A Numerical Method for Solving the Third Boundary Value Problem for the Convection-Diffusion Equation with a Fractional Time Derivative in a Multidimensional Domain

2021 ◽  
Vol 42 (7) ◽  
pp. 1630-1642
Author(s):  
M. Kh. Beshtokov
Keyword(s):  
Diffusion Equation ◽  
Boundary Value Problem ◽  
Numerical Method ◽  
Time Derivative ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
The Third ◽  
Fractional Time Derivative ◽  
Third Boundary Value Problem ◽  
Multidimensional Domain

scholarly journals New High-Order Compact ADI Algorithms for 3D Nonlinear Time-Fractional Convection-Diffusion Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Shuying Zhai ◽  
Xinlong Feng ◽  
Zhifeng Weng
Keyword(s):  
Diffusion Equation ◽  
Time Derivative ◽  
Three Dimensional ◽  
Cost Effective ◽  
Linearization Method ◽  
High Order ◽  
Thomas Algorithm ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Alternating Direction

Numerical approximations of the three-dimensional (3D) nonlinear time-fractional convection-diffusion equation is studied, which is firstly transformed to a time-fractional diffusion equation and then is solved by linearization method combined with alternating direction implicit (ADI) method. By using fourth-order Padé approximation for spatial derivatives and classical backward differentiation method for time derivative, two new high-order compact ADI algorithms with ordersO(τmin(1+α,2−α)+h4)andO(τ2−α+h4)are presented. The resulting schemes in each ADI solution step corresponding to a tridiagonal matrix equation can be solved by the Thomas algorithm which makes the computation cost effective. Numerical experiments are shown to demonstrate the high accuracy and robustness of two new schemes.

scholarly journals Analytical solution of the time fractional diffusion equation and fractional convection-diffusion equation

2018 ◽  
Vol 65 (1) ◽  
pp. 82 ◽  
Author(s):  
Francisco Gomez ◽  
Victor Morales ◽  
Marco Taneco
Keyword(s):  
Diffusion Equation ◽  
Analytical Solutions ◽  
Fourier Transforms ◽  
Time Derivative ◽  
Fractional Diffusion ◽  
Transport Processes ◽  
Diffusion Equations ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Diffusion And Convection

In this paper, we obtain analytical solutions for the time-fractional diffusion and time-fractional convection-diffusion equations. These equations are obtained from the standard equations by replacing the time derivative with a fractional derivative of order $\alpha$. Fractional operators of type Liouville-Caputo, Atangana-Baleanu-Caputo, fractional conformable derivative in Liouville-Caputo sense and Atangana-Koca-Caputo were used to model diffusion and convection-diffusion equation. The Laplace and Fourier transforms were applied to obtain the analytical solutions for the fractional order diffusion and convection-diffusion equations. The solutions obtained can be useful to understand the modeling of anomalous diffusive, subdiffusive systems and super-diffusive systems, transport processes, random walk and wave propagation phenomenon.

scholarly journals Novel Cubic Trigonometric B-Spline Approach Based on the Hermite Formula for Solving the Convection-Diffusion Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-17
Author(s):  
Aatika Yousaf ◽  
Thabet Abdeljawad ◽  
Muhammad Yaseen ◽  
Muhammad Abbas
Keyword(s):  
Approximate Solution ◽  
Diffusion Equation ◽  
Time Derivative ◽  
High Accuracy ◽  
Spline Method ◽  
B Spline ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Numerical Tests ◽  
Piecewise Continuous

This paper introduces a cubic trigonometric B-spline method (CuTBM) based on the Hermite formula for numerically handling the convection-diffusion equation (CDE). The method utilizes a merger of the CuTBM and the Hermite formula for the approximation of a space derivative, while the time derivative is discretized using a finite difference scheme. This combination has greatly enhanced the accuracy of the scheme. A stability analysis of the scheme is also presented to confirm that the errors do not magnify. The main advantage of the scheme is that the approximate solution is obtained as a smooth piecewise continuous function empowering us to approximate a solution at any location in the domain of interest with high accuracy. Numerical tests are performed, and the outcomes are compared with the ones presented previously to show the superiority of the presented scheme.

scholarly journals Difference scheme for the convection-diffusion equation of fractional order

2021 ◽  
pp. 146-154
Author(s):  
Е.М. Казакова
Keyword(s):  
Diffusion Equation ◽  
Boundary Value Problem ◽  
Difference Scheme ◽  
Fractional Order ◽  
Boundary Value ◽  
Stability And Convergence ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
The Difference ◽  
The Stability

Построена разностная схема, аппроксимирующая первую краевую задачу для уравнения конвекции-диффузии дробного порядка с постоянными коэффициентами. Исследована устойчивость и сходимость разностной схемы. A difference scheme is constructed that approximates the first boundary value problem for the fractional-order convection-diffusion equation. The stability and convergence of the difference scheme.

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