Thread: Using fourier and laplace transform to solve PDE

1. 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??

3. 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,
$\displaystyle u_{t}=c^2u_{xx}$
Now the way you have it is,
$\displaystyle u_{t} = -u_{xx}$
How is that possible?

4. O...Sorry...correct it>_<

5. And you forgot to mention the initial value problem.

6. Originally Posted by ThePerfectHacker
And you forgot to mention the initial value problem.
yES///i Want to have some example^^ But I can not find related this >_<

7. 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,
$\displaystyle u_{xx} = u_t \mbox{ for } -\infty< x < \infty \mbox{ and } t>0 \mbox{ with }u(x,0)=f(x)=x$

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

8. 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,
$\displaystyle u_{xx} = u_t \mbox{ for } -\infty< x < \infty \mbox{ and } t>0 \mbox{ with }u(x,0)=f(x)=x$

The solution is given by,
$\displaystyle u(x,t) = \frac{1}{\sqrt{\pi}}\int_{-\infty}^{\infty}f(x+2y\sqrt{t})e^{-y^2}dy= \frac{1}{\sqrt{\pi}}\int_{-\infty}^{\infty}(x+2y\sqrt{t})e^{-y^2} dy$
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 $\displaystyle -\infty< x < \infty$ and then inverse laplace transform?

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

Thanks