Study of one‐dimensional contaminant transport in soils using fractional calculus

Abstract

In this communication, the soil contamination transport phenomenon is studied using fractional calculus. Specifically, a variant of non‐integer fractional derivative known as Antangana–Baleanu fractional derivative is applied on one‐dimensional advection diffusion equation to obtain a fractional model for the contaminant transport in soils. This fractional model is examined under the hypothesis of initial and generalized Robin type boundary conditions. The analytical expression of solution for contaminant transport in soil is derived for the fractional initial‐boundary value problem for partial differential equation (PDE) of mixed type using integral transforms, namely, Laplace and finite sine–cosine Fourier transforms. The solution of the ordinary advection diffusion equation arises as a case of the obtained solution. The numerical results are obtained for initial concentrated loading and Heaviside unit step function as a boundary conditions, from the analytical solution for various parameters of interest, and the corresponding graphs are plotted with the help of software MATHCAD. The graphs illustrate that the memory effects are remarkable for small values of time and ordinary for large values of the time.

Fractional modelling of the reverse osmosis process used for dam water desalination

“…However, the majority of previous modelling studies (Ruth et al, 1933;Hermans and Bredée, 1935;Ho and Zydney, 2000;Jamal et al, 2004 ;Fouladitajar et al, 2013;Tien et al, 2014;Heidari et al, 2017;Goldrick et al, 2017;Debnath et al, 2019;Tong et al, 2020;Xu et al, 2020;Heidari et al, 2021;Azizi et al, 2022;Bchiti et al, 2022) have relied on a limited range of experimental data, thus limiting their range of validity. On the other hand, the classical models cannot best represent all the phenomena that occur during the membrane separation process, unlike fractional models that have proven their performance for other processes (Kashchenko and Nikitin, 2014;Zhai et al, 2015;Padrino, 2017;Obembe et al, 2018;Kumar et al, 2019;Ramírez et al, 2017;De Souza Matias et al, 2019;Lemus-Mondaca et al, 2021;Mahdad et al, 2021a;Mahdad et al, 2021b;Friesen et al, 2015; Nikan et al, 2020;Mirza et al, 2021;El-Gazar et al, 2021).…”

Complexity Analysis of Three-Dimensional Fractional-Order Chaotic System Based on Entropy Theory

“…However, due to its long-term lack of application background, its development is slow [3,4]. In recent years, with the development of computer science and the discovery of more and more fractional-order phenomena, people have done a lot of work in the field of fractional calculus [5][6][7]. Based on the study of integer-order chaotic system, the fractional differential operator is introduced into the system.…”