I want to solve a heat equation (assumed a.o. that the time derivative equals zero): $u_{xx}=-(H(x-0.3)-H(x-0.7))$. I let some constants out, because these don't matter for the solving method. Note $u(0)=200, u(1)=400, u_x(0)=0$.
To obtain an analytical expression I used LaPlace transformation and got $U(s)=\frac{\exp{-0.7s}}{s^3}-\frac{\exp{-0.3s}}{s^3}+\frac{200}{s}$.
Doing the inverse LaPlace transformation yields $u(x)=\frac{1}{2}(x-0.7)^2H(x-0.7)-\frac{1}{2}(x-0.3)^2H(x-0.3)+200.$
This last expression gives a straight line. However, this doesn't make sense, since we start with a heat source (the heaviside function) and we expect to heat the intervals which started at '0 degrees'. My numerical simulation confirmed this expectation.
Did I make a miscalculation somewhere, an error in the transformation or some other mistake? Thank you in advance!