Partial Differential Equations
Math 118A, Fall 2013
Midterm 1: Solutions
1. [15%] Is the PDE
yuxx + uyy = 0
linear or nonlinear? For what constants A, B , C is the function
u(x, y ) = Ax2 y + Bxy 2 + Cy 4
a solution of the PDE?
Solution.
The PDE is linea
Final: Solutions
Math 118A, Fall 2013
1. [20 pts] For each of the following PDEs for u(x, y ), give their order and say
if they are nonlinear or linear. If they are linear, say if they are homogeneous
or nonhomogeneous and if they have constant or variabl
Math 118 Fall 2013
Homework 5 Solutions
2.3.2 (a) increases or stays the same. (b) decreases or stays the same.
Consider the two rectangles R1 := cfw_0 x l, 0 t T1 and R2 := cfw_0 x l, 0 t T2
with T1 < T2 . We have R1 R2 . Thus, M (T1 ) M (T2 ) and m(T
Math 118: PDE
HW 4 Solutions
2.4.1
The solution of the IVP is
1
4kt
1
=
4kt
e(xy)
u(x, t) =
l
2 /4kt
2 /4kt
e(xy)
(y ) dy
dy.
l
Substitute p = (x y )/ 4kt in this integral, we obtain
1
u(x, t) =
x
+l
4kt
x
l
4kt
x
+l
4kt
1
=
=
2
ep dp
0
1
Erf
2
x
Math 118: PDE
HW 3 Solutions
1.3.3
By the law of conservation of energy, we have
rate of change of thermal energy = heat ux in heat ux out. (1)
Let u(x, t) denote the temperature of the rod at postition x and time
t. The thermal energy density e(x, t) p
Math 118 Fall 2013
Homework 2 Solutions
1.3.1
Applying Newtons Law F = ma in both x- and y -directions. Refer to class notes for assumptions.
For x-direction, no motion implies no resistance force. We still have as in class
T (b, t) cos(b, t) T (a, t) c
Math 118: PDE
HW 6 Solutions
4.1.1
A violin string is modeled exactly by the problem (1), (2), and (3).
Its general solution is given as a Fourier expansion in (9).
We nd that the frequencies (or note produced by the violin string) are
n T
for n = 1, 2
Math 118: PDE
HW 7 Solutions
4.1.4
Using the separated solution in the PDE, we get
F G = c2 F G rF G.
Separation of variables gives
G + rG
F
=
= ,
2G
c
F
where is a separation constant.
The eigenvalue problem for F (x) is
F + F = 0, F (0) = 0, F (l) = 0
Midterm 2: Sample solutions
Math 118A, Fall 2013
1. Find all separated solutions u(r, t) = F (r )G(t) of the radially symmetric
heat equation
u
k
u
r
.
=
t
r r
r
Solve for G(t) explicitly. Write down an ODE for F (r ) but dont try to
solve it. (What makes
Midterm 2: Solutions
Math 118A, Fall 2013
1. [25%] Find all separated solutions u(x, t) = F (x)G(t) of the advection
equation
ut + cux = 0
where c is a constant. Show that the separated solutions have the same form
as the general solution u(x, t) = f (x c
Midterm 2: Sample questions
Math 118A, Fall 2013
1. Find all separated solutions u(r, t) =
heat equation
u
k
=
t
r r
F (r )G(t) of the radially symmetric
r
u
r
.
Solve for G(t) explicitly. Write down an ODE for F (r ) but dont try to
solve it. (What makes
Midterm 1: Sample questions
Math 118A, Fall 2013
1. Say whether the following operators acting on functions u(x, y ) are linear
or nonlinear. Justify your answers. (a) Lu = uxx + uyy + 1; (b) Lu =
yuxx + uyy + u; (c) Lu = uuxx + uyy .
2. Solve the followi
Midterm 1: Sample solutions
Math 118A, Fall 2013
1. Say whether the following operators acting on functions u(x, y ) are linear
or nonlinear. Justify your answers. (a) Lu = uxx + uyy + 1; (b) Lu =
yuxx + uyy + u; (c) Lu = uuxx + uyy .
Solution.
(a) Nonli