Solution for PDE
∂u/∂t=k ∂/∂x (∂u/∂x)
I.C.→ u(x,0)=12 cos(9πx⁄L)-7 sin(4πx⁄L)
B.C.→ u(-L,t)=u(L,t) & ∂u/∂x (-L,t)=∂u/∂x(L,t)
I expect you can solve this with separation of variables.
Assume you can write , then we have and .
Substituting into the DE gives
Now notice that the LHS is only a function of t, and the RHS is only a function of x, so for them to be equal, they must be equal to the same constant value .
We also have
Now see what you can do with the Boundary and Initial conditions to evaluate .
This might help you further...
Dear Prove IT,
Thanks for your kind guidance.
There may be type error, and I corrected as below. Sorry for that.
I.C.→ u(x,0)=12 cos(9πx⁄L)-7 sin(4πx⁄L)
Could you let me know how to get root-square of (lambda)
Thanks