Use "Leibniz' rule":
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Hi all, I wish to verify
with the shorthand notations being and for the following expression
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 ? 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
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
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 ). A function must be continous to be differentiable....
bugatti79
So you just want to verify that ?
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.
For I think an argument using the dominated convergence theorem will give you the interchanging of integral and partial derivative. For just note that if is fixed your integral is just a convolution of functions, and so you can derive only the exponential inside the integral.
Still cant get
I dont know wheter im getting my algebra in a twist or my derivatives are still wrong.
I recalculate
and for I use product rule in the numerator
and qoutient rule for
If these derivatives can be confirmed correct then I know the rest is basic algebra....
Thanks
bugatti79