8 03/05 F

1. 3.38. This does not require solution of the problem using the Poisson integral formula. You can just examine what symmetry properties the solution $u(x,y)$ should have to figure out the value at the origin. You might want to draw some isotherms to guide your thoughts. However, feel free to check your result against the Poisson integral.

2. 3.39. Again this does not require solution of the problem using the Poisson integral formula. You should be able to find the complete solution $u(x,y)$ by mere inspection. However, feel free to check your result against the Poisson integral; in that case, first write the boundary values in the form
   
   $$f(\phi) = A + B \cos(\phi-\theta) + C \sin(\phi-\theta)$$
   
   
   The integrals for $A$ and $B$ can be found in a table of definite integrals.

3. 3.41. This is similar to the Laplace version discussed earlier in class. Describe the reason that there is no solution physically, considering it as a heat conduction problem in a circular plate.

4. 3.44. This is mostly the uniqueness proof given in class, which can also be found in solved problems 3.14-3.16. However, here you will want to write out the two parts of the surface integral separately since the boundary conditions are a mixture of the two cases 3.14 and 3.15 (with $c=0$).

5. Show that the following Laplace equation problem has a unique solution, $u=0$:
   
   $$\mbox{PDE: } \nabla^2 u = 0\qquad \mbox{BC: } u(0,y)=u_y(x,0)=u_y(x,1)=u(1,y)+u_x(1,y)=0$$
   
   
   This is essentially the uniqueness proof given in class, which can also be found in solved problems 3.14-3.16. However, you will want to write the four parts of the surface integral out separately since the boundary conditions are a mixture of the three cases 3.14-3.16.

6. Show that the following Laplace equation problem has infinitely many solutions beyond $u=0$:
   
   $$\mbox{PDE: } \nabla^2 u = 0\qquad \mbox{BC: } u(0,y)=u_y(x,0)=u_y(x,1)=u(1,y)-u_x(1,y)=0$$
   
   
   Hint: Guess a very simple nonzero solution and check that it satisfies all boundary conditions and that its second order derivatives are zero. Since the equations are linear, any arbitrary multiple of this solution is also a solution. Verify whether or not the uniqueness proof of the previous section conflicts with the nonunique solution of this problem. Why would a slight difference in one boundary condition make a difference?