Wikipedia:Reference desk/Archives/Mathematics/2009 February 3

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February 3

Solving the diffusion equation

The temperature θ(x, t) in a very long rod is governed by the one-dimensional diffusion equation

${\frac {\partial \theta }{\partial t}}=D{\frac {\partial ^{2}\theta }{\partial x^{2}}}$

where D is constant. At time t = 0, the point x = 0 is heated to a high temperature. At all later times, conservation of energy implies

$\int _{-\infty }^{\infty }\theta (x,t)\,dx=Q$ where Q is some constant.

Having shown by dimensional analysis θ(x, t) can be written in the form

$\theta (x,t)={\frac {Q}{(Dt)^{\frac {1}{2}}}}F(z)$ where $z={\frac {x}{(Dt)^{\frac {1}{2}}}}$,

I need to show that ${\frac {d^{2}F}{dz^{2}}}+{\frac {z}{2}}{\frac {dF}{dz}}+{\frac {F}{2}}=0$ and integrate it to obtain a first order D.E. I haven't tried the second integrating part yet because I'm stubborn and don't want to give up on the first bit, but how on earth do I show that the big F/z mess sums to 0 without calculating god knows how many derivatives?

Any help would be -greatly- appreciated,

Mathmos6 —Preceding unsigned comment added by 131.111.8.98 (talk) 06:09, 3 February 2009 (UTC)

It's not that bad. Start with

${\frac {\partial \theta }{\partial x}}={\frac {Q}{(Dt)^{\frac {1}{2}}}}{\frac {dF}{dz}}{\frac {\partial z}{\partial x}}$

$\Rightarrow {\frac {\partial ^{2}\theta }{\partial x^{2}}}={\frac {Q}{(Dt)^{\frac {1}{2}}}}\left({\frac {d^{2}F}{dz^{2}}}\left({\frac {\partial z}{\partial x}}\right)^{2}+{\frac {dF}{dz}}{\frac {\partial ^{2}z}{\partial x^{2}}}\right)$

which becomes simpler once you notice that

${\frac {\partial z}{\partial x}}={\frac {1}{(Dt)^{\frac {1}{2}}}}{\text{ and }}{\frac {\partial ^{2}z}{\partial x^{2}}}=0$

$\Rightarrow {\frac {\partial ^{2}\theta }{\partial x^{2}}}={\frac {Q}{(Dt)^{\frac {3}{2}}}}{\frac {d^{2}F}{dz^{2}}}$

Then we have

${\frac {\partial \theta }{\partial t}}={\frac {-Q}{2t(Dt)^{\frac {1}{2}}}}F+{\frac {Q}{(Dt)^{\frac {1}{2}}}}{\frac {\partial F}{\partial t}}$

$={\frac {-Q}{2t(Dt)^{\frac {1}{2}}}}F+{\frac {Q}{(Dt)^{\frac {1}{2}}}}{\frac {dF}{dz}}{\frac {\partial z}{\partial t}}$

Now you just need to work out ${\frac {\partial z}{\partial t}}$ (I'll leave that bit to you) and the rest is just algebra. Gandalf61 (talk) 11:05, 3 February 2009 (UTC)

Get rid of the D by using Dt as a new independent variable. Also set Q=1 in order to simplify the formulas. Bo Jacoby (talk) 14:17, 3 February 2009 (UTC).

Anti-derivatives vs Riemann sums

I was cleaning up my room and came across my old course notes and calculus textbook (James Stewart). Reading up the section on integrals and finding areas, I thought up the following problems:

1. Is there any example of a function for which one can compute the integral as the limit of the Riemann sum in closed form, but for which it is difficult (or impossible) to find an anti-derivative as an elementary function, except at a finite/countable set of discontinuities? (Note: italicised text added on 13:04, 3 February 2009 (UTC))
2. How does one prove that a given function does not have an anti-derivative in terms of the elementary functions? Just some links to relevant articles or a proof outline would be nice. Thanks. Zunaid 08:13, 3 February 2009 (UTC)

For 2: According to our Risch algorithm article, Risch's approach can be used to prove that $e^{-x^{2}}$ does not have an elementary antiderivative. I haven't checked it. -- Jao (talk) 08:51, 3 February 2009 (UTC)

See differential Galois theory. — Emil J. 11:54, 3 February 2009 (UTC)

Addressing part 1 of your question, the fundamental theorem of calculus guarantees that if f is Riemann integrable with indefinite integral F then f = dF/dx - in other words, F is an anti-derivative of f. So if f is Riemann integrable then we always know an anti-derivative exists - but it is not necessarily expressible in terms of functions that we already know and love. But mathematicians are always happy to expand their circle of acquaintances, so since we know that the anti-derivative exists, we can just use the Riemann integral as a definition of this new function - this is how functions such as the error function, the logarithmic integral function and the Fresnel integrals are defined. Gandalf61 (talk) 12:23, 3 February 2009 (UTC)

What I should have said was "impossible to find an anti-derivative in terms of elementary functions. Zunaid 13:04, 3 February 2009 (UTC)

The fundamental theorem of calculus only holds for continuous functions. For example, the indicator function of the interval [0,1] is Riemann integrable, but it does not have an antiderivative (as derivatives are Darboux). — Emil J. 12:37, 3 February 2009 (UTC)

Ah dammit. I think my question 1 is poorly phrased, and in fact is impossible. The correct phrasing is as corrected above, hopefully it is more water-tight. However if it were the case that there is a closed form for the Riemann sum, then integrating f from a to x would automatically give an anti-derivative in the form of F(x), obtained as the closed form expression of the summation. For example, I thought of exp(-x²) which doesn't have an elementary anti-derivative. I thought perhaps one could compute a closed form Riemann sum for it, or perhaps for any other example. Thanks for the answers. Zunaid 13:04, 3 February 2009 (UTC)

You may transform integrals into sums, if that appears more elementary to you:

$\int _{0}^{x}e^{-t^{2}}dt=\int _{0}^{x}\sum _{i=0}^{\infty }{\frac {(-t^{2})^{i}dt}{i!}}=\sum _{i=0}^{\infty }\int _{0}^{x}{\frac {(-t^{2})^{i}dt}{i!}}=\sum _{i=0}^{\infty }{\frac {(-1)^{i}x^{2i+1}}{i!(2i+1)}}=x-{\frac {x^{3}}{3}}+{\frac {x^{5}}{10}}-{\frac {x^{7}}{42}}+{\frac {x^{9}}{216}}-{\frac {x^{11}}{1320}}+\cdots$

Your questions are treated in Concrete Mathematics. Bo Jacoby (talk) 22:15, 3 February 2009 (UTC).

Surds Question

Well..i thought i knew this subject.. we learnt it last early last year but i have a bad memory and it seems to of all fallen out of my head along with my babelfish for understanding math.

I almost completely dont understand it.. i managed to do one but i was working backwards from the answer..I tried a multitude of different ways..none of them worked.. i hope you guys will be able to help (sorry i dont have much idea on the method.. i really wish i was able to do this..but i need help unfortunately)

without futher delay

Simplify: $-3{\sqrt {28}}+2{\sqrt {180}}-7{\sqrt {63}}-2{\sqrt {245}}$

Try factoring the numbers under the radicals and see if that gives you any ideas. -- Jao (talk) 11:39, 3 February 2009 (UTC)

i tried doing that but i didnt end up with any similar numbers under the square root sign :S.. so i was not able to add them

i know i have to add/subtract them but i cant get the radicals to become similar numbers (this is actually like question no.2.. i was only able to do the first one as i said by working backwards.. im sure once i understand the fundamentals ill be able to do the rest)—Preceding unsigned comment added by 124.180.230.234 (talk) 11:44, 3 February 2009 (UTC)

28=2*2*7, 180=2*2*3*3*5, I see some common factors there (in fact, I see square factors, so you can deal with them pretty easily, you don't even need to worry about combining surds). Remember, you won't necessarily get it all simplified down to one term, just get it as simple as possible. --Tango (talk) 11:53, 3 February 2009 (UTC)

(Explicit answer removed -I did not even realized that the previous answers were maieutics, sorry) pma (talk) 11:57, 3 February 2009 (UTC)

The numbers infront of the radicals dont matter? —Preceding unsigned comment added by 124.180.230.234 (talk) 12:26, 3 February 2009 (UTC)

They only matter that when you find a perfect square inside the radical, you take it's square root, and remove the result from the under the radical, multiplying it by the coefficient in front of the radical. StuRat (talk) 13:06, 3 February 2009 (UTC)

OK, let's start from the basics:

1) Find all the factors for each number under the radical symbol. Do this by trying to divide by 2 until you can't anymore (and still get an integer result). Then try to divide by 3, then 5, then 7. I think that's as high as any of those factors go.

2) If the factors contain two 2's, you can take those out and multiply the coefficient in front of the radical by one of those 2's. Same thing for if it contains two 3's, 5's, or 7's.

3) Multiply any remaining factors back together and leave them under the radical.

4) After you've done all of this, if any of the radicals contain the same values, these may be combined together by adding the coefficients in front of them. Remember that when you add a negative number, that's the same as changing it to positive, then subtracting it.

Show us your work and we will tell you if you did it right. StuRat (talk) 13:06, 3 February 2009 (UTC)