Heat (diffusion) equation in a 3D sphere

[PCS_Exponent]

This one's giving me a headache.

du/dt = K*laplacian(u) inside a 3D sphere, r<a. Boundary conditions u(r=a) = 0. Initial conditions are u(t=0, 0<r<a/2) = 1, u(t=0, a/2<r<a) = 0.

 

Obviously this is spherically symmetric, so u doesn't depend on the angles (phi and theta). Then by separation of varibales, u(r,t) = R®*T(t), R is a Bessel function, R = J(1/2, sqrt(lambda)*r). Possible eigen values for lambda are the zeros of J when r = a. T is a simple exponential. Total solution is given with a superposition for all possible lambdas, from 1 to infinity.

 

But how do I know find the correct coefficients for each term in the superposition series? Surely I cannot easily derive them from the initial conditions, u(t=0, 0<r<a/2) = 1, u(t=0, a/2<r<a) = 0 ???? This is, after all, a Bessel function, not a fundamental trigonometric function. One way or the other, I'm stuck. Any help would be greatly appreciated.

 

PS I suspect I should not have used separation of variables here, and the solution is actually given with the use of conviolution and the correct kernel, like diffusion on one infinite line. I could obviously be wrong, though.

 

PPS Feel free to move this to homework forum - I'm never sure whether I'm posting stuff to the most appropriate forum when I post here, since usually more than one forum applies.

 

Thanks in advance!!!

[Jay-qu]

Moved to homework :D

 

Sorry I cant really help with this - I only just started my differential equations course, and havent seen anything that hairy yet ;)

[Erasmus00]

Your initial conditions are best thought of as many sources. You should solve for (or look up) the spherically symmetric Green's function (probably called the heat kernel) for the heat operator, and then integrate it against your initial conditions. If you need more, let me know.

-Will

 

Edit:if you're interested, I could type up a 1-d analogy and you could try to do the same in 3d?

 

Further edit: Your bessel expansion is certainly on the right track. Think of doing something like a fourier expansion, but with the bessel modes.

[PCS_Exponent]

Thanks for your replies, guys. I have solved it, at final last. For anyone interested, the solution for R was NOT R = J(1/2, s) as I had erroneously written (s = sqrt(lambda)*r). It is R = s^(-1/2)*J(1/2, s), which, since J(1/2, x) is constant times sin(x)*x^(-1/2), simply a constant times sin(s)/s. Then the coefficients are simply given by a regular sine Fourier series, seeing as series(An*sin(s)/s) = 1 at r 0<r<a/2, or series(An*sin(s)) = s.

 

Erasmus00, you are absolutely correct that even if it had been a (integer index) Bessel function (but not Bessel multiplied with a polynomial!!!!!) an expansion would have worked. I had totally forgotten about that, but Bessel functions form a complete basis of functions. The correct weight function of Bessel(n, r) is r. Just to illustrate: if y® = 1 on r 0<r<a, then a fully convergent expansion would be series(Bn*Bessel(0, r*beta(n))), and Bn coefficients given with int(r*Bessel(0, r*beta(n)))/int(r*(Bessel(0,r*beta(n)))^2). beta(n) is the n-th zero of Bessel(0,x). Please take this with a grain of salt, though- I can't remember whether it's always the zeroth Bessel function (Bessel(0, r.....)) or whether it's rather Bessel(n, r*beta(n)). I'll check this and edit as necessary.

 

 

Jay-qu - don't fear the mighty PDEs. I feel this is, on the whole, easier than dealing with ordinary DE, where developing solutions is the main problem. I think most "partial" stuff is that complicated that the student is hardly required to do more than follow given solutions and 'recipes' for solutions both in the final exam and in home work assignments. Of course, if you're thinking of working or doing research in the field of PDEs, rather than just taking a course about them, then scrap what I just said....