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I need to calculate solution of Bessel-like equation having general form: $\frac{d^2F}{dr^2}+\frac{1}{r}\frac{dF}{dr}+Q(r)F(r)=0$. Problems come from the points near $r=0$ leading to numeric errors.

For example, simple Bessel equation can be solved using DSolve:

DSolve[{y''[x] + 1/x y'[x] + y[x] == 0, y[0] == 1, y'[0] == 0}, y[x], x]

with output:

{{y[x] -> BesselJ[0, x]}}

However same equation using NDSolve produces errors:

sol = NDSolve[{y''[x] + 1/x y'[x] + y[x] == 0, y[0] == 1, y'[0] == 0}, y[x], {x, 0, 10}]

Power::infy: Infinite expression 1/0. encountered. >>
Infinity::indet: Indeterminate expression 0. ComplexInfinity encountered. >>
NDSolve::ndnum: Encountered non-numerical value for a derivative at x == 0.`. >>

The question is: how to overcome erros in NDSolve routine and get numeric solution?

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Is this the whole story? I get errors when I run your DSolve, though I also get the answer you show. –  bill s Oct 20 '13 at 17:09
    
You can substitute the 0 making trouble with a small number : sol[x_] = First[y[x] /. NDSolve[{y''[x] + 1/x y'[x] + y[x] == 0, y[$MachineEpsilon] == 1, y'[$MachineEpsilon] == 0}, y[x], {x, $MachineEpsilon, 10}]]. –  b.gatessucks Oct 20 '13 at 17:18
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1 Answer

There is a problem with the initial conditions. At $x=0$ there is no solution. The analytical solution given by Mathematica seems to have used the standard solution for standard Bessel ODE http://mathworld.wolfram.com/BesselDifferentialEquation.html, which is a Bessel function. But I am not sure for $n=0$ one can just use Bessel(0,x), I think BesselY(0,x) is also needed. This is what I get.

But to answer you, I think you need to use series solution for this. Or avoid $x=0$ as was suggested in the comment for a numerical solution.

\begin{align*} y^{\prime\prime}\left( x\right) +\frac{1}{x}y^{\prime}\left( x\right) +y\left( x\right) & =0\\ y\left( 0\right) & =1\\ y^{\prime}\left( 0\right) & =0 \end{align*}

Let $x=e^{z}$ or $\ln\left( x\right) =z$, hence $\frac{dy}{dx}=\frac{dy} {dz}\frac{dz}{dx}=\frac{dy}{dz}\frac{1}{x}$ and $\frac{d^{2}y}{dx^{2}} =\frac{d^{2}y}{dz^{2}}\frac{dz}{dx}\frac{1}{x}+\frac{dy}{dz}\left( \frac {-1}{x^{2}}\right) =\frac{d^{2}y}{dz^{2}}\frac{1}{x^{2}}-\frac{1}{x^{2}} \frac{dy}{dz}$

Substituting all these back into the original ODE gives

\begin{align*} \frac{d^{2}y}{dz^{2}}\frac{1}{x^{2}}-\frac{1}{x^{2}}\frac{dy}{dz}+\frac{1} {x}\frac{dy}{dz}\frac{1}{x}+y\left( z\right) & =0\\ \frac{d^{2}y}{dz^{2}}+x^{2}y\left( z\right) & =0\\ \frac{d^{2}y}{dz^{2}}+e^{2z}y\left( z\right) & =0 \end{align*}

The solution to the above is

$$ y\left( z\right) = BesselJ_{0}\left( \sqrt{e^{2z}}\right) c_{1}+2 BesselY_{0}\left( \sqrt{e^{2z}}\right) c_{2} $$

Replacing back

\begin{equation} y\left( x\right) = BesselJ_{0}\left( x\right) c_{1} +2 BesselY_{0}\left( x\right) c_{2}\tag{1} \end{equation}

You see that there is a BesselY[0,x] function in the solution which do not show up the solution given by Mathematica for some reason.

Now to find the constants $c_{1},c_{2}$ we use initial conditions. At $x=0,y=1\,\ $,hence

$$ 1=c_{1}-\infty $$

For the derivative at $x=0$,

$$ y^{\prime}\left( x\right) =- BesselJ_{1}\left( x\right) c_{1}-2 BesselY_{1}\left( x\right) c_{2} $$

At $x=0$

$$ 0=0-\infty $$

sol = DSolve[{y''[z] + Exp[2 z] y[z] == 0}, y[z], z]

Mathematica graphics

Btw, Maple 17 gives a numerical solution for this, without the 1/0 issue. But I am not sure how it avoided the point $x=0$ now. It uses RK45 standard numerical method. Here it is:

Mathematica graphics

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Thanks so much! Brilliant idea to make change of variables, it helped a lot. –  denkorw Oct 21 '13 at 12:41
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