Analytical solutions to the advection-diffusion equation with Atangana-Baleanu time-fractional derivative and a concentrated loading

Alexandria Engineering Journal

Waqas Imtiaz¹  Muhammad Saeed Akram¹  Nehad Ali Shah²  Itrat Abbas Mirza²  Jae Dong Chung³ 

[1] Information Technology, Rahim Yar Khan, Pakistan;;Department of Mathematics, Khwaja Fareed University of Engineering &Informetrics Research Group, Ton Duc Thang University, Ho Chi Minh City, Viet Nam;

关键词: Advection-diffusion;    Fractional partial differential equation;    Integral transforms;    Robin type boundary conditions;   

DOI  :  

来源: DOAJ

【 摘 要 】

In this communication we have studied well known physical process of two-dimensional advection–diffusion phenomena. The advection–diffusion equation is time-fractionalized by exploiting Atangana-Baleanu fractional derivative operator. This fractionalization is achieved in the generalized constitutive equation of the mass flux density vector. The fractionalized two-dimensional advection–diffusion equation turns out to be a two-dimensional nonlinear fractional partial differential equation. This partial differential equation is considered under the hypothesis of an initial concentrated loading and Robin type boundary conditions. The analytical expression of the solution is determined for this boundary value problem by employing the integral transforms method, namely, the Laplace transform, sine-Fourier transform and finite sine–cosine Fourier transform. The effects of fractional parameter α on the concentration obtained from the analytical solution, for various parameters of interest, are illustrated graphically 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.

【 授权许可】

Unknown