Heat Equation - Trouble Finding a General Solution

The_Chromer

1. The problem statement, all variables and given/known data
Solve:

U_t=kU_xx
U(x,0)=e^3x


2. Relevant equations
The Heat Equation:


3. The attempt at a solution

g(y) in the heat equation for this problem is e^3y. I'm having serious trouble solving this because my professor hasn't taught us the method, and it isn't in the book. I've considered trying a change of variables by taking z=x-y, but this has led me to nowhere. I am lost, and I am not just fishing for a result.

I actually want to know how to solve this damn thing. There's a few pages worth of **** in my notebook, and now I need some guidance.

Thank you for your time.

fzero

The integral that you have to do is known as a Gaussian integral. A standard result, explained, for example, at http://en.wikipedia.org/wiki/Gaussian_integral is

[itex] \int_{-\infty}^\infty \exp\left( - A u^2\right) du = \sqrt{\frac{\pi}{A}}.~~~(1)[/itex]

In your case, you must integrate

[itex] \int_{-\infty}^\infty \exp\left( - \frac{(x-y)^2}{4kt} + 3y \right) dy.[/itex]

The technique needed is known as "completing the square," namely we attempt to write the quadratic expression in [itex]y[/itex] as a sum of a square plus a constant term:

[itex] -A y^2 + B y + C = -a (y+b)^2 + c . ~~~(2)[/itex]

Here, [itex]a,b,c[/itex] will possibly be functions of [itex]x,t,k[/itex], but do not depend on [itex]y[/itex], so the resulting integral can be done by substituting [itex] u = y +b [/itex] and then using the formula (1).

You should probably start with equation (2) and determine [itex]a,b,c[/itex] in terms of [itex] A, B,C[/itex].

The_Chromer

I just wanted to let you know that I haven't given up on the problem and am still working at it.

I'll be back with results whenever I solve it.

The_Chromer

Thank you so much for your help. I reached the final solution. I'll upload the results to show you as soon as I can.

I have a final question as to what happened with a negative sign when I did a substitution, if you don't mind.