Partial differential equations Function depends on two or more independent variables This is a very simple one - there are many more complicated ones

Order of the PDE is given by its highest derivative is 2nd order is 4th order

Linear PDE is linear in dependent variable, and all coefficients depend on independent variables only Nonlinear PDEs violate these rules

PDE that often appears in engineering is second order, linear PDE General form: A, B, C are functions of x and y D is function of x,y,u and and

Can use values of coefficients A,B,C to characterize the PDE

Why categorize? Different methods to solve different types Different types describe different engineering problems Elliptic - steady state Parabolic - propagation Hyperbolic - vibrations

Analytic solutions - there aren’t many Often can use analytic tools to get idea of behavior of a PDE, especially as parameters are changed Important for limiting cases

Elliptic PDEs Steady-state two-dimensional heat conduction equation is prototypical elliptic PDE This is the Laplace equation This is the Poisson equation

Think of a small box q x, in q y, out q x, out q y, in At steady state, net change in heat is 0, so

Shrink to differential size Fourier’s law of heat conduction

Substitute We will solve with finite differences

Discretize PDE so that we have a mesh of grid points with boundary conditions Index for x is i Index for y is j

Use finite differences for the derivatives T i-1,j T i,j T i+1,j

Now the y derivative T i,j+1 T i,j T i,j-1

Substitute these expressions back into original elliptic PDE Assume  x=  y. Can rearrange to get True for all interior points

Need to define values on ALL boundaries - Dirichlet boundary condition (Neumann BC fix flux at boundary)

Each interior point has an equation - for 9 x 9 interior points - 81 equations Adds up quickly Example: 4 x 4 grid - 2 x 2 interior points

Let i=1, j=1

Fill in the matrix Generally, we get a sparse matrix (big, too) Technique most often used is Gauss-Seidel or some variation of it - matrix is always diagonally dominant - also called Liebmann’s rule

Apply these steps iteratively until T converges Rewrite equation in Gauss-Seidel form Use overrelaxation (if desired)

Solving our example - the four equations are

Rewrite them in Gauss Seidel form and assume initial values for T

Run without overrelaxation

End result

What about derivative (flux) boundary conditions I.E. if we insulate one side of the plate, is 0 there Create an imaginary point outside boundary T 0,j+1 T -1,j T 1,j T 0,j T 0,j-1

Equation becomes Now consider finite difference for derivative at 0

Substitute Derivative BC now included in equation

Irregular domains (funny shapes) What do you do with a domain like?

Your book uses ,  to scale the  x,  y Different  x,  y 1x1x 2x2x 1 y1 y 2 y2 y

Can develop equations for edge points Now use a Gauss-Seidel or other matrix approach