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January 3

Generalized Heat Equation

Studying for a PDE exam, I came across the following practice problem

$T_{t}=(k(x)T_{x})_{x}+F(x),\,0<x<L,\,t>0$

where F(x)>0, k(x)>0, T(x,0)=f(x) with T(0,t)=T(L,t)=0. My question is how to even get started on this? Can someone please at least point me in the correct direction? The domain in x is finite so I am thinking separation of variables or the method of eigenfunction expansion and I have no idea what k(x) is. In fact, part of the problem asks for the conditions on F(x),f(x), and k(x) that are needed. I can't find it in any of my books either. Thanks! 75.171.178.10 (talk) 08:34, 3 January 2010 (UTC)[reply]

Start by studying a simple special case such as L=1, k(x)=1, F(x)=0. Bo Jacoby (talk) 13:30, 3 January 2010 (UTC).[reply]

But what's exactly your question? Yours is a second order parabolic equation. There are several approach to it (what PDE course was yours? Classical PDE's? Distributions? Semigroups? I think you are not supposed to rewrite the whole theory by yourself. If you are not given a precise problem, I suggest to go and look any textbook on PDE (a tame starting point is e.g. Evans, Partial Differential Equations, Chapt.7). --pm a (talk) 16:59, 3 January 2010 (UTC)[reply]

The question is "For the boundary conditions given, find the solution T(x,t) in terms of suitable functions and stipulate any conditions on F(x), k(x), and f(x) that are needed". Specific cases I can do. For the past couple of weeks I have doing a plethora of such problems with all their variations on the forcing terms and the damping terms and the boundary conditions in different coordinate systems. And I know that sometimes even if a function (like the forcing term in $u_{t}=u_{xx}+F(x,t)$ with u(0,t)=u(L,t)=0) is not explicitly given you can still write down the solution (in this case using the method of eigenfunction expansion) in terms of F. But in the problem I mentioned above, k(x) is inside the partial derivative. So I have two unknown functions and even if I expand the partial, it looks just as formidable. So what method would I use on this to solve this PDE? This is just a classical PDEs course. Thanks!75.171.178.10 (talk) 00:20, 4 January 2010 (UTC)[reply]

"So I have two unknown functions"? No! T(x,t) is one unknown function. The functions F(x), f(x), and k(x) are supposed to be known, and so is the derivative k'(x). What is the PDE for G(x,t)=k(x)T(x,t)? Bo Jacoby (talk) 14:38, 4 January 2010 (UTC).[reply]

Diffusion Equation

Hi. I need some help with a homework question but I'll show you what I've done so far.

"The function $\theta (x,t)$ obeys the diffusion equation ${\frac {\partial ^{2}\theta }{\partial x^{2}}}={\frac {\partial \theta }{\partial t}}$ . Find, by substitution, solutions of the form $\theta (x,t)=f(t)\exp {\frac {-(x+a)^{2}}{4(t+b)}}$ , where a and b are arbitrary constants and the function f is to be determined. Hence find a solution that satisfies the initial condition $\theta (x,0)=exp(-(x+2)^{2})-exp(-(x-2)^{2})$ and sketch its behaviour for t≥0."

So, after some tedious algebra, I get $f(t)=k(t+b)^{-0.5}$ for some constant k. But I can't see how to get the initial condition part for the entire function - it seems to me that you would need to consider two separate cases, when a=2 and a=-2, to get it but surely you want to solve this for one particular a value not two. Help anyone? Thanks 92.0.129.48 (talk) 17:18, 3 January 2010 (UTC)[reply]

Hint: the heat equation u_t=u_xx being linear, if you are able to solve the initial value problems

v(x,0)=\exp(-(x+2)^{2})

and

w(x,0)=\exp(-(x-2)^{2}),

then you solve the given one with the function

\theta (x,t):=v(x,t)-w(x,t).

Remark: on the first part (that you have already done) you could have considered just the case

a=b=0

and found

\scriptstyle f(t)=k/{\sqrt {t}}.

Then it is clear that if

u(x,t)

solves the heat equation so does

u(x+a,\,t+b).

This remark is useful for the previous hint too: you just have to solve the heat equation with initial condition

u(x,0)=\exp(-x^{2}),

after all. Further remark: yours is the initial value problem for the homogeneous heat equation, whose solution admits an integral representation. You do not need it here of course, but you may notice that the general integral formula is based on the same idea of your exercise: finding solutions as (limits of) linear combinations of translations of the "fundamental solution" (that is, convolutions). --pm a (talk) 18:29, 3 January 2010 (UTC) Was I clear? You should have found[reply]

\scriptscriptstyle \theta (x,t)={\frac {1}{\sqrt {4t+1}}}\left(\exp \left({\frac {-(x+2)^{2}}{4t+1}}\right)-\exp \left({\frac {-(x-2)^{2}}{4t+1}}\right)\right)

--pm a (talk) 21:32, 3 January 2010 (UTC)[reply]

Yes, thank you, I got that. Your comments were very helpful. 92.0.129.48 (talk) 23:13, 3 January 2010 (UTC)[reply]

Formal sums for quotients of primorials

What are the formal sums, with lower limit m and upper limit n, for (P(j)-1)#/(P(j))# and for (P(j+1)-1)#/(P(j))#, where P(j) means the j'th prime?

For (P(j-1)-1)#/(P(j))# I obtained the sum as -(P(j-1)-1)#/(P(j-1))#, but those other two have me stumped.

(Assuming that this answer for (P(j-1)-1)#/(P(j))# is correct, then, e.g., instead of (P(j+1)-1)#/(P(j))#, the answer for P(j)*(P(j)-1)#/(P(j))#, or (P(j)-1)#/(P(j-1))#, should be just as useful.)

I tried summation by parts and Abel's transformation, to no avail (but then again maybe I didn't apply them correctly).

FWIW: Last week I traveled a few dozen miles to the library of a university that awards graduate degrees in mathematics, and looked for books on primorials. CWOT. Yes, they have books on number theory and primes and finite differences and summation, but nothing on primorials.ImJustAsking (talk) 22:43, 3 January 2010 (UTC)[reply]