# Thread: Tricky Partial Differential problem

1. ## Tricky Partial Differential problem

Hi all, I wish to verify

$\displaystyle u_t=ku_x_x$ with the shorthand notations being $\displaystyle u_t=\frac{\partial u}{\partial t}$ and $\displaystyle u_x_x=\frac{\partial^2 u}{\partial x^2}$ for the following expression

$\displaystyle u(x,t)=\frac{1}{2\sqrt {\pi kt}} \int^{\infty}_{-\infty} e^{-(x-\xi)^2/4kt} f(\xi)d\xi$ where k is a constant and f is continous on the real line ( I actually dont understrand what that 'f' bit means)

Can anyone give me a clue as how to tackle this one? Do I integrate the integrand first and then find the derivatives $\displaystyle u_t=ku_x_x$? If so, how does one tackle the integrand with the infinity signs etc?

Thanks, tips will be appreciated.
Looking forward to hearing from some one.
Bugatti79

2. Use "Leibniz' rule":
$\displaystyle \frac{\partial}{\partial x} \int_{\alpha(x,t)}^{\beta(x,t)} f(x, t, \xi)d\xi= \frac{\partial\beta}{\partial x} f(x, t, \beta(x,t))- \frac{\partial\alpha}{\partial x} f(x, t, \alpha(x,t))+ \int_{\alpha(x,t)}^{\beta(x,t)}\frac{\partial f}{\partial x} d\xi$.

3. Hi HallsofIvy,

This exercise is taken out of 'beginning partial differential equations' by Peter V O Neil. It states the solution is achieved using chain rule method. No mention of Leibniz method unless both are same thing?
1) Is this equivalent to the chain rule method?

2) Does anyone know this book? I find it difficult to grasp. I was hoping to find an easier book on beginning partial differential equations. Any suggestions will be appreciated.

Thanks
bugatti79

4. ## Tricky Partial Differential problem

Loooking at that Leibniz expression...i dont know how one would tackle the problem because I believe the derivative of infinity is undefined ( I am referring to $\displaystyle \frac{\partial\beta}{\partial x}$ ). A function must be continous to be differentiable....
bugatti79

5. Originally Posted by bugatti79
Hi all, I wish to verify

$\displaystyle u_t=ku_x_x$ with the shorthand notations being $\displaystyle u_t=\frac{\partial u}{\partial t}$ and $\displaystyle u_x_x=\frac{\partial^2 u}{\partial x^2}$ for the following expression

$\displaystyle u(x,t)=\frac{1}{2\sqrt {\pi kt}} \int^{\infty}_{-\infty} e^{-(x-\xi)^2/4kt} f(\xi)d\xi$ where k is a constant and f is continous on the real line ( I actually dont understrand what that 'f' bit means)

Can anyone give me a clue as how to tackle this one? Do I integrate the integrand first and then find the derivatives $\displaystyle u_t=ku_x_x$? If so, how does one tackle the integrand with the infinity signs etc?

Thanks, tips will be appreciated.
Looking forward to hearing from some one.
Bugatti79
So you just want to verify that $\displaystyle u_t=ku_{xx}$?

You should not take the integral, you should just differentiate. Since you're differentiating with respect to t,x then you can move the derivatives inside the integral. WRT to t, you just have simple product rule. And then similarly WRT to x. Compute it out (it will be slightly long/messy) and see that they're equal.

EDIT: Leibniz' Rule gives you the same thing here, since the first two terms in the RHS are zero, leaving just what I said.

6. Originally Posted by bugatti79
Loooking at that Leibniz expression...i dont know how one would tackle the problem because I believe the derivative of infinity is undefined ( I am referring to $\displaystyle \frac{\partial\beta}{\partial x}$ ). A function must be continous to be differentiable....
bugatti79
Why do you think this?
$\displaystyle \frac{d}{dx}\lim _ {a \to \infty} a=\lim _ {a \to \infty}\frac{d}{dx}a=\lim _ {a \to \infty}0=0$

7. Originally Posted by bugatti79
Hi all, I wish to verify

$\displaystyle u_t=ku_x_x$ with the shorthand notations being $\displaystyle u_t=\frac{\partial u}{\partial t}$ and $\displaystyle u_x_x=\frac{\partial^2 u}{\partial x^2}$ for the following expression

$\displaystyle u(x,t)=\frac{1}{2\sqrt {\pi kt}} \int^{\infty}_{-\infty} e^{-(x-\xi)^2/4kt} f(\xi)d\xi$ where k is a constant and f is continous on the real line ( I actually dont understrand what that 'f' bit means)

Can anyone give me a clue as how to tackle this one? Do I integrate the integrand first and then find the derivatives $\displaystyle u_t=ku_x_x$? If so, how does one tackle the integrand with the infinity signs etc?

Thanks, tips will be appreciated.
Looking forward to hearing from some one.
Bugatti79
For $\displaystyle \partial _{t}$ I think an argument using the dominated convergence theorem will give you the interchanging of integral and partial derivative. For $\displaystyle \partial _{xx}$ just note that if $\displaystyle t$ is fixed your integral is just a convolution of functions, and so you can derive only the exponential inside the integral.

8. Im confused as to where to start
If i dont need to evalute the integral then i can re-arrange
$\displaystyle u(x,t)=\frac{1}{2\sqrt {\pi kt}} \int^{\infty}_{-\infty} e^{-(x-\xi)^2/4kt} f(\xi)d\xi$ to

$\displaystyle u(x,t)= \int^{\infty}_{-\infty} \frac{e^{-(x-\xi)^2/4kt}}{2\sqrt {\pi kt}} f(\xi)d\xi$ which looks likes solving using quotient rule? Is that correct? How is this a product?

Thanks

9. The quotient and product rule are the same, which ever you prefer.

10. Originally Posted by bugatti79
Hi all, I wish to verify

$\displaystyle u_t=ku_x_x$ with the shorthand notations being $\displaystyle u_t=\frac{\partial u}{\partial t}$ and $\displaystyle u_x_x=\frac{\partial^2 u}{\partial x^2}$ for the following expression

$\displaystyle u(x,t)=\frac{1}{2\sqrt {\pi kt}} \int^{\infty}_{-\infty} e^{-(x-\xi)^2/4kt} f(\xi)d\xi$ where k is a constant and f is continous on the real line ( I actually dont understrand what that 'f' bit means)

Can anyone give me a clue as how to tackle this one? Do I integrate the integrand first and then find the derivatives $\displaystyle u_t=ku_x_x$? If so, how does one tackle the integrand with the infinity signs etc?

Thanks, tips will be appreciated.
Looking forward to hearing from some one.
Bugatti79
BTW - $\displaystyle f(x)$ is the intial condition so $\displaystyle u(x,0) = f(x).$

11. ## Tricky Partial Differential problem

I dont know where I am gone wrong. $\displaystyle u(x,t)= \int^{\infty}_{-\infty} \frac{e^{-(x-\xi)^2/4kt}}{2\sqrt {\pi kt}} f(\xi)d\xi$

Using the chain rule I calculate for $\displaystyle u_x, u_x_x$
$\displaystyle u_x=\frac{\frac{-2(x-\xi)e^{-(x-\xi)^2/4kt}}{4kt}}{2\sqrt {\pi k t}}$ and

$\displaystyle u_x_x=\frac{\frac{4(x-\xi)e^{-(x-\xi)^2/4kt}}{4kt}}{2\sqrt {\pi k t}}$

and the product rule for $\displaystyle u_t$ since t is on both numerator and denominator
$\displaystyle \frac {2\sqrt {\pi k t} (\frac{(x-\xi)^2}{4kt^2}e^{-(x-\xi)^2/4kt}) - (e^{-(x-\xi)^2/4kt} {(\pi k )}{\pi k t}^{-1/2})}{4\pi kt}$

Can anyone confirm im going correct so far?

12. I think you forgot some parentheses in $\displaystyle u_t$
I didn't check $\displaystyle u_{xx}$, but it looks like it simplified nicely for you.

13. Originally Posted by bugatti79
I dont know where I am gone wrong. $\displaystyle u(x,t)= \int^{\infty}_{-\infty} \frac{e^{-(x-\xi)^2/4kt}}{2\sqrt {\pi kt}} f(\xi)d\xi$

Using the chain rule I calculate for $\displaystyle u_x, u_x_x$
$\displaystyle u_x=\frac{\frac{-2(x-\xi)e^{-(x-\xi)^2/4kt}}{4kt}}{2\sqrt {\pi k t}}$ and

$\displaystyle u_x_x=\frac{\frac{4(x-\xi)e^{-(x-\xi)^2/4kt}}{4kt}}{2\sqrt {\pi k t}}$

and the product rule for $\displaystyle u_t$ since t is on both numerator and denominator
$\displaystyle \frac {2\sqrt {\pi k t} (\frac{(x-\xi)^2}{4kt^2}e^{-(x-\xi)^2/4kt}) - (e^{-(x-\xi)^2/4kt} {(\pi k )}{\pi k t}^{-1/2})}{4\pi kt}$

Can anyone confirm im going correct so far?
Check your $\displaystyle u_{xx}$ again (product rule).

14. ## Tricky Partial Differential problem

Still cant get $\displaystyle u_t=ku_x_x$

I dont know wheter im getting my algebra in a twist or my derivatives are still wrong.
$\displaystyle u(x,t)= \int^{\infty}_{-\infty} \frac{e^{-(x-\xi)^2/4kt}}{2\sqrt {\pi kt}} f(\xi)d\xi$

I recalculate
$\displaystyle u_x=\frac{\frac{-2(x-\xi)e^{-(x-\xi)^2/4kt}}{4kt}}{2\sqrt {\pi k t}}$
and for $\displaystyle u_x_x$ I use product rule in the numerator

$\displaystyle u_x_x=\frac{\frac{(-2(x-\xi))}{4kt}\frac{(-2(x-\xi))}{4kt}(e^\frac{-(x-\xi)^2}{4kt})+e^\frac{-(x-\xi)^2}{4kt}(\frac{-2}{4kt})}{2(\pi kt)^\frac{1}{2}}$
and qoutient rule for $\displaystyle u_t$
$\displaystyle \frac {2\sqrt {\pi k t} (\frac{(x-\xi)^2}{4kt^2}e^{-(x-\xi)^2/4kt}) - (e^{-(x-\xi)^2/4kt} {(\pi k )}({\pi k t})^{-1/2})}{4\pi kt}$

If these derivatives can be confirmed correct then I know the rest is basic algebra....
Thanks
bugatti79

15. I can - I got the same.

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