Heat equation (Separation of variables): solve on domain -1<=x<=1, 0<=t<=1, and...

[Meph]

Heat equation (Separation of variables): solve on domain -1<=x<=1, 0<=t<=1, and...

Hi,

The problem is to solve the heat equation:

. . . . .\(\displaystyle \dfrac{\partial u}{\partial t}\, =\, D\, \dfrac{\partial^2 u}{\partial x^2}\)

...in the domain:

. . . . .\(\displaystyle -1\, \leq\, x\, \leq\, 1,\, 0\, \leq\, t\, \leq\, 1,\)

...where the boundary and initial condtions are:

. . . . .\(\displaystyle u(-1,\, t)\, =\, u(1,\, t)\, =\, 0,\, u(x,\, 0) \,= \,\sin(\pi x)\)

...and \(\displaystyle D\, =\, \dfrac{1}{10}\)

The solution should be written in exact form (no sum).

Thanks in advance!

EDIT: I have separated the variables like this:

X''/X = T'/DT = -lambda

X'' + lambda*X = 0
T' + D*lambda*T = 0

But when applying the boundary conditions for the different cases of lambda (lambda = 0, lambda = -a^2 < 0, lambda = a^2 > 0) I get stuck.
If I'm not mistaken the equations for the given lambdas above are respectively:

lambda = 0: X(x) = c1+c2*x
lambda = -a^2: X(x) = c1*cosh(ax) + c2*sinh(ax)
lambda = a^2: X(x) = c1*cos(ax) + c2*sin(ax)