Numerical solutions of the space-time fractional advection-dispersion equation

2008 ◽  
Vol 24 (6) ◽  
pp. 1416-1429 ◽  
Author(s):  
Shaher Momani ◽  
Zaid Odibat
Keyword(s):  
Dispersion Equation ◽  
Numerical Solutions ◽  
Space Time ◽  
Advection Dispersion

scholarly journals Analytical Solution of 1D Advection-Dispersion Equation In Finite Groundwater Reservoir With Spatially Dependent Dispersivity And Two Inputs Sources With Source-Sink Term Impact

2021 ◽  
Author(s):  
Thomas TJOCK-MBAGA ◽  
Patrice Ele Abiama ◽  
Jean Marie Ema'a Ema'a ◽  
Germain Hubert Ben-Bolie
Keyword(s):  
Analytical Solution ◽  
Linear Function ◽  
Dispersion Equation ◽  
Source Term ◽  
Numerical Solutions ◽  
Integral Transform ◽  
Liouville Problem ◽  
Finite Domain ◽  
Advection Dispersion ◽  
Sink Term

Abstract This study derives an analytical solution of a one-dimensional (1D) advection-dispersion equation (ADE) for solute transport with two contaminant sources that takes into account the source term. For a heterogeneous medium, groundwater velocity is considered as a linear function while the dispersion as a nth-power of linear function of space and analytical solutions are obtained for and . The solution in a heterogeneous finite domain with unsteady coefficients is obtained using the Generalized Integral Transform Technique (GITT) with a new regular Sturm-Liouville Problem (SLP). The solutions are validated with the numerical solutions obtained using MATLAB pedpe solver and the existing solution from the proposed solutions. We exanimated the influence of the source term, the heterogeneity parameters and the unsteady coefficient on the solute concentration distribution. The results show that the source term produces a solute build-up while the heterogeneity level decreases the concentration level in the medium. As an illustration, model predictions are used to estimate the time histories of the radiological doses of uranium at different distances from the sources boundary in order to understand the potential radiological impact on the general public.

scholarly journals Generalized Analytical Solutions of The Advection-Dispersion Equation with Variable Flow and Transport Coefficients

2021 ◽  
Vol 13 (14) ◽  
pp. 7796
Author(s):  
Abhishek Sanskrityayn ◽  
Heejun Suk ◽  
Jui-Sheng Chen ◽  
Eungyu Park
Keyword(s):  
Dispersion Equation ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Transient Flow ◽  
Dispersion Coefficient ◽  
Transport Coefficients ◽  
Variable Coefficients ◽  
Dispersion Coefficients ◽  
Transform Method ◽  
Advection Dispersion

Demand has increased for analytical solutions to determine the velocities and dispersion coefficients that describe solute transport with spatial, temporal, or spatiotemporal variations encountered in the field. However, few analytical solutions have considered spatially, temporally, or spatiotemporally dependent dispersion coefficients and velocities. The proposed solutions consider eight cases of dispersion coefficients and velocities: both spatially dependent, both spatiotemporally dependent, both temporally dependent, spatiotemporally dependent dispersion coefficient with spatially dependent velocity, temporally dependent dispersion coefficient with constant velocity, both constant, spatially dependent dispersion coefficient with spatiotemporally dependent velocity, and constant dispersion coefficient with temporally dependent velocity. The spatial dependence is linear, while the temporal dependence may be exponential, asymptotical, or sinusoidal. An advection–dispersion equation with these variable coefficients was reduced to a non-homogeneous diffusion equation using the pertinent coordinate transform method. Then, solutions were obtained in an infinite medium using Green’s function. The proposed analytical solutions were validated against existing analytical solutions or against numerical solutions when analytical solutions were unavailable. In this study, we showed that the proposed analytical solutions could be applied for various spatiotemporal patterns of both velocity and the dispersion coefficient, shedding light on feasibility of the proposed solution under highly transient flow in heterogeneous porous medium.

scholarly journals Improved reproducing kernel method to solve space-time fractional advection-dispersion equation

2021 ◽  
Author(s):  
Tofigh Allahviranloo ◽  
Hussein Sahihi ◽  
Soheil Salahshour ◽  
D. Baleanu
Keyword(s):  
Dispersion Equation ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Approximate Solutions ◽  
Variable Coefficients ◽  
Space Time ◽  
Finite Domain ◽  
Reproducing Kernel Method ◽  
Advection Dispersion ◽  
Schmidt Orthogonalization

In this paper, we consider the Space-Time Fractional Advection-Dispersion equation on a finite domain with variable coefficients. Fractional Advection- Dispersion equation as a model for transporting heterogeneous subsurface media as one approach to the modeling of the generally non-Fickian behavior of transport. We use a semi-analytical method as Reproducing kernel Method to solve the Space-Time Fractional Advection-Dispersion equation so that we can get better approximate solutions than the methods with which this problem has been solved. The main obstacle to solve this problem is the existence of a Gram-Schmidt orthogonalization process in the general form of the reproducing kernel method, which is very time-consuming. So, we introduce the Improved Reproducing Kernel Method, which is a different implementation for the general form of the reproducing kernel method. In this method, the Gram-Schmidt orthogonalization process is eliminated to significantly reduce the CPU-time. Also, the present method increases the accuracy of approximate solutions.

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