Math 5C 2005
Practice Final

1. Compute the Fourier series of the function
   
   $$f(x)=\left\{ \begin{array}{lcr} -x, \:\:\:\:\:\: -\pi < &x& < 0\\ x, \:\:\:\:\:\:\:\:\:\:\: 0 < &x& < \pi \end{array}\right.$$
   
   

2. Find a solution of the IVP,
   $$\begin{eqnarray*} u_{t}&=&{1\over 2}u_{xx},\; 0 \le x \le 1, \; t\in{\rm I\!R}... ...\ u(x,0)&=&{1\over 2}\cos (3 \pi x) - {3\over 5} \cos (7\pi x), \end{eqnarray*}$$
   
   
   
   
   
   for the heat distribution in a rod with insulated ends,
   
   $$u_x(0,t) = 0 = u_x(1,t), \; t\in{\rm I\!R}^+.$$
   
   
   Express the solution as a Fourier series.
3. Solve the IVP,
   $$\begin{eqnarray*} u_{tt}-u_{xx}&=&0 , \; 0 \le x \le 1, \; t\in{\rm I\!R}^+ ,\\ u(x,0)&=&\sin (\pi x)\\ u_t(x,0)&=&-3 \sin (3 \pi x), \end{eqnarray*}$$
   
   
   
   
   
   with the vanishing BC,
   
   $$u(0,t)=0= u(1,t), \; \; t\in{\rm I\!R}^+.$$
   
   
   Express the solution as a Fourier series.
4. Solve the BVP
   $$\begin{eqnarray*} u_{xx}+u_{yy}&=&0 ,\ \; 0 \le x \le 1,\ \; 0 \le y \le 1\\ u(... ...e y \le 1\\ u(x,0)&=&0,\;\; u(x,1) \;=\; x^2,\ \; 0 \le x \le 1 \end{eqnarray*}$$
   
   
   
   
   

5. Solve the BVP,
   
   $$u_{xx}+u_{yy}= \sin (6\pi x)\cos (2\pi y) + 4 \cos (4\pi x)\cos (10\pi y), \; 0 \le x, y \le 1$$
   
   
   with periodic BC,
   
   $$u(x+1,y)=u(x,y), \; u(x,y+1)=u(x,y), 0 \le x, y \le 1.$$