18.336 spring 2009
lecture 5
02/19/09
Finite Dierence (FD) Approximation
Consider u C l .
Goal: Approximate derivative by nitely many function values:
m
ku
(x0 )
ai u(xi ) (k l)
xk
i=0
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Vector of coecients a = (a0 , a1 , , am
18.336 spring 2009
lecture 4
02/12/09
Heat equation
ut = 2 u
Physics:
Ficks law: ux F
= au
d
mass balance:
u dx = b
F n dS = b
divF dx
dt V
V
V
ut = b div(au) = c 2 u
simple: c = 1
Fundamental Solution
|x|2
1
(x, t) =
e 4t
(4t)n/2
ut = 2 u
in Rn ]
18.336 spring 2009
lecture 3
Four Important Linear PDE
Laplace/Poisson equation
2 u = f in
u = g on 1 Dirichlet boundary condition
u
= h on 2 Neumann boundary condition
n
1 2 =
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f 0 Laplace equation
2 u = 0
u = harmonic func
18.336 spring 2009
lecture 2
02/05/09
Well-Posedness
Def.: A PDE is called well-posed (in the sense of Hadamard), if
(1) a solution exists
(2) the solution is unique
(3) the solution depends continuously on the data
(initial conditions, boundary condition
18.336 spring 2009
lecture 1
02/03/09
18.336 Numerical Methods for Partial Dierential Equations
Fundamental Concepts
Domain Rn with boundary
PDE in
b.c. on
PDE = partial dierential equation
b.c. = boundary conditions
(if time involved, also i.c. = in