How to integrate an ordinary differential equation around a singular point?

[wdlang]

i have a second order ordinary differential equation of f(x):

f''+(E-A(A+1)/x^2)f=0, where A is a positive integer, E is a real constant

the domain is [0, \infty).

the boundary condition is f(x=0)=0

since this is a linear equation, i only need to determine f up to a overall constant

how to do this numerically from the origin and outward?

we can prove that in the neighborhood of the origin, f is on the order of x^(A+1)

 

[wdlang]

a naive idea is to use the boundary condition mentioned above and the 4th-order Runge-Kutta method to do it

my concern is that in near the origin, the A(A+1)/x^2 term is too large

and therefore i have no idea whether the RK method will work, or if it does, how to control the error by shrinking the step length.

 

[kosovtsov]

If I do not mistaken, the solution

$$f(x) = C\sqrt{x}J_{A+1/2}(\sqrt{E}x)$$

where C is an arbitrary constant fit your ODE and the boundary condition f(x=0)=0.

 

[wdlang]

If I do not mistaken, the solution

$$f(x) = C\sqrt{x}J_{A+1/2}(\sqrt{E}x)$$

where C is an arbitrary constant fit your ODE and the boundary condition f(x=0)=0.

thanks a lot

however, i am not interested in the analytical solution

i want a general numerical method to deal with such singular points

the equation above is just an example, for which, luckily, an analytical solution is available

 

[blue_raver22]

There are a few different approaches to integrating an ordinary differential equation around a singular point, such as the one described in the content above. One common method is to use a power series expansion around the singular point, in this case the origin. This allows us to approximate the solution of the differential equation in the neighborhood of the singular point and then extend it outward.

Another approach is to use a numerical integration method, such as the Runge-Kutta method, to solve the differential equation numerically. This involves breaking the domain into smaller intervals and approximating the solution at each interval. The accuracy of this method depends on the size of the intervals, so it may be necessary to use smaller intervals near the singular point to get a more accurate solution.

Additionally, it is important to consider the boundary conditions when integrating around a singular point. In this case, the boundary condition is f(x=0)=0, which means that the solution must approach 0 as x approaches 0. This can be taken into account in the numerical integration method by adjusting the initial conditions or using a shooting method to find the appropriate solution.

In summary, integrating an ordinary differential equation around a singular point requires careful consideration of the domain, boundary conditions, and choice of numerical method. It may also be helpful to use analytical techniques, such as a power series expansion, to approximate the solution near the singular point.