# Using fourier and laplace transform to solve PDE

• Aug 5th 2007, 07:10 AM
cyw1984
Using fourier and laplace transform to solve PDE
PDE is type of heat equation.
Where have above reference notes?

Many book only gives an example of solving heat equation using fourier transform.
An exercise asks me to solve it for using fourier and laplace transform:
u_xx = u_t -inf < x < +inf, t >0
u(x,o) = x
u(o,t) = 0

In the heat equation, we'd take the fourier transform with respect to x for
each term in the equation. How to combine it with using fourier and laplace transform

Can anyone suggest some example and notes to me??
• Aug 5th 2007, 07:16 AM
cyw1984
• Aug 5th 2007, 08:57 AM
ThePerfectHacker
Quote:

Originally Posted by cyw1984
An exercise asks me to solve it for using fourier and laplace transform:
u_xx + u_t = 0 -inf < x < +inf, t >0

The heat equation is,
$u_{t}=c^2u_{xx}$
Now the way you have it is,
$u_{t} = -u_{xx}$
How is that possible?
• Aug 5th 2007, 09:02 AM
cyw1984
O...Sorry...correct it>_<
• Aug 5th 2007, 09:07 AM
ThePerfectHacker
And you forgot to mention the initial value problem. :eek:
• Aug 5th 2007, 09:29 AM
cyw1984
Quote:

Originally Posted by ThePerfectHacker
And you forgot to mention the initial value problem. :eek:

yES///i Want to have some example^^ But I can not find related this >_<
• Aug 5th 2007, 09:50 AM
ThePerfectHacker
Quote:

Originally Posted by cyw1984
An exercise asks me to solve it for using fourier and laplace transform:
u_xx = u_t -inf < x < +inf, t >0
u(x,o) = x
u(o,t) = 0

The condition u(0,t)=0 are only used in finite regions when a boundary value problem exists.

A valid problem would be,
$u_{xx} = u_t \mbox{ for } -\infty< x < \infty \mbox{ and } t>0 \mbox{ with }u(x,0)=f(x)=x$

The solution is given by,
$u(x,t) = \frac{1}{\sqrt{\pi}}\int_{-\infty}^{\infty}f(x+2y\sqrt{t})e^{-y^2}dy= \frac{1}{\sqrt{\pi}}\int_{-\infty}^{\infty}(x+2y\sqrt{t})e^{-y^2} dy$
Is the solution.
• Aug 5th 2007, 10:05 AM
cyw1984
Quote:

Originally Posted by ThePerfectHacker
The condition u(0,t)=0 are only used in finite regions when a boundary value problem exists.

A valid problem would be,
$u_{xx} = u_t \mbox{ for } -\infty< x < \infty \mbox{ and } t>0 \mbox{ with }u(x,0)=f(x)=x$

The solution is given by,
$u(x,t) = \frac{1}{\sqrt{\pi}}\int_{-\infty}^{\infty}f(x+2y\sqrt{t})e^{-y^2}dy= \frac{1}{\sqrt{\pi}}\int_{-\infty}^{\infty}(x+2y\sqrt{t})e^{-y^2} dy$
Is the solution.

umum.....the question mention us to use fourier and laplace transform. Is there mention us to use fourier transform with respect to t (t>0) and then inverse fourier transform to solve the solution; Separately, solve the question using laplace transform with respect to x $-\infty< x < \infty$ and then inverse laplace transform?

Can there combines two transformation in one??
What is the result compared with two method??

Thanks