7 02/29 F

1. 2.28c, find the transformed equation.
2. Solve the transformed equation and convert the solution back to physical space.
3. 2.28k.
4. 2.28b. You could also diagonalized this equation by rotating the coordinate system and then stretching the axes. Is the two-dimensional canonical transformation equivalent to this? in particular, are the lines of constant $\xi$ and $\eta$ orthogonal?
5. 3.41. This is similar to the Laplace version discussed in class. Describe the reason that there is no solution physically, considering it as a heat conduction problem in a circular plate.
6. 3.44. 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 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 $c=0$).
7. 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.
8. 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.
9. Find the Green’s function in three-dimensional unbounded space $R^3$. Use either the method of section 2.1 or 2.2 of the web page example as you prefer.