Introduction to Fourier Methods and Partial Differential Equations
MATH 418

Fall 2009
Math 418
Homework 1 Solutions
1.1.2 (a) linear
(b) nonlinear (due to the product uu y )
(c) nonlinear (due to u2 )
y
(d) linear
(e) linear
1.1.3 (a) Order 2, Linear inhomogeneous: rewrite the equation as L(u) = 1,
where L(u) = ut uxx is linear.
(b) Order
Introduction to Fourier Methods and Partial Differential Equations
MATH 418

Fall 2009
Math 418
Homework 1 Solutions
1.1.2 (a) linear
(b) nonlinear
(c) nonlinear
(d) linear
(e) linear
1.1.3 (a) Linear inhomogeneous: rewrite the equation as L(u) = 1, where
L(u) = ut uxx is linear.
(b) Linear homogeneous: rewrite the equation as L(u) = 0, whe
Introduction to Fourier Methods and Partial Differential Equations
MATH 418

Fall 2009
Math 418
Homework 2 Solutions (continued)
2.4.1 The solution is
1
4kt
1
=
4kt
u(x, t) =
e(xy)
l
e(xy)
2
2
/4kt
/4kt
(y)dy
dy.
l
Now, let p = (x y)/ 4kt. Then dp = dy)/ 4kt, so the integral takes
the form
1
u(x, t) =
xl
4kt
x+l
4kt
xl
4kt
2
ep dp
x+l
1