# Laplace transform heat equation $u(0,t)=f(t),u(x,0)=u_0$

Solve the heat equation $$\frac{\partial u}{\partial{t}}=\nu\frac{\partial^2{u}}{\partial{x}^2},t>0,x>0$$ where $$u(x,0)=f(t),u(x,0)=u_0$$ using Laplace transforms.

My trouble is dealing with the general initial condition to find constants. Applying the Laplace transform I got $$U(s)=A(s)e^{\sqrt{\frac{s}{\nu}}x}+B(s)e^{-\sqrt{\frac{s}{\nu}}x}.$$ The issue is finding out $$A$$ and $$B$$. Using $$u(0,s)=F(s)$$, I get $$A(s)+B(s)+\frac{u_0}{s}=F(s).$$ The only way I can think of determining them is by forcing $$U$$ is bounded. Then $$A=0$$ and I can write $$B(s)=F(s)-\frac{u_0}{s}$$ and the solution is $$u(t,x)=L^{-1}(F(s)e^{\sqrt{\frac{s}{\nu}}x})-u_0\text{erfc}(\frac{x}{2\sqrt{\nu t}})+u_0.$$ Is this the way to proceed?

• Look a little closely at your problem set up, you’ve a typo. – DaveNine Nov 17 '18 at 10:06