Heat Equation - Trouble Finding a General Solution

In summary, the conversation is about solving the heat equation with a given initial condition. The person is struggling to solve it and is considering a change of variables. Another person suggests using a Gaussian integral technique, which involves completing the square. The person continues to work on the problem and eventually finds the solution. They have a final question about a negative sign in their substitution.
  • #1
The_Chromer
15
0

Homework Statement


Solve:

Ut=kUxx
U(x,0)=e^3x


Homework Equations


The Heat Equation:
48b739a375dd1fc340c8fe456f9e165c.png


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.
 
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  • #2
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].
 
  • #3
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.
 
  • #4
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.
 

1. What is the Heat Equation?

The Heat Equation is a mathematical model that describes the flow of heat through a given medium over time. It is a partial differential equation that relates the rate of change of temperature with respect to time and space.

2. Why is it important to find a general solution for the Heat Equation?

Finding a general solution for the Heat Equation is important because it allows us to predict the temperature at any point in space and time within the given medium. This can be useful in various fields such as physics, engineering, and meteorology.

3. What are the difficulties in finding a general solution for the Heat Equation?

One of the main difficulties in finding a general solution for the Heat Equation is that it is a second-order partial differential equation, which can be challenging to solve analytically. Additionally, the boundary conditions and initial conditions of the problem may also complicate the solution process.

4. What methods can be used to find a general solution for the Heat Equation?

There are several methods that can be used to find a general solution for the Heat Equation, including separation of variables, Fourier series, and Laplace transforms. Each method has its advantages and limitations, and the choice of method depends on the specific problem at hand.

5. How can I verify if my general solution for the Heat Equation is correct?

To verify the correctness of a general solution for the Heat Equation, you can substitute it back into the original equation and see if it satisfies the equation. Additionally, you can also compare your solution with numerical or experimental results to check for consistency.

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