Analytical and Numerical Solutions of Riesz Space Fractional Advection-Dispersion Equations with Delay

The study of pollution movement is an important basis for solving water quality problems, which is of vital importance in almost every country. This research proposes the motion of flowing pollution by using a mathematical model in one-dimensional advection-dispersion equation which includes terms of decay and enlargement process. We are assuming an added pollutant sources along the river in two cases: uniformly and exponentially increasing terms. The unsteady state analytical solutions are obtained by using the Laplace transformation, and the finite difference technique is utilized for numerical solutions. Solutions are compared by relative error values. The result appears acceptable between the analytical and numerical solutions. Varying the value of the rate of pollutant addition along the river (q) and the arbitrary constant of exponential pollution source term (λ) is displayed to explain the behavior of the incremental concentration. It is shown that the concentration increases as q and λ increase, and the exponentially increasing pollution source is a suitable model for the behavior of incremental pollution along the river. The results are presented and discussed graphically. This work can be applied to other physical situations described by advection-dispersion phenomena which are affected by the increase of those source concentrations.

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An upscaled approach for transport in media with extended tailing due to back-diffusion using analytical and numerical solutions of the advection dispersion equation

Journal of Contaminant Hydrology ◽

10.1016/j.jconhyd.2015.09.008 ◽

2015 ◽

Vol 182 ◽

pp. 157-172 ◽

Cited By ~ 9

Author(s):

Jack C. Parker ◽

Ungtae Kim

Keyword(s):

Dispersion Equation ◽

Numerical Solutions ◽

Back Diffusion ◽

Analytical And Numerical Solutions ◽

Advection Dispersion

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An advanced numerical modeling for Riesz space fractional advection–dispersion equations by a meshfree approach

Applied Mathematical Modelling ◽

10.1016/j.apm.2016.03.036 ◽

2016 ◽

Vol 40 (17-18) ◽

pp. 7816-7829 ◽

Cited By ~ 14

Author(s):

Z.B. Yuan ◽

Y.F. Nie ◽

F. Liu ◽

I. Turner ◽

G.Y. Zhang ◽

...

Keyword(s):

Numerical Modeling ◽

Riesz Space ◽

Dispersion Equations ◽

Advection Dispersion

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Novel Second-Order Accurate Implicit Numerical Methods for the Riesz Space Distributed-Order Advection-Dispersion Equations

Advances in Mathematical Physics ◽

10.1155/2015/590435 ◽

2015 ◽

Vol 2015 ◽

pp. 1-14 ◽

Cited By ~ 27

Author(s):

X. Wang ◽

F. Liu ◽

X. Chen

Keyword(s):

Numerical Methods ◽

Quadrature Rule ◽

Second Order ◽

Riesz Space ◽

Stability And Convergence ◽

Dispersion Equations ◽

Advection Dispersion ◽

Alternating Direction ◽

The Stability ◽

Distributed Order

We derive and analyze second-order accurate implicit numerical methods for the Riesz space distributed-order advection-dispersion equations (RSDO-ADE) in one-dimensional (1D) and two-dimensional (2D) cases, respectively. Firstly, we discretize the Riesz space distributed-order advection-dispersion equations into multiterm Riesz space fractional advection-dispersion equations (MT-RSDO-ADE) by using the midpoint quadrature rule. Secondly, we propose a second-order accurate implicit numerical method for the MT-RSDO-ADE. Thirdly, stability and convergence are discussed. We investigate the numerical solution and analysis of the RSDO-ADE in 1D case. Then we discuss the RSDO-ADE in 2D case. For 2D case, we propose a new second-order accurate implicit alternating direction method, and the stability and convergence of this method are proved. Finally, numerical results are presented to support our theoretical analysis.

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