partial derivitives hard problem

[oohaysomeone]

i need to take the following partial derivatives: u[sub:38lcx38b]t[/sub:38lcx38b] and u[sub:38lcx38b]xx[/sub:38lcx38b] for the following function:

u(x,t) = erf(x/(2t^(1/2))) when the error function is defined by the first function in this article: http://en.wikipedia.org/wiki/Error_function

i know what im trying to do here is take the partial derivitives, but i dont know how to carry out this problem. the answer book says u[sub:38lcx38b]t[/sub:38lcx38b] and u[sub:38lcx38b]xx[/sub:38lcx38b] are equal.

 

[tkhunny]

Where are you stumbling? It seems to me that you could be confused by the usual defintion of erf(x), since it includes an integral that uses the variable "t". If you confuse this dummy variable with the "t" in your u(x,t), you WILL be confused. Switch the dummy variable in the definion of erf(x) to 's' and see if is any more clear. Do NOT switch the 't' argument in u(x,t).

\(\displaystyle erf(x) = \frac{2}{\sqrt{\pi}} \cdot \int_{0}^{x} e^{-s^{2}}\;ds\)

\(\displaystyle u(x,t) = erf\left(\frac{x}{2\sqrt{t}}\right) = \frac{2}{\sqrt{\pi}} \cdot \int_{0}^{\left(\frac{x}{2\sqrt{t}}\right)} e^{-s^{2}}\;ds\)

 

[oohaysomeone]

i think where i keep making mistakes is taking the partial derivitives of u(x,t). i dont know how to do this step.

 

[tkhunny]

That seems reasonable, since that is the ONLY step there is.

She us this derivative: \(\displaystyle \frac{d}{dx}\int_{0}^{x}\;f(t)\;dt\)

What say you?

 

[oohaysomeone]

got it now. thanks a bunch.