any drafts tonight?

A thin rod of unit length has its lateral surface insulated against the flow of heat. The material comprising the rod has thermal conductivity 5.2, specific heat 4.0, and density 1.3. The left end of the rod is insulated, and it's right end radiates freely into air of constant temperature zero, and the initial temperature distribution in the rod is uniform.

u_t-ku_xx=0
u_x(0,t) = 0 for t >= 0
u_x(1,t) + 0.5(1,t) = 0 for t >= 0
u(x,0)=100 for 0 =< x =< 1

(a) Use separation of vairbales and deduce the corresponding eigenvalue problem.
(b) Find the condition(s) satisfied by the eigenvalues of this problem. Please show the details for the calculation in each "case".
(c) Show that there exists an infinite sequence of eigenvalues for this problem and numerically approximate to eight decimal place accuracy for the first three eigenvalues.
(d) Write the corresponding eigenfunctions and the "asymptotic behavior" of the eigenvalues.
(e) Find a series expression for the temperature u=u(x,t) at any point x in the rod at any subsequent time t. Although you do not have to evaluate them, be sure to give explicit formulas for the coefficients in you series expansion for the temperature function.
(f) Approximate to eight decimal place accuracy the first three coefficients of your series expression.
(g) Truncate your series expression for the temperature to the first three terms and use this to estimate the temperature of the rod at position x=0.5 and time t=1.0. How accurate is your answer? How do you know this?


Just got that bad boy for homework tonight. I am going to want to let off some stress after I finish it. If you want to pug/draft just send me a xfire. I'll stomp anyone we play and might possible freestyle (I will be drinking).