Q:

Non homogeneous Heat Equation PDE

$$A_t-A_{xx} = \sin(\pi x)$$
$$A(0,t)=A(1,t)=0$$
$$A(x,t=0)=0$$
Find $A$.
I know I need to find the homogeneous and particular solutions. Im just not sure on this PDE.

A:

The solution may be accomplished using a Laplace transform.  Defining
$$\hat{A}(x,s) = \int_0^{\infty} dt \, A(x,t) \, e^{-s t}$$
and applying the initial condition, we get an ordinary differential equation in $x$:
$$\frac{d^2}{dx^2} \hat{A} - s \hat{A} = -\frac{1}{s} \sin{\pi x}$$
The zero boundary conditions in $x$ mean that the homogeneous solution is zero.  The solution then takes the form $\hat{a}(x,s) = P \sin{\pi x}$.  Plugging this into the equation, we get the solution
$$\hat{A}(x,s) = \frac{\sin{\pi x}}{s (\pi^2 + s)}$$
You can use partial fractions, or simply look up in a table of inverse LT's; the solution is
$$A(x,t) = \frac{1}{\pi^2} \sin{\pi x} \, (1-e^{-\pi^2 t})$$

