New idea of Atangana‐Baleanu time‐fractional derivative to advection‐diffusion equation

2019 ◽  
Vol 44 (3) ◽  
pp. 2521-2531
Author(s):  
Iskander Tlili ◽  
Nehad Ali Shah ◽  
Saif Ullah ◽  
Humera Manzoor
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Advection Diffusion Equation ◽  
Advection Diffusion ◽  
Time Fractional Derivative

scholarly journals On Numerical Solution Of The Time Fractional Advection-Diffusion Equation Involving Atangana-Baleanu-Caputo Derivative

Open Physics ◽  
2019 ◽  
Vol 17 (1) ◽  
pp. 816-822 ◽  
Author(s):  
Mohammad Partohaghighi ◽  
Mustafa Inc ◽  
Mustafa Bayram ◽  
Dumitru Baleanu
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Numerical Solution ◽  
Caputo Derivative ◽  
Initial Guess ◽  
Advection Diffusion Equation ◽  
Advection Diffusion ◽  
Time Fractional Derivative ◽  
Group Preserving Scheme ◽  
New Equation

Abstract A powerful algorithm is proposed to get the solutions of the time fractional Advection-Diffusion equation(TFADE): $^{ABC}\mathcal{D}_{0^+,t}^{\beta}u(x,t) =\zeta u_{xx}(x,t)- \kappa u_x(x,t)+$ F(x, t), 0 < β ≤ 1. The time-fractional derivative $^{ABC}\mathcal{D}_{0^+,t}^{\beta}u(x,t)$is described in the Atangana-Baleanu Caputo concept. The basis of our approach is transforming the original equation into a new equation by imposing a transformation involving a fictitious coordinate. Then, a geometric scheme namely the group preserving scheme (GPS) is implemented to solve the new equation by taking an initial guess. Moreover, in order to present the power of the presented approach some examples are solved, successfully.

Two-Dimensional Legendre Wavelets for Solving Variable-Order Fractional Nonlinear Advection-Diffusion Equation with Variable Coefficients

2018 ◽  
Vol 19 (7-8) ◽  
pp. 793-802 ◽  
Author(s):  
M. Hosseininia ◽  
M. H. Heydari ◽  
Z. Avazzadeh ◽  
F. M. Maalek Ghaini
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Main Idea ◽  
Variable Coefficients ◽  
Variable Order ◽  
Two Dimensional ◽  
Legendre Wavelets ◽  
Advection Diffusion Equation ◽  
Advection Diffusion ◽  
Nonlinear Advection

AbstractThis article studies a numerical scheme for solving two-dimensional variable-order time fractional nonlinear advection-diffusion equation with variable coefficients, where the variable-order fractional derivative is in the Caputo type. The main idea is expanding the solution in terms of the 2D Legendre wavelets (2D LWs) where the variable-order time fractional derivative is discretized. We describe the method using the matrix operators and then implement it for solving various types of fractional advection-diffusion equations. The experimental results show the computational efficiency of the new approach.

Sign in / Sign up

Export Citation Format

Share Document