Oh, if the second one is partial as well then substitute it into the first one.
So assuming k2 is non-zero, you have dC/dx = [k1*k3/k2]*C
The solution to this is the standard exponential as in the ODE case, only your constant is now a function of time.
Then just plug this back into the...
I'm assuming for your PDE you mean both as partial derivatives and for the ODE you mean total derivative?
In which case (writing D for total derivative and d for partial),
the ODE gives you: DC/Dt = dC/dt + [dC/dx][dx/dt]
Further, you know dC/dx in terms of dC/dt from the PDE, and you can...
Exactly, the only difference is the operation i.e. sigma means to sum and U means to take the union. So Sigma over n means summing over all n in the given range, and U over n means taking the union of the sets over the given range.
Well, your Fourier series is periodic. If you look at the sketch posted above you can see that the fourier series replicates the function in the period given and then just repeats itself as a periodic function outside that interval (in the same way the simple trig functions do).
Moreover, if...
I think (and I may be wrong) that if they're using standard convention, then := means "define".
In other words, that statement means "Define A as the union of the sets A(n) over all n"
And I believe that the up arrow indeed means the limit.
The best way to derive these results is to use summation convention. Everything drops out quite nicely.
The vector identity you've stated is also derived in the same way but you'll notice the subtleties of the fact that you're dealing with an operator when doing the derivation.
Here's a simple example:
Suppose you have a function f(t) defined on the range 0<t<a
Then two possible fourier series for f(t) are the fourier sine series and the fourier cosine series.
You can obtain the sine series by defining f(t) to be an odd function i.e. define it on -a<t<a with f(-t) =...
Oh, I'm not doing this from anywhere near a mathematically rigorous point of view (far to many subtleties as you've mentioned), wouldn't know where to begin with constructing anything sensible. I'm simply stating the best you can do with the information you've been given (and even then there...
So I'm intending to teach myself some Particle Physics and Standard Model type stuff, I was wondering if someone who's already covered this could give me some advice.
I did some Group Theory a few years back and looking over content pages of lecture notes I occasionally spot references to...
The weird thing about maths is that once you get to higher level stuff, the first time you see it you have no idea what's going on. Then after a while you get your head around what the definitions actually mean, but still can't seem to answer the questions. After a long time spent of having...
It's been a while since I did this, but as far as I remember you don't necessarily have to derive the operator A, you just need to find one which works.
I think I always used A = 4[(d/dx)^3] - 3 (u[d/dx] + [d/dx]u), or something along those lines
Hmmm... let's see what we can do.
Suppose you have a dot product a.x = A where a and x are vectors and A is some number.
Now, given a and A, what can we say about x?
Well a.x = |a||x|cos(theta), where theta is the angle between a and x.
So |a||x|cos(theta) = A
For a non-zero, we can say...
I think what you're referring to as nabla is what I call grad.
Just do the vector taylor expansion as I mentioned, this was the box standard thing to do back in electrodynamics exams. Oh, and use summation convention to make life easier.
Use the vector form of Taylor Expansion i.e.
f(x+h) = f(x) + (h.grad)f(x) + [(h.grad)^2]f(x) + ...
where x and h are vectors, grad is the usual gradient operator and "." indicates the dot product.
Well, first things first, find your green's function for the operator given.
In other words, solve -G''(z;z') + [a^2]G(z;z') = diracdelta(z - z')
Then simply integrate the product of G(z;z') with f(z') w.r.t z' and you're done :)
If you know a bit about quantum then this should be nice way of explaining things:
Given a wavefuntion W(x,y) of two identical particles in states x and y. we say that the particles are bosons if the wavefunction is symmetric under particle exchange i.e. W(x,y) = W(y,x), and they are fermions if...
For the first one, I'd say suppose that y1(x) and y2(x) are both solutions to the IVP. Then consider y(x) = y1(x) - y2(x) and show that y=0 in which case you have uniqueness.
The main idea here is the concept of "poles". This is basically when your function ends up dividing by zero. At these points the function isn't defined and hence your taylor series diverges.
As an example, consider 1/(1+x).
This has a pole at x=-1, which is where the function diverges but for...
Ah, right, so you actually want to be using difference equations, not differential equations!
So, ODE's are when you have a continuous model i.e. at every infinitesimal point in time there is synthesis and death, but here you clearly have a discrete problem.
In other words, you want to solve...
You'll have to explain your reasoning behind that, I don't quite see why D(t) = r*C(t-1)
Unless you're dealing with finite time intervals, but if that were the case you'd be solving difference equations, not differential equations
This is because the D that we've found is the total population that have died.
For example, we'd expect that over the history of the Earth, more people have died in total than are currently living now.
As such if you want to find the population that died in the interval [T, T+K],
then...
Right, so basically, here are the general ideas:
ODE's are when you have a differential equation where the function you're dealing with only depends on one variable e.g. y(x).
The other types of differential equations are PDE's where you deal with functions of more than one variable e.g...
I'd say before I start explaining things, it would help if you tell me what you already know about differential equations and how to solve them. It seems to me like you might want to review a lot of the material as there are some fundamental concepts you seem to have missed out
Firstly, I'd have expected the equation to be dC/dt = (S-r)*C
because otherwise if you have r=0, i.e. no people dying,
then your equation says dC/dt = S which means C = Co + St
which is counter-intuitive as you would expect exponential growth
so you might wanna just double check that!
Anyways...
I'm pretty sure you just appeal to usual Sturm Liouville Orthogonality, just work out the weight function which I think is e^(-x) but haven't looked at in a while.
Have we ignored the x^2 and the x?
Clearly y = x or y = 1/x are solutions to (x^2)*y'' + x*y' - y = 0
Sorry, you'll have to explain carefully where it is that you're getting confused...