Analytical solutions of the \(^{222}\)Rn radon diffusion-advection equation through soil using Atangana–Baleanu time fractional derivative

Abstract

In this paper, we propose a mathematical model based on the fractality of time to describe and predict the radon diffusion-advection process through soil. For a given decay constant, \(\lambda _{R_n}\), and a uniform diffusion coefficient, \(D_0\), and assuming that radon flow occurs in only one direction, the process is modeled by the one-dimensional diffusion-advection equation. This equation is generalized to the fractional order \(\alpha \), based on the newly proposed Atangana-Baleanu derivative in the Caputo sense. Analytical solutions are obtained using Laplace and sine-Fourier transforms, and the key points for choosing the Atangana-Baleanu fractional derivative are highlighted. The role of the fractional order parameter is significant in this research. It is observed that, at a given time t, each radon concentration profile increases with soil depth, regardless of the value of \(\alpha \). However, the radon concentration profile reaches higher values as the fractional order parameter increases. A memory effect is observed in the system each time the value of \(\alpha \) is changed, providing evidence of the fractal nature of the process. The obtained pattern reveals concentration levels that are not accessible in classical studies. This work goes beyond previous studies in the literature by showing that the investigated fractality of time captures the memory effects inherent in the radon diffusion process.