Analytical Solution of Two-Dimensional Advection-Diffusion Equation with Caputo-Fabrizio Derivative and Time Dependent Velocity
DOI:
https://doi.org/10.11113/mjfas.v22n2.4531Keywords:
Advection-Diffusion Equation, Caputo-Fabrizio Fractional Derivative, Time Dependent, Laplace Transform, Fourier TransformAbstract
The advection-diffusion equation (ADE) is a fundamental mathematical model that is widely used to describe the transport of substances, such as the transport of pollutants in rivers, groundwater or soil. Incorporating fractional derivative into the ADE allows for non-integer orders which have been demonstrated to capture more complex dynamics that cannot be described by classical ADE. This study investigates a two-dimensional ADE with time-fractional Caputo-Fabrizio derivative while considering a time-dependent velocity. The velocity function varies temporally, while the diffusion coefficient remains constant. By introducing appropriate transformations, the equation is reformulated and reduced to an equation with constant coefficient. Analytical solutions are obtained using the Laplace transform in time and the Fourier transform in spatial coordinates. The derived solutions encompass classical and fractional advection-diffusion processes, highlighting the impact of fractional-order derivatives. Numerical simulations illustrate the influence of time-dependent velocity on concentration profiles, providing a comparative analysis. The results offer valuable insights into the role of fractional calculus in modeling transport processes with evolving velocity fields, contributing to both theoretical advancements and practical applications in environmental and engineering sciences.
References
Manitcharoen, N., & Pimpunchat, B. (2020). Analytical and numerical solutions of pollution concentration with uniformly and exponentially increasing forms of sources. Journal of Applied Mathematics, 2020, 1-9.
Paudel, K., Kafle, J., & Bhandari, P. S. (2022). Advection-Dispersion Equation for Concentrations of Pollutant and Dissolved Oxygen. Journal of Nepal Mathematical Society, 5(1), 30-40.
Hadhouda, M. K., & Hassan, Z. S. (2022). Mathematical model for unsteady remediation of river pollution by aeration. Inf. Sci. Lett, 11-323.
Gbenro, S. O., & Nchejane, J. N. (2022). Numerical Simulation of the Dispersion of Pollutant in a Canal. Asian Research Journal of Mathematics, 18(4), 25-40.
Ruishi, L., & Isa, Z. M. (2023). Two-Dimensional Heavy Metal Migration in Soil with Adsorption and Instantaneous Injection. Malaysian Journal of Fundamental and Applied Sciences, 19(6), 980-988.
Liang, R., & Isa, Z. M. (2024). Heavy metal transport with adsorption for instantaneous and exponential attenuation of concentration. Scientific Reports, 14(1), 537.
Chaudhary, M., Thakur, C. K., & Singh, M. K. (2020a). Analysis of 1-D pollutant transport in semi-infinite groundwater reservoir. Environmental Earth Sciences, 79, 1-23.
Jiang, J., Luo, H. H., Wang, S. F., Su, J., & Yu, Y. D. (2023). A two-dimensional analytical model for heavy metal contaminants transport in permeable reactive barrier. Water Science & Technology, 87(2), 393-406.
Mustafa, S., Bahar, A., Abidin, A. R. Z., Aziz, Z. A., & Darwish, M. (2021). Three-dimensional model for solute transport induced by groundwater abstraction in river-aquifer systems. Alexandria Engineering Journal, 60(2), 2573-2582.
Mustafa, S., Bahar, A., Aziz, Z. A., & Darwish, M. (2020). Solute transport modelling to manage groundwater pollution from surface water resources. Journal of contaminant hydrology, 233, 103662.
Mustafa, S., Bahar, A., Aziz, Z. A., & Darwish, M. (2022). Analytical solutions of contaminant transport in homogeneous and isotropic aquifer in three-dimensional groundwater flow. Environmental Science and Pollution Research, 29(58), 87114-87131.
Agarwal, R., Yadav, M. P., Agarwal, R. P., & Baleanu, D. (2019). Analytic solution of space time fractional advection dispersion equation with retardation for contaminant transport in porous media. Progress in Fractional Differentiation and Applications, 5(4), 1-13.
Mainardi, F. (2023). Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. World Scientific.
Mirza, I. A., & Vieru, D. (2017). Fundamental solutions to advection–diffusion equation with time-fractional Caputo–Fabrizio derivative. Computers & Mathematics with Applications, 73(1), 1-10.
Ahmed, N., Shah, N. A., & Vieru, D. (2019). Two-dimensional advection–diffusion process with memory and concentrated source. Symmetry, 11(7), 879
Mirza, I. A., Akram, M. S., Shah, N. A., Imtiaz, W., & Chung, J. D. (2021). Analytical solutions to the advection-diffusion equation with Atangana-Baleanu time-fractional derivative and a concentrated loading. Alexandria Engineering Journal, 60(1), 1199-1208.
Jannelli, A. & Speciale, M. P (2021). Exact and numerical solutions of two-dimensional time-fractional diffusion–reaction equations through the Lie symmetries. Nonlinear Dynamics, 105, 2375–2385.
Purohit, M., & Mushtaq, S. (2020). Applications of Laplace-Adomian decomposition method for solving time-fractional advection dispersion equation. J. Math. Comput. Sci., 10(5), 1960-1968.
Sene, N. (2021). Fractional advection-dispersion equation described by the Caputo left generalized fractional derivative. Palestine J. Math, 10(2), 562-579.
Ahmed, M., Zainab, Q. U. A., & Qamar, S. (2017). Analysis of One-Dimensional Advection–Diffusion Model with Variable Coefficients Describing Solute Transport in a Porous medium. Transport in Porous Media, 118, 327-344.
Djordjevich, A., Savović, S., & Janićijević, A. (2017). Explicit finite-difference solution of two-dimensional solute transport with periodic flow in homogenous porous media. Journal of Hydrology and Hydromechanics, 65(4), 426-432.
Das, P., Akhter, A., & Singh, M. K. (2018). Solute transport modelling with the variable temporally dependent boundary. Sādhanā, 43, 1-11.
Yadav, R. R., & Kumar, L. K. (2019). Solute transport for pulse type input point source along temporally and spatially dependent flow. Pollution, 5(1), 53-70.
Kumar, R., Chatterjee, A., Singh, M. K., & Tsai, F. T. (2022). Advances in analytical solutions for time-dependent solute transport model. Journal of Earth System Science, 131(2), 131.
Jaiswal, D. K., Kumar, N., & Yadav, R. R. (2022). Analytical solution for transport of pollutant from time-dependent locations along groundwater. Journal of Hydrology, 610, 127826.
Yang, S., Zhou, H., Zhang, S., & Wang, L. (2019). Analytical solutions of advective–dispersive transport in porous media involving conformable derivative. Applied Mathematics Letter, 92, 85-92.
Yang, S., Chen, X., Ou, L., Cao, Y., & Zhou, H. (2020). Analytical solutions of conformable advection–diffusion equation for contaminant migration with isothermal adsorption. Applied Mathematics Letters, 105, 106330.
Prudnikov, A.P., Brychkov, Yu. A., & Marichev, O.I. (1986). Integrals and Series Vol 1 Elementary Functions. Gordon and Breach Science Publishers
Fazli, T. (2006). A Case Study of Sg. Skudai in the Comparison of River Water Quality Models in Total Maximum Daily Load Assessment. Master's thesis, Universiti Teknologi Malaysia.
Qiao, C., Xu, Y., Zhao, W., Qian, J., Wu, Y., & Sun, H. (2020). Fractional derivative modeling on solute non-fickian transport in a single vertical fracture. Frontiers in Physics, 8, 378.
Downloads
Published
Issue
Section
License
Copyright (c) 2026 Purani Kunasegaran, Zaiton Mat Isa
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
