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Laplace's Equation: Curvature Operator and PDEs in Physics

This text provides an overview of Laplace's equation and its applications in physics. It covers the Laplacian operator, boundary conditions, numerical solutions, and the classification of hyperbolic, elliptic, and parabolic PDEs. The text also discusses external and internal boundary conditions and the uniqueness theorem.

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Laplace's Equation: Curvature Operator and PDEs in Physics

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  1. §3.1Laplace’s equation Christopher Crawford PHY 416 2014-10-20

  2. Outline • OverviewSummary of Ch. 2Intro to Ch. 3, Ch. 4 • Laplacian – curvature (X-ray) operatorPDEs 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 – HW6

  3. Summary of Ch. 2

  4. Laplacian in physics • Derivative chain: potential to conservative flux to source • 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 Conic Sections • Quadratic bilinear form: matrix of coefficients • Elliptic – 2 positive eigenvalues, det > 0 • Hyperbolic – 1 negative eigenvalue, det < 0 • Parabolic – 1 null eigenvalue, det = 0

  8. 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

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

  10. 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:

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