Several approaches to modeling diffusion in combination with other processes are illustrated.

## Description

The following approaches are detailed: 1-D Diffusion modeled as a partial differential equation: Model # 0330 Partial differential equation with no flux boundary conditions (Neumann), initialized with a centered spike. 1-D Diffusion with asymmetrical Consumption modeled as a partial differential equation: Model # 0364 Same as 1-D Diffusion with consumption of metabolite as a function of x. 1-D diffusion-advection equation with Robin boundary condition: Model # 0169 Similar to 1-D diffusion with added advection term. Initial value is zero. There is an inflow concentration, Cin. Uses a Robin condition for inflow boundary condition. Random Walks of multiple particles in 1 dimension: Model # 0184 Multiple realizations of 1-D random walks. Final positions are binned and compared to theoretical calculations. Random Walk of single particle in 2 dimensions: Model # 0372 A single random walk in two dimensions is plotted wth with green circles, marked with starts every nsteps. Three kinds of steps may be chosen: (1) fixed step sizes (2) random step sizes (3) random step size with random angle. Fractional Brownian Motion Walk in 2 dimensions: Model # 0374 Uses increments of fractional Gaussian noise to create fractional Brownian Motion using the Davies-Harte algorithm. (See FGP model for details.) Illustrates that steps can be Gaussian, but the Hurst Coefficient measuring correlation of steps is highly important. Diffusion in a uniform slab: Model # 176 Similar to 1-D Diffusion with addition of a partition coefficient lambda, the ratio of the concentration immediately inside the region to that outside. Uses Dirichlet boundary conditions. Diffusion in two uniform slabs with different diffusivities: Model 0212 Similar to Diffusion in a uniform slab, but with the diffusion coefficient containing a discontinuity and the boundary of two media. Laplace's equation in 2-D with Dirichlet Boundary conditions: Model # 0363 Laplace's equation in two dimension is solved using ordinary different equations and solved again used 1-d partial differential equations. both methods use second order accurate finite difference approximations.

## References

None.

## Models Referenced

- Diffusion Tutorial,
- 1-D Diffusion modeled as a partial differential equation,
- 1-D Diffusion with asymmetrical Consumption modeled as a partial differential equation,
- 1-D diffusion-advection equation with Robin boundary condition
- Random Walks of multiple particles in 1 dimension
- Random Walk of single particle in 2 dimensions
- Fractional Brownian Motion Walk in 2 dimensions
- Diffusion in a uniform slab
- Diffusion in two uniform slabs with different diffusivities
- Heat equation in two dimensions with Dirichlet boundary conditions

## Key Terms

diffusion tutorial, diffusion, tutorial, 1d, 1-d, 1D, 1_d, one dimension, 2d, 2-d, 2D, 2-D, two dimension, PDE, Dirichlet, Neumann, Robin, boundary condition, Random Walk, fractional Brownian motion, fBm, fGn, FGP

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## Acknowledgements

Please cite **https://www.imagwiki.nibib.nih.gov/physiome** in any publication for which this software is used and send an email with the citation and, if possible, a PDF file of the paper to:

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The National Simulation Resource, Director J. B. Bassingthwaighte, Department of Bioengineering, University of Washington, Seattle WA 98195-5061.

**Model development and archiving support at https://www.imagwiki.nibib.nih.gov/physiome provided by the following grants:** NIH U01HL122199 Analyzing the Cardiac Power Grid, 09/15/2015 - 05/31/2020, NIH/NIBIB BE08407 Software Integration, JSim and SBW 6/1/09-5/31/13; NIH/NHLBI T15 HL88516-01 Modeling for Heart, Lung and Blood: From Cell to Organ, 4/1/07-3/31/11; NSF BES-0506477 Adaptive Multi-Scale Model Simulation, 8/15/05-7/31/08; NIH/NHLBI R01 HL073598 Core 3: 3D Imaging and Computer Modeling of the Respiratory Tract, 9/1/04-8/31/09; as well as prior support from NIH/NCRR P41 RR01243 Simulation Resource in Circulatory Mass Transport and Exchange, 12/1/1980-11/30/01 and NIH/NIBIB R01 EB001973 JSim: A Simulation Analysis Platform, 3/1/02-2/28/07.