Applied PDEs Basic (for Reference)

PBA

Background

  • Nalba operator ()
  • Laplace operator () (Divergence of gradient)

Mostly, we'll deal with order linear PDEs.

Classification of order liner PDEs

Basic form

Hyperbolic order PDEs

() * Typically time dependent * Retain & propagate disturbances present in initial data

Prototype: Wave Equation

where, is the amplitude, is the propagation speed.

Parabolic PDEs

() * Time dependent * Solutions smooth out as time increases

Prototype: Heat Equation

where, is the temperature, is the thermal diffusivity.

Elliptic PDEs

() * Describe static problems (systems in equilibrium) * Solution is smooth (if coefficients are)

Prototype: Laplace Equation

Steady-state solutions to hyperbolic and parabolic PDEs.

Boundary Conditions

Assume function in Domain

  • Initial condition:
  • Dirichlet condition: (assume on boundary ) Value prescribed.
  • Neumann condition: (on boundary ) Derivative prescribed.

Poisson's equation(Prototype of elliptical PDE)

if , it is Laplace equation. 2D Laplace operator:

Finite Difference Solution

  • 1D problem:
  • Use regular grid:
  • Approximate derivatives with finite differences:
  • Equation for every grid point :

Finite Differences

  • Approximate derivatives with difference formulas on regular grid
  • Approximate continuous solution in grid points

Finite Elements

  • Decompose domain into simple elements
  • Construct simple function space and compute approximate solution(integrally on domain)