scholarly journals Numerical Methods for the Variable-Order Fractional Advection-Diffusion Equation with a Nonlinear Source Term

2009 ◽  
Vol 47 (3) ◽  
pp. 1760-1781 ◽  
Author(s):  
P. Zhuang ◽  
F. Liu ◽  
V. Anh ◽  
I. Turner
Keyword(s):  
Numerical Methods ◽  
Diffusion Equation ◽  
Source Term ◽  
Variable Order ◽  
Nonlinear Source ◽  
Advection Diffusion Equation ◽  
Advection Diffusion

scholarly journals SPACE-TIME SPECTRAL COLLOCATION ALGORITHM FOR THE VARIABLE-ORDER GALILEI INVARIANT ADVECTION DIFFUSION EQUATIONS WITH A NONLINEAR SOURCE TERM

2017 ◽  
Vol 22 (1) ◽  
pp. 1-20 ◽  
Author(s):  
Mohamed A. Abd-Elkawy ◽  
Rubayyi T. Alqahtani
Keyword(s):  
Source Term ◽  
Dimensional Space ◽  
Space Time ◽  
Variable Order ◽  
Spectral Collocation ◽  
Nonlinear Source ◽  
Advection Diffusion Equation ◽  
Advection Diffusion ◽  
Collocation Technique ◽  
Collocation Scheme

This paper presents a space-time spectral collocation technique for solving the variable-order Galilei invariant advection diffusion equation with a nonlinear source term (VO-NGIADE). We develop a collocation scheme to approximate VONGIADE by means of the shifted Jacobi-Gauss-Lobatto collocation (SJ-GL-C) and shifted Jacobi-Gauss-Radau collocation (SJ-GR-C) methods. We successfully extend the proposed technique to solve the two-dimensional space VO-NGIADE. The discussed numerical tests illustrate the capability and high accuracy of the proposed methodologies.

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.

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