Solve the Partial Differential equation:
$$ \frac{\partial u}{\partial t} = \frac{1}{r}\frac{\partial }{\partial r}\big(r\frac{\partial u}{\partial r}\big) $$
for $r \in [0,1]$ and $t \gt 0$, with the initial condition $u(r,0)=g(r), \text{ and boundary condition } u(1,t)=0.$
My Approach
Rewriting the pde we get:
$$\frac{\partial u}{\partial t} =\frac{\partial^2 u}{\partial r^2}+\frac{1}{r}\frac{\partial u}{\partial r} $$
Let $u(r,t)= R(r)T(t)$, be the solution of the given pde, which reduces the pde:$$rR''+R'-\lambda rR=0, T'-\lambda T=0$$, which can be reduced to Strum-Livouille problem $-(-rR')'=\lambda rR$, my question is how to find the function $R(r)$?
Is my approach right?- Any help is appreciated.
Best Answer
Your approach is correct but far to lead to the result. As already pointed out in the comments, one can express $R(r)$ in terms of Bessel functions : $$R(r)=c_1J_0(\sqrt{-\lambda}\:\:r)+c_2Y_0(\sqrt{-\lambda}\:\:r)\qquad\text{if}\quad \lambda<0$$ $$R(r)=c_1I_0(\sqrt{\lambda}\:\:r)+c_2K_0(\sqrt{\lambda}\:\:r)\qquad\text{if}\quad \lambda>0$$ http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html
http://mathworld.wolfram.com/BesselFunctionoftheSecondKind.html
http://mathworld.wolfram.com/ModifiedBesselFunctionoftheFirstKind.html
http://mathworld.wolfram.com/ModifiedBesselFunctionoftheSecondKind.html
Then you get some particular solutions of the PDE, but not a particular solution fitting the condition $u(r,0)=g(r)$ where $g(r)$ is a given function which is not necessarily a Bessel function of the above kind.
A more general solution of the PDE can be (for exqample in case $\lambda<0$) on the form : $$u(x,t)=\sum_{\lambda}\left(f_1(\lambda)I_0(\sqrt{\lambda}\:\:r)+f_2(\lambda)K_0(\sqrt{\lambda}\:\:r)\right)e^{\lambda t}$$ With condition $g(r)=\sum_{\lambda}\left(f_1(\lambda)I_0(\sqrt{\lambda}\:\:r)+f_2(\lambda)K_0(\sqrt{\lambda}\:\:r)\right)$
$f_1(\lambda)$ and $f_2(\lambda)$ have to be determined in order t0 fit $g(r)$.
The difficulty is to expend the given fonction $g(r)$ into series of Bessel functions. This is a problem similar to the expansion into Fourier series of sinusoidal functions, but more arduous. One can find relevant papers on the web with key words "Neumann Bessel series" or "Fourier Bessel series". For example : http://mathworld.wolfram.com/Fourier-BesselSeries.html . Good luck!