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§3.1 Laplace’s equation

§3.1 Laplace’s equation. Christopher Crawford PHY 311 2014-02-19. Outline. Overview Summary of Ch. 2 Intro to Ch. 3, Ch. 4 Laplacian – cur vature (X-ray) operator PDE’s in physics with Laplacian Laplacian in 1-d, 2-d, 3-d

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§3.1 Laplace’s equation

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  1. §3.1Laplace’s equation Christopher Crawford PHY 311 2014-02-19

  2. Outline • OverviewSummary of Ch. 2Intro to Ch. 3, Ch. 4 • Laplacian – curvature (X-ray) operatorPDE’s in physics with LaplacianLaplacian in 1-d, 2-d, 3-d • Boundary conditionsClassification of hyperbolic, elliptic, parabolic PDE’sExternal boundaries: uniqueness theorem Internal boundaries: continuity conditions • Numerical solution – real-life problems solved on computerRelaxation methodFinite differenceFinite element analysis – HW5

  3. Summary of Ch. 2

  4. Laplacian in physics • The source of a conservative flux • Example: electrostatic potential, electric flux, and charge

  5. Laplacian in lower dimensions • 1-d Laplacian • 2nd derivative: curvature • Flux: doesn’t spread out in space • Solution: • Boundary conditions: • Mean field theorem • 2-d Laplacian • 2nd derivative: curvature • Flux: spreads out on surface • 2nd order elliptic PDE • No trivial integration • Depends on boundary cond. • Mean field theorem • No local extrema

  6. Laplacian in 3-d • Laplace equation: • Now curvature in all three dimensions – harder to visualize • All three curvatures must add to zero • Unique solution is determined by fixing V on boundary surface • Mean value theorem:

  7. Classification of 2nd order PDEs • Same as conic sections (where ) • Elliptic – Laplacian • Spacelike boundary everywhere • 1 boundary condition at each point on the boundary surface • Hyperbolic – wave equation • Timelike (initial) and spacelike (edges) boundaries • 2 initial conditions in time, 1 boundary condition at each edge • Parabolic – diffusion equation • Timelike (initial) and spacelike (edges) boundaries • 1 initial condition in time, 1 boundary condition at each edge

  8. External boundary conditions • Uniqueness theorem – difference between any two solutions of Poisson’s equation is determined by values on the boundary • External boundary conditions:

  9. Internal boundary conditions • Possible singularities (charge, current) on the interface between two materials • Boundary conditions “sew” together solutions on either side of the boundary • External: 1 condition on each side Internal: 2 interconnected conditions • General prescription to derive any boundary condition:

  10. Solution: relaxation method • Discretize Laplacian • Fix boundary values • Iterate adjusting potentials on the grid until solution settles

  11. Solution: finite difference method • Discretize Laplacian • Fix boundary values • Solve matrix equation for potential on grid

  12. Solution: finite element method • Weak formulation: integral equation • Approximate potential by basis functions on a mesh • Integrate basis functions; solve matrix equation

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