Study of one‐dimensional contaminant transport in soils using fractional calculus. (18th February 2021)

• Record Type: Journal Article
• Authors: Mirza, Itrat Abbas; Akram, Muhammad Saeed; Shah, Nehad Ali; Akhtar, Shehraz; Muneer, Mirfat
• Is Part Of: Mathematical methods in the applied sciences. Volume 44:Number 8(2021)
• Page Start: 6839
• Page End: 6856
• Publication Date: 2021-02-18
• Subjects: Atangana–Baleanu fractional derivative -- fractional derivatives and integrals -- initial‐boundary value problems for PDEs of mixed type -- integral transforms; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.7225
• Languages: English
• ISSNs: 0170-4214

Abstract

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.