Solve initial-value problem for heat equation and find relaxation time

[mmo115]

Homework Statement

Solve the initial-value problem for the heat equation u_t = K\nabla²u in the column 0< x < L₁, 0< y < L₂ with the boundary conditions u(0,y;t)=0, u_x(L₁,y,t)=0, u(x,0;t)=0, u_y(x,L₂;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.. =/

Homework Equations

We are told that :

where m,n=1 to infinity
u(x,y;t)= \Sigma B_mnsin (m*pi*x / L₁) sin (n*pi*y)/L₂) * e^-lambda_mnKt

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

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/pi² \Sigma_m,n=1 [ sin[(m-(1/2))(\pix/L₁)] / (m-1/2) ] * [ sin[n-1/2)(pi y/L₂)\ / (n-1/2) ] * [ e^-lambda_mnKt ]

lambda_mn= (m-1/2)² (pi/L₁)² + (n-1/2)²(pi/L₁)²

relaxation time = (4/pi² K)[L1²L2²/L1²+L2²)]

 

[member 553083]

I would really appreciate it if someone could help me understand how to get this solution. Thank you very much in advance.