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Wave Equation, stuck on a partial calculation 
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#1
Aug3008, 04:21 PM

P: 225

Hey everybody,
My professor started our PDE I class in Chapter six, so I am having a hard time with the really basic stuff to get the theory down. One of my questions to answer is to verify a solution by using direct substitution. [tex]u(x,t) \ = \ \frac{1}{2}\left[\phi(x+t) \ + \ \phi(xt) \right] \ + \ \frac{1}{2} \int^{x+t}_{xt}\Psi(s)ds [/tex] With initial conditions [tex] u(x,t_{0}) = \phi(x) \ , \ \frac{\partial u}{\partial t} (x,t_{0}) = \Psi(x), \ and \ t_{0} = 0 [/tex] satisfies [tex]\frac{\partial^{2}u}{\partial x^{2}}  \frac{\partial^{2}u}{\partial t^{2}} = 0 [/tex] It was easy for me to plug and chug to show that [tex] u(x,t_{0}) = \phi(x) \ and \ \frac{\partial u}{\partial t} (x,t_{0}) = \Psi(x) [/tex] Clearly my next step is to find [tex] \frac{\partial^{2}u}{\partial x^{2}} [/tex] but that's the step on which I'm stuck. Can someone get me started? If someone can show me how to do the second partial w.r.t x, it would be a good exercise for me to figure out the second partial w.r.t t. I apply the chain rule and just get a bunch of garbage back, which means I'm messing it up somewhere. Any input is appreciated. 


#2
Aug3008, 04:37 PM

Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,348

You need "Leibniz' formula":
[tex]\frac{d}{dx}\int_{\alpha(x)}^{\beta(x)} \phi(x,t)dt= \frac{d\beta}{dx}\phi(x,\beta(x)) \frac{d\alpha}{dx}\phi(x,\alpha(x))+ \int_{\alpha(x)}^{\beta(x)} \frac{\partial \phi}{\partial x}dt[/tex] It's really just applying the chain rule correctly. 


#3
Aug3108, 02:13 PM

P: 225

[tex]
\frac{d}{dx}\int_{\alpha(x)}^{\beta(x)} \phi(x,t)dt= \frac{d\beta}{dx}\phi(x,\beta(x)) \frac{d\alpha}{dx}\phi(x,\alpha(x))+ \int_{\alpha(x)}^{\beta(x)} \frac{\partial \phi}{\partial x}dt [/tex] I'm having trouble understanding this. It looks like the integrand you gave is a function of 2 variables, but the integrand I have is a function of one variable. I'm not sure what to do with that. Also: [tex] \frac{\partial}{\partial x} (\phi(x+t)) = ? [/tex] is it [tex] = \phi '(x + t), \ or \ \phi_{x}(x+t) \ ? [/tex] In the notation of a function, how do I write that? I guess I'm having a notational brain fart. 


#4
Aug3108, 03:04 PM

P: 225

Wave Equation, stuck on a partial calculation
I found this worked out in Walter Strauss's book, so I guess I don't need help anymore. Thanks for looking :)



#5
Sep108, 03:29 AM

Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,348

[tex]\frac{\partial\phi}{\partial x}= 0[/tex] and the formula becomes [tex] \frac{d}{dx}\int_{\alpha(x)}^{\beta(x)} \phi(x,t)dt= \frac{d\beta}{dx}\phi(x,\beta(x)) \frac{d\alpha}{dx}\phi(x,\alpha(x)) [/tex] [tex]\frac{\partial\phi(x+t)}{\partial x}= \phi'(x+t)(1)= \phi'(x+t)[/tex] 


#6
Sep308, 07:38 AM

P: 225

HallsofIvy, thanks for helping me.
If anyone finds this post and needs help with taking the derivative of an integral, I found this website that helped as well: http://mathmistakes.info/facts/Calcu...n/doi/doi.html 


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