I'm trying to learn how to use Laplace transforms to solve PDE's. I'm now trying to find the general solution of the heat equation.
$$\partial_{xx}u=\partial_{t}u$$ Taking the Laplace transform w.r.t $t$ gives: $$\partial_{xx}T(x,s)=sT(x,s)-u(x,0)$$ $$\text{Where } T(x,s):=\mathcal L_t[u(x,t)](s)$$
Solving this ODE, again using laplace transforms gives me:
$$T(x,s)=(\frac{1}{2}T(0,s)+T_x(0,s)\frac{1}{2 \sqrt s})e^{\sqrt s x}+(\frac{1}{2}T(0,s)-T_x(0,s)\frac{1}{2 \sqrt s})e^{-\sqrt s x}-(e^{\sqrt s x}-e^{-\sqrt s x})*u(x,0)$$ Where the last $*$ denotes the convolution operator, and $T_x=\partial _xT$
Taking the inverse laplace transform w.r.t $t$ of this equation seems like a mess. I don't know how to deal with all the $\sqrt s$ in the exponentials, nor the one in the first coefficient.
Am I taking the right approach? How do I continue solving this heat equation?