The Diffusion Equation in 2d rectangular coordinates is:
dc/dt = D(d^2c/dx^2 + d^2c/dy^2), where c is the concentration, and D is the Diffusion Constant.
A fundamental solution of this 2d Diffusion Equation in rectangular coordinates is DiracDelta[x - xo]DiracDelta[y - yo], which can be further expanded as an explicit function of space and time as
1/(4piDt) e^(( - (x - xo)^2 - (y - yo)^2))/4Dt).
The Diffusion Equation in 2d Polar Coordinates is
dc/dt = D(d^2c/dr^2 + (1/r)dc/dr + (1/r^2)d^2c/dtheta^2).
What is the fundamental solution to this 2d Polar Diffusion Equation written in terms of Delta Functions, and also as an explicit function of the 2d polar coordinates r and theta?
Your assistance with this question is greatly appreciated.
May the following paper is useful
A compact ADI scheme for the two dimensional time fractional diffusion-wave equation in polar coordinates
Vong, Seakweng, Wang, Zhibo
https://booksc.xyz/book/38671160/db9dcb
Your question is a little bit confusing. The Delta function is NOT the fundamental solution. It is probably your initial value at t=0. I can only guess that you are interested in the solution of
dc/dt - D \Delta c = 0 in R^2
c(t=0) = DiracDelta(x-x0)
The solution of this problem is the fundamental solution, see also https://en.wikipedia.org/wiki/Heat_equation#Homogeneous_heat_equation.
As the representations of your equation in Cartesian and polar coordinates are equivalent, so should be the solution. In Cartesian coordinates x=(x,y) (slight abuse of notation) the fundamental solution is
U(t,x) = 1/(4 pi D t) * exp(-|x-x0|^2 / (4 D t))
Due to |x-x0|=r, the fundamental solution in polar coordinates centered in the point x0 where the Dirac acts reads
U(t,r,phi) = 1/(4 pi D t) * exp(-r^2 / (4 D t)).
I hope this explanation helps.
Best Regards
You have to see these four books, if exists it must be there:
1- Carslaw HS, Jaeger JC (1959) Conduction of heat in solids, 2nd edn. Clarendon Press, Oxford
2- Cannon JR (2008) The one-dimensional heat equation, encyclopedia of mathematics and its applications. Cambridge University Press, Cambridge.
3- Crank J (1979) The mathematics of diffusion, 2nd edn. Oxford University Press, Oxford.
4- Polyanin AD, Zaitsev VF (2003) Handbook of exact solutions for ordinary differential equations. Chapman & Hall/CRC, London/Boca Raton, Florida.
If you do nto find it there you can be %100 sure it does not exist!
This is an old problem of more than 200 years age. The problem is the so-called Laplace operator, which should be avoided by using the older Euler operator. The Euler operator can be formulated in cartesian, cylindrical or sphere coordinates and will always yield identical solutions - independently to the coordinate system. The Laplace operator, which you use, cannot manage this.
Especially some universities forbid to think on alternatives to the hypotheses of Laplace - he seems to have avoided to check his results himself.
If there are questions or comments, please feel free to contact me.
With kind regards
Norbert Südland
You might want to check
https://www.amazon.com/Distributions-Differential-Equations-Harmonic-Universitext/dp/3030032957/ref=sr_1_1?keywords=Dorina+Mitrea&qid=1564099918&s=gateway&sr=8-1
Chapter 8 if I remember correctly.
A fundamental solution to the 2d Diffusion Equation in rectangular coordinates is
1/(4(pi)Dt)exp[( - (x-xo)^2 - (y-yo)^2)/(4Dt)].
Now convert this solution to 2d polar coordinates (r, ro, theta, thetao):
1/(4(pi)Dt)exp[( - (rCos(theta) - roCos(thetao))^2 - (rSin(theta) - roSin(thetao))^2)/(4Dt)]
= 1/(4(pi)Dt)exp[ - (r^2 + ro^2 - 2rroCos(theta-thetao))/(4Dt)].
This is the fundamental solution to the Diffusion Equation in 2d polar coordinates.
A fundamental ring solution of the 2d Diffusion Equation which is centered at the origin can be found by integrating the fundamental solution shown above over thetao from 0 to 2pi, and dividing by 2pi for normalization:
1/(2pi)Integral from 0 to 2pi[dthetao(1/(4(pi)Dt)exp[ - (r^2 + ro^2 - 2rroCos(theta-thetao))/(4Dt)]
= 1/(4(pi)Dt)exp[ - (r^2 + ro^2)/(4Dt)]BesselI[0, rro/(2Dt)],
where BesselI[0, x] is the modified Bessel function of the first kind of order 0.
The fundamental solution and the fundamental ring solution that are shown above are confirmed in Carslaw and Jaeger, Conduction of Heat in Solids, 2nd edition, page 260, Item V.
To my knowledge, this is a formal solution only, because the 2d Laplace operator already is wrong.
If you consider a fractional diffusion wave equation, then you will get by your way solutions with partial negative density, i.e. a theoretical state, where you must put in something to get an absolute vacuum.
If you are interested in the older Euler operator to avoid such nonsense, please give a notice to me, thus I can tell you the details.
The big fallacies of Mathematical Physics are not solved by coordinate transform only.
God bless you and your family.
With kind regards,
Norbert Südland
This might be of interest.
https://www.ndsu.edu/pubweb/~novozhil/Teaching/483%20Data/24.pdf
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