Physics 212, 2018: Lecture 24

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Back to Physics 212, 2018: Computational Modeling.

The heat diffusion equation -- one of many partial differential equations (differential equations involving partial derivatives).

• Fick's first law of diffusion Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J=-D\frac{d\phi}{dx}} or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{J}=-D\nabla \phi} (also known as Newton's law of cooling if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi=T} , or the Fourier law of heat conduction)
• Fick's second law of diffusion Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \partial_t \phi = -D\nabla^2 \phi} .
• Finite difference form of the heat diffusion equation; need boundary conditions to complete the solution
• Boundary conditions (implemented with extending the grid matrix):
  • Absorbing
  • Reflecting
  • Peeriodic
• Solution of the equation using Python code.