- #1

- 89

- 1

first off the heat equation is given by:

[tex]\frac{\partial u}{\partial t}=k^{2}\frac{\partial^{2}u}{\partial x^{2}}[/tex]

then take the fourier transform of this w.r.t.x, where in this notation the Ftransform of u(x,t) is denoted by U(alpha,t):

[tex]\frac{\partial U}{\partial t}=-\alpha^{2}k^{2}U[/tex]

***it is this step I don't quite follow**** integrate with respect to t:

[tex]U=c(\alpha)e^{-k^{2}\alpha^{2}t}[/tex]

Can someone please explain this step ie. how do you know what limits to integrate between and how do you know that the rights side of the equation ends up in the form?:

[tex]c(\alpha)e^{-k^{2}\alpha^{2}t}[/tex]

obviously an intermediate step is:

[tex]\partial U=-k^{2}\alpha^{2}Udt[/tex]