Partial differential equations

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Partial differential equations Function depends
on two or more independent variables
This is a very simple one - there are many more
complicated ones
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Order of the PDE is given by its highest
derivative
is 4th order
is 2nd order
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Linear PDE is linear in dependent variable, and
all coefficients depend on independent variables
only
Nonlinear PDEs violate these rules
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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
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Can use values of coefficients A,B,C to
characterize the PDE
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• Why categorize?
• Different methods to solve different types
• Different types describe different engineering
  problems
• Elliptic - steady state
• Parabolic - propagation
• Hyperbolic - vibrations

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Analytic solutions - there arent many Often can
use analytic tools to get idea of behavior of a
PDE, especially as parameters are
changed Important for limiting cases
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Elliptic PDEs Steady-state two-dimensional heat
conduction equation is prototypical elliptic PDE
This is the Laplace equation
This is the Poisson equation
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Think of a small box
qy, in
qx, in
qx, out
qy, out
At steady state, net change in heat is 0, so
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Shrink to differential size
Fouriers law of heat conduction
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Substitute
We will solve with finite differences
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Discretize PDE so that we have a mesh of grid
points with boundary conditions
Index for y is j
Index for x is i
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Use finite differences for the derivatives
Ti-1,j
Ti1,j
Ti,j
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Now the y derivative
Ti,j1
Ti,j
Ti,j-1
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Substitute these expressions back into original
elliptic PDE
Assume ?x?y. Can rearrange to get
True for all interior points
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Need to define values on ALL boundaries -
Dirichlet boundary condition (Neumann BC fix flux
at boundary)
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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
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0
25
50
75
10
75
20
75
75
30
60
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Let i1, j1
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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 Liebmanns rule
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Rewrite equation in Gauss-Seidel form
Use overrelaxation (if desired)
Apply these steps iteratively until T converges
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Solving our example - the four equations are
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Rewrite them in Gauss Seidel form
and assume initial values for T
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Run without overrelaxation
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End result
31.79
53.39
56.72
38.39
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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
T0,j1
T-1,j
T1,j
T0,j
T0,j-1
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Equation becomes
Now consider finite difference for derivative at 0
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Substitute
Derivative BC now included in equation
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Irregular domains (funny shapes) What do you do
with a domain like?
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Different ?x, ?y
?2 ?x
?2 ?y
?1 ?y
?1 ?x
Your book uses ?, ? to scale the ?x, ?y
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Can develop equations for edge points
Now use a Gauss-Seidel or other matrix approach