9 Due March 30, 2016 1. Solve 1 @T @2 T @2 T D C 2; @t @x 2 @y with boundary and initial conditions as shown. 35 points T D0 yDb T D0 initially T D0 k @T h @x C T D T0 xDa yD0 T D0 Decompose as T(x,y,t)=T1(x,y)+T2(x,y,t) and follow the recent class example. 2. Solve @2 @2 @ C C2 C 2 2 @x @y @y 30 points D 0; with boundary conditions as shown. @ @y D q0 yDb D 0 De y xDa yD0 D 0 For part requiring eigenfunction expansion in y, you will need to carefully identify the y-operator. There is a first derivative in y along with the second. It will be convenient to bundle up everything @2 @ other than the d2/dx2 into the y-operator, i.e., @y 2 C2 @y C The traditional separation of variables may be instructive in this case. 3. Obtain the 'general' solution for the differential equation: 3 @2 u @t 2 4 @2 u @2 u C 2 D 0: @t@x @x 25 points Proceed as follows: Assume D x C t; D x C t; and write the differential equation in terms of and . Choose D D 1 and obtain the values of and so that there are only the cross derivative terms left. Then obtain the general solution in terms of and . Now apply the initial conditions u.x; 0/ D .x/ and @u D 0: @t tD0 Consider the special case when .x/ has the form shown below, and plot u.x; t/ for t D 1 and t D 2. .x/ 2 1 rr r r r rr r x 2 20 points Carry out an eigenfunction expansion based on the homogeneous boundary conditions in x