineering and Science, 8, 217–225. https://doi.org/10.1016/j.joes.2022.09.004
Ren, J., Sun, Z. and Dai, W. (2016). New approximation for solving the Caputo-type fractional differential equation. Applied Mathematical Modelling, 40(4), 2625–2636. https://doi.org/10.1016/j.apm.2015.09.077
Singh, B. K. and Kumar, P. (2017). Fractional variational iteration method for solving fractional partial differential equations with proportional delay. International Journal of Differential Equations, 2017, 1–11. https://doi.org/10.1155/2017/2797892
Srivastava, H., Badal, D. and Srivastava, K. (2021). Fractional reduced differential transform method to analytical solution of fractional order biological population model. International Journal of Mathematics Trends and Technology, 67(1), 9–16. https://doi.org/10.14445/22315373/IJMTT-V67I1P502
Wu, G. and Lee, E. (2010). Fractional variational iteration method and its application. Physics Letters A, 374(25), 2506-2509. https://doi.org/10.1016/j.physleta.2010.04.034
Yadav, L. K., Agarwal, D., Suthar, D. L. and Purohit, S. D. (2022). Time-fractional partial differential equations: A novel technique for analytical and numerical solutions. Arab Journal of Basic and Applied Sciences, 29(1), 86–98. https://doi.org/10.1080/25765299.2022.2032679
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