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Diffusion Equation.pdf

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Preview of Diffusion Equation
🔗 Source: uni-muenster.de
📊 Size: 268 KB
📄 Pages: 15 pages
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Summary

The diffusion equation is a partial differential equation that describes density fluctuations in a material undergoing diffusion, given by ∂u(r,t)/∂t = ∇·D(u(r,t),r)∇u(r,t), where u(r,t) is the density of the diffusing material and D(u(r,t),r) is the collective diffusion coefficient. If D is constant, the equation reduces to the linear equation ∂u(r,t)/∂t = D∇2u(r,t), also known as the heat equation. The one-dimensional diffusion equation ∂u(x,t)/∂t = D∂2u(x,t)/∂x2 can be derived from the continuity equation and Fick's first law. The analytical solution to this equation with Dirichlet boundary conditions is given by u(x,t) = ∑[An sin(πnx/L) exp(-D(πn/L)2t)], where An are constants determined by the initial condition. The numerical treatment of the diffusion equation can be done using the FTCS explicit method, which is conditionally stable with a stability condition of α ≤ 1/2, where α = DΔt/Δx2. However, this method has limitations, such as requiring small time steps for high-resolution calculations and introducing numerical dispersion.

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Description

The diffusion equation is a partial differential equation that describes density fluctuations in a material undergoing diffusion, given by ∂u(r,t)/∂t =...

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  • File Format: PDF
  • File Size: 268 KB
  • Pages: 15
  • Language: EN
  • Total Downloads: 188
  • Last Updated: 2 hours ago

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