(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Solve the initial-value problem for the heat equation u_{t}= K[tex]\nabla[/tex]^{2}u in the column 0< x < L_{1}, 0< y < L_{2}with the boundary conditions u(0,y;t)=0, u_{x}(L_{1},y,t)=0, u(x,0;t)=0, u_{y}(x,L_{2};t)=0 and the initial condition u(x,y;0)=1. Find the relaxation time.

Can anyone please explain how to get this solution.. I really don't understand how to arrive at the solution. I'm hoping that if i can learn specifically how to do this i can apply it to other similar problems. I have searched around the internet for some information, but it seems like most of it is using actual situations as opposed to theoretical. i mean the book doesn't even explain what relaxation time is or how to derive it.. =/

2. Relevant equations

We are told that :

where m,n=1 to infinity

u(x,y;t)= [tex]\Sigma[/tex] B_{mn}sin (m*pi*x / L_{1}) sin (n*pi*y)/L_{2}) * e^-lambda_{mn}Kt

We can use this to solve initial-value problems for the heat equation.

3. The attempt at a solution

I really don't get how to solve for the B_{mn}... the book really doesn't give a good explanation.

The solution is u(x,y;t) = 4/pi^{2}[tex]\Sigma[/tex]_{m,n=1}[ sin[(m-(1/2))([tex]\pi[/tex]x/L_{1})] / (m-1/2) ] * [ sin[n-1/2)(pi y/L_{2})\ / (n-1/2) ] * [ e^-lambda_{mn}Kt ]

lambda_{mn}= (m-1/2)^{2}(pi/L_{1})^{2}+ (n-1/2)^{2}(pi/L_{1})^{2}

relaxation time = (4/pi^{2}K)[L1^{2}L2^{2}/L1^{2}+L2^{2})]

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# Homework Help: Solve initial-value problem for heat equation and find relaxation time

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