Math 432
Eli Sadoff
January 27, 2016
General Linear Second Order Equation (2.15)
(
ut + (k(x)ux (x, t)x = f (x)
u
t = 0
Can be generalized to
f (x) = a(x)u00 (x) + b(x)u0 (x) + c(x)u(x)
u(a) =
u(b) =
a = x0
b = xm+1
We can discritize this by making a m
Numeric Differential Equations
Prof: Dr. Zhao
January 21, 2016
1
Method of Undeteremined Coefficients
If you want to approximate a derivative,
k
X
u0 (
x) Du(
x) =
aj u(
x = jh)
j=k
We then do a Taylor expansion, and then we collect the coefficients.
Du(x
We have the Dirichlet Boundary Condition for 0 x 1
00
u (x) = f (x)
u(0) =
u(1) =
And we let
D2 u =
1
(u(x h) 2u(x) + u(x + h)
h2
Now we can say
2
1
1
A= 2
h
1
2
.
.
1
.
.
1
.
.
2
1
U1
U2
U = .
.
Um
f (x1 ) h2
f (x2 )
.
F =
.
f (xm1 )
f (
Numeric Differential Equations
Prof: Dr. Zhao
January 11, 2016
Office Hours: Monday and Friday 1:00pm1:50pm
5 homeworks throughout the semester
Attendance is required and is recorded randomly
Grading: Attendance 5%, Homework 40%, Midterm 25%, Final Exam 3
Elliptical Equation:
* Steady-state heat equation (time independent)
00
u (x) = f (x)
u(a) =
u(b) =
1
f (x) = (x)
We can discretize our domain, into smaller intervals. Let [, ] = [0, 1] and
we can divide the interval into equal sub-intervals.
1
Width,