scholarly journals A Discrete Method Based on the CE-SE Formulation for the Fractional Advection-Dispersion Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
Silvia Jerez ◽  
Ivan Dzib
Keyword(s):  
Numerical Simulation ◽  
Analytical Solution ◽  
Fractional Derivative ◽  
Dispersion Equation ◽  
Numerical Algorithm ◽  
Initial Condition ◽  
Discrete Method ◽  
One Dimensional ◽  
Advection Dispersion ◽  
The One

We obtain a numerical algorithm by using the space-time conservation element and solution element (CE-SE) method for the fractional advection-dispersion equation. The fractional derivative is defined by the Riemann-Liouville formula. We prove that the CE-SE approximation is conditionally stable under mild requirements. A numerical simulation is performed for the one-dimensional case by considering a benchmark with a discontinuous initial condition in order to compare the results with the analytical solution.

ONE-DIMENSIONAL TEMPORALLY DEPENDENT ADVECTION-DISPERSION EQUATION IN POROUS MEDIA: ANALYTICAL SOLUTION

2010 ◽  
Vol 23 (4) ◽  
pp. 521-539 ◽  
Author(s):  
R. R. YADAV ◽  
DILIP KUMAR JAISWAL ◽  
HAREESH KUMAR YADAV ◽  
GUL RANA
Keyword(s):  
Porous Media ◽  
Analytical Solution ◽  
Dispersion Equation ◽  
One Dimensional ◽  
Advection Dispersion

Advection-dispersion modelling tools: what about numerical dispersion?

2005 ◽  
Vol 52 (3) ◽  
pp. 19-27 ◽  
Author(s):  
R. Bouteligier ◽  
G. Vaes ◽  
J. Berlamont ◽  
C. Flamink ◽  
J.G. Langeveld ◽  
...  
Keyword(s):  
Dispersion Equation ◽  
Model Calibration ◽  
Suspended Particles ◽  
Numerical Dispersion ◽  
Model Parameters ◽  
One Dimensional ◽  
Advection Dispersion ◽  
The One ◽  
Calibration Tool ◽  
Tracer Experiments

In general the transport of dissolved substances and fine suspended particles is governed by the one-dimensional advection-dispersion equation. In order to model the transport of dissolved substances and fine suspended particles, the advection-dispersion equation is incorporated into commonly used urban drainage modelling tools such as InfoWorks CS (Wallingford Software, United Kingdom) and MOUSE (DHI Software, Denmark). Two examples show the use of InfoWorks CS and MOUSE using standard model settings. Modelling results using tracer experiments show that numerical model parameters need to be altered in order to calibrate the model. Using tracer experiments as a model calibration tool, it is shown that a non-negligible amount of dispersion is generated by InfoWorks CS and MOUSE and that it is in fact the numerical dispersion that is calibrated.

scholarly journals Numerical Simulation of Non-Equilibrium Stationary Currents Through Nano-scale One-Dimensional Chains

2020 ◽  
Vol 233 ◽  
pp. 05010
Author(s):  
João Pedro dos Santos Pires ◽  
Bruno Amorim ◽  
João Manuel Viana Parente Lopes
Keyword(s):  
Numerical Simulation ◽  
Size Effects ◽  
Tight Binding ◽  
Finite Size ◽  
Initial Condition ◽  
Zero Temperature ◽  
Finite Size Effects ◽  
One Dimensional ◽  
Open Boundaries ◽  
The One

Using a method based on the time-evolution of the occupied states at zero temperature, we observe the onset of a quasi-uniform and quasisteady state current across a disordered tight-binding chain, coupled between two finite (but large) clean leads with open boundaries. This current is seen to match the one obtained in the Landauer-Büttiker formalism and is also independent of the initial condition considered (partitioned or non-partitioned). Finite-size effects are also reported and briefly discussed.

Initial-boundary value problems for the one-dimensional linear advection–dispersion equation with decay

2020 ◽  
Vol 75 (8) ◽  
pp. 713-725 ◽  
Author(s):  
Guenbo Hwang
Keyword(s):  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
One Dimensional ◽  
Advection Dispersion ◽  
Asymptotically Periodic ◽  
The One ◽  
Initial Boundary Value

AbstractInitial-boundary value problems for the one-dimensional linear advection–dispersion equation with decay (LAD) are studied by utilizing a unified method, known as the Fokas method. The method takes advantage of the spectral analysis of both parts of Lax pair and the global algebraic relation coupling all initial and boundary values. We present the explicit analytical solution of the LAD equation posed on the half line and a finite interval with general initial and boundary conditions. In addition, for the case of periodic boundary conditions, we show that the solution of the LAD equation is asymptotically t-periodic for large t if the Dirichlet boundary datum is periodic in t. Furthermore, it can be shown that if the Dirichlet boundary value is asymptotically periodic for large t, then so is the unknown Neumann boundary value, which is uniquely characterized in terms of the given asymptotically periodic Dirichlet boundary datum. The analytical predictions for large t are compared with numerical results showing the excellent agreement.

scholarly journals Analytical solution of one-dimensional Pennes’ bioheat equation

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 1084-1092
Author(s):  
Hongyun Wang ◽  
Wesley A. Burgei ◽  
Hong Zhou
Keyword(s):  
Analytical Solution ◽  
Radiofrequency Energy ◽  
Bioheat Equation ◽  
Surface Cooling ◽  
One Dimensional ◽  
The Fourier Transform ◽  
The One ◽  
Parameter Values ◽  
Mathematical Derivation ◽  
Effect Of Surface

Abstract Pennes’ bioheat equation is the most widely used thermal model for studying heat transfer in biological systems exposed to radiofrequency energy. In their article, “Effect of Surface Cooling and Blood Flow on the Microwave Heating of Tissue,” Foster et al. published an analytical solution to the one-dimensional (1-D) problem, obtained using the Fourier transform. However, their article did not offer any details of the derivation. In this work, we revisit the 1-D problem and provide a comprehensive mathematical derivation of an analytical solution. Our result corrects an error in Foster’s solution which might be a typo in their article. Unlike Foster et al., we integrate the partial differential equation directly. The expression of solution has several apparent singularities for certain parameter values where the physical problem is not expected to be singular. We show that all these singularities are removable, and we derive alternative non-singular formulas. Finally, we extend our analysis to write out an analytical solution of the 1-D bioheat equation for the case of multiple electromagnetic heating pulses.

Diffusion-Limited Cell Dehydration: Analytical and Numerical Solutions for a Planar Model

1999 ◽  
Author(s):  
Alexander V. Kasharin ◽  
Jens O. M. Karlsson
Keyword(s):  
Analytical Solution ◽  
Numerical Solutions ◽  
Planar System ◽  
One Dimensional ◽  
Diffusion Limited ◽  
Dimensional Diffusion ◽  
Constant Diffusivity ◽  
A Cell ◽  
Cell Dehydration ◽  
The One

Abstract The process of diffusion-limited cell dehydration is modeled for a planar system by writing the one-dimensional diffusion-equation for a cell with moving, semipermeable boundaries. For the simplifying case of isothermal dehydration with constant diffusivity, an approximate analytical solution is obtained by linearizing the governing partial differential equations. The general problem must be solved numerically. The Forward Time Center Space (FTCS) and Crank-Nicholson differencing schemes are implemented, and evaluated by comparison with the analytical solution. Putative stability criteria for the two algorithms are proposed based on numerical experiments, and the Crank-Nicholson method is shown to be accurate for a mesh with as few as six nodes.

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