Math 412501
September 22, 2006
Exam 1
Problem 1 (40 pts.) Let f (x) = 2x for 0 x .
(i) Find the Fourier sine series of f (with [0, ] as the basic interval).
(ii) Over the interval [2.5, 2.5 ], sketch the function to which the series converges.
(iii) Roug
Math 412501 Theory of Partial Dierential Equations Lecture 5: Linearity and homogeneity.
Linearity Linear space = a set V of objects that can be summed and multiplied by scalars (real numbers). That is, for any u, v V and r R expressions u + v and ru sho
Math 412501 Theory of Partial Differential Equations Lecture 2: Diffusion equation. Wave equation. Boundary conditions.
heat equation:
2u u =k 2 t x
2 2u 2 u =c t 2 x 2
wave equation:
Laplace's equation:
2u 2u + =0 x 2 y 2
heat equation:
u =k t 2u = c2
Math 412501 Theory of Partial Differential Equations Lecture 6: Separation of variables.
How do we solve a linear homogeneous PDE? Step 1: Find some solutions. Step 2: Form linear combinations of solutions obtained on Step 1. Step 3: Show that every solu
Solutions for homework assignment #4
Problem 1. Solve Laplace's equation inside a rectangle 0 x L, 0 y H, with the following boundary conditions: u (0, y) = 0, x
Solution: y u(x, y) = b0 + H where
n=1
u (L, y) = 0, x
u(x, 0) = 0,
u(x, H) = f (x).
bn
nH s
Math 412501 Sample problems for the final exam
Any problem may be altered or replaced by a different one!
Fall 2006
Some possibly useful information
Parseval's equality for the complex form of the Fourier series on (, ):
f (x) =
n=
cn einx
=

f (x
Math 412501 Theory of Partial Dierential Equations Lecture 4: DAlemberts solution (continued).
Wave equation 2u 2u = c2 2 , t 2 x < x < , < t <
Change of independent variables: w = x + ct, z = x ct. Jacobian:
w x z x w t z t
=
1 c 1 c
How does the equat
Math 412501 Exam 3: Solutions
November 17, 2006
Problem 1 (40 pts.) Solve the initialboundary value problem for the wave equation in a semicircle (in polar coordinates r, ) 2u = t2
2
u
(0 < r < 1,
0 < < ), (0 < r < 1, 0 < < ),
u(r, , 0) = f (r) sin 3,
u
Math 412501 Exam 2: Solutions
October 20, 2006
Problem 1 (50 pts.) Solve the heat equation in a rectangle 0 < x < , 0 < y < , 2u 2u u = + t x2 y 2 subject to the initial condition u(x, y, 0) = (sin 2x + sin 3x) sin y and the boundary conditions u(0, y, t
Math 412501 Exam 1: Solutions
September 22, 2006
Problem 1 (40 pts.) Let f (x) = 2x for 0 x . (i) Find the Fourier sine series of f (with [0, ] as the basic interval).
Take L = in the usual formulas:
2x where Bn = 2
n=1
Bn sin nx, 4 n
2x sin nx dx =
0
4
Math 412501 Fall 2006 Sample problems for the final exam: Solutions
Any problem may be altered or replaced by a different one!
Some possibly useful information
Parseval's equality for the complex form of the Fourier series on (, ):
f (x) =
n=
cn ein
Math 412501 Exam 1
September 22, 2006
Problem 1 (40 pts.) Let f (x) = 2x for 0 x . (i) Find the Fourier sine series of f (with [0, ] as the basic interval). (ii) Over the interval [2.5, 2.5], sketch the function to which the series converges. (iii) Roug
Solutions for homework assignment #2
Problem 1. Show that the equation u 2u = k 2 + Q(u, x, t) t x is linear if Q(u, x, t) = (x, t)u + (x, t) and in addition homogeneous if (x, t) = 0.
Solution: The equation has the form L(u) = (x, t), where L(u) = u  k
Math 412501 Theory of Partial Dierential Equations Lecture 1: Introduction. Heat equation
Denitions
A dierential equation is an equation involving an unknown function and certain of its derivatives. An ordinary dierential equation (ODE) is an equation in
Math 412501 Theory of Partial Dierential Equations Lecture 3: Steadystate solutions of the heat equation. DAlemberts solution of the wave equation.
Onedimensional heat equation c u = t x K0 u x +Q
K0 = K0 (x), c = c(x), = (x), Q = Q(x, t). Assuming K0
Math 412501 Exam 3
November 17, 2006
Problem 1 (40 pts.) Solve the initialboundary value problem for the wave equation in a semicircle (in polar coordinates r, ) 2u = t2
2
u
(0 < r < 1,
0 < < ), (0 < r < 1, 0 < < ),
u(r, , 0) = f (r) sin 3,
u (r, , 0) =
Math 412501
September 22, 2006
Exam 1: Solutions
Problem 1 (40 pts.) Let f (x) = 2x for 0 x .
(i) Find the Fourier sine series of f (with [0, ] as the basic interval).
Take L = in the usual formulas:
2x
where
Bn =
2
=
2x sin nx dx =
0
4
x cos nx
n
n=1
4
Math 412501
October 20, 2006
Exam 2
Problem 1 (50 pts.) Solve the heat equation in a rectangle 0 < x < , 0 < y < ,
u
2u 2u
=
+
t
x2 y 2
subject to the initial condition
u(x, y, 0) = (sin 2x + sin 3x) sin y
and the boundary conditions
u(0, y, t) = u(, y,
Math 412501
October 20, 2006
Exam 2: Solutions
Problem 1 (50 pts.) Solve the heat equation in a rectangle 0 < x < , 0 < y < ,
2u 2u
u
=
+
t
x2 y 2
subject to the initial condition
u(x, y, 0) = (sin 2x + sin 3x) sin y
and the boundary conditions
u(0, y,
Math 412501
November 17, 2006
Exam 3
Problem 1 (40 pts.) Solve the initialboundary value problem for the wave equation in
a semicircle (in polar coordinates r, )
2u
=
t2
2
u
(0 < r < 1,
0 < < ),
u
(r, , 0) = 0
t
u(r, , 0) = f (r) sin 3,
(0 < r < 1,
0 <
Math 412501
November 17, 2006
Exam 3: Solutions
Problem 1 (40 pts.) Solve the initialboundary value problem for the wave equation in
a semicircle (in polar coordinates r, )
2u
=
t2
2
u
(0 < r < 1,
0 < < ),
u
(r, , 0) = 0
t
u(r, , 0) = f (r) sin 3,
(0 <
Math 412501
Sample problems for the nal exam
Fall 2006
Any problem may be altered or replaced by a dierent one!
Some possibly useful information
Parsevals equality for the complex form of the Fourier series on (, ):
cn einx
f (x) =
f (x)2 dx = 2
=
n=
Math 412501
Fall 2006
Sample problems for the nal exam: Solutions
Any problem may be altered or replaced by a dierent one!
Some possibly useful information
Parsevals equality for the complex form of the Fourier series on (, ):
cn einx
f (x) =
f (x)2 d
Homework assignment #1
(due Friday, September 8)
Problem 1. Determine the equilibrium temperature distribution for a onedimensional
rod with constant thermal properties (K0 = const) with the following sources and boundary
conditions:
(i)
Q = 0,
(ii)
Q =
Solutions for
homework assignment #2
Problem 1. Show that the equation
u
2u
= k 2 + Q(u, x, t)
t
x
is linear if Q(u, x, t) = (x, t)u + (x, t) and in addition homogeneous if (x, t) = 0.
2
Solution: The equation has the form L(u) = (x, t), where L(u) = u k
Solutions for
homework assignment #4
Problem 1. Solve Laplaces equation inside a rectangle 0 x L, 0 y H , with the
following boundary conditions:
u
(0, y ) = 0,
x
u
(L, y ) = 0,
x
u(x, 0) = 0,
Solution:
y
u(x, y ) = b0 +
H
n=1
bn
where
nH
sinh
L
1
sinh
u(
Solutions for
homework assignment #5
Problem 1. Consider the nonSturmLiouville dierential equation
d2
d
+ (x)
+ ( (x) + (x) = 0.
2
dx
dx
Multiply this equation by H (x). Determine H (x) such that the equation may be reduced to
the standard SturmLiouvi
Math 412501 Exam 2
October 20, 2006
Problem 1 (50 pts.) Solve the heat equation in a rectangle 0 < x < , 0 < y < , u 2u 2u = + t x2 y 2 subject to the initial condition u(x, y, 0) = (sin 2x + sin 3x) sin y and the boundary conditions u(0, y, t) = u(, y,
Solutions for homework assignment #5
Problem 1. Consider the nonSturmLiouville differential equation d2 d + (x) + (x) + (x) = 0. 2 dx dx Multiply this equation by H(x). Determine H(x) such that the equation may be reduced to the standard SturmLiouville