User chad - MathOverflow

Answer by Chad for Differential equation for a ratio of consecutive Bessel functions

2011-12-02T02:24:51Z

I do not know how helpful this would be to you however it was very helpful to understand the physics and numerics of Bessel. if you are studying elastic wave propagation. The solution of the differential equations of potential wave is the cylindrical Bessel:

$ r^2 \frac{d^2 R}{dr^2} + r \frac{dR}{dr} + (r^2 - \alpha^2)R = f(r)$

for an arbitrary real integer number α (the '''order''' of the Bessel function). In solving problems in cylindrical coordinate systems, Bessel functions are of integer order (α = ''n''). Since this is a second-order differential equation, there must be two [[linearly independent]] solutions. My solutions use Bessel J(n,.) and Hankel H(n,.) (as previously mentioned)

The potential is assumed for each media to be:

$\phi=\left(a_{1}J_{n}(K*r)+a_{2}*H_{n}(K*r)\right)*e^{in\theta} ,$

$\psi(t) = \left(a_{3}J_{n}(k*r)+a_{4}*H_{n}(k*r)\right)*e^{in\theta} \, $

However for numerical stability: They (ref. 1) normalize the potential for each layer and for each nth iteration The potential will have Hankel function equal to 1 at the inner radius, while Bessel J will be multiplied by Hankel at outer radius.

$\phi=\left(a_{1}J_{n}(K*r)*H_{n}(K*r_{out})+a_{2}*\frac{H_{n}(K*r)}{H_{n}(K*r_{in})}\right)*e^{in\theta},$

$\psi(t)=\left(a_{3}J_{n}(k*r)*H_{n}(k*r_{out})+a_{4}*\frac{H_{n}(k*r)}{H_{n}(K*r_{in})}\right)*e^{in\theta},$

For more details: http://mathoverflow.net/questions/82303/bessel-functions-in-wave-propagation-and-scattering Reference: David C. Ricks and Henrik Schmidt, "A numerically stable global matrix method for cylindrically layered shells excited by ring forces" 1994

Bessel functions in wave propagation and scattering

2011-11-30T19:01:39Z

Is there a way to scale Bessel J(n,.) (Bessel of first kind) and Bessel H(n,.) (Bessel of third kind or Hankel)? I am having computer problems with higher orders (higher vlaues of n) and small arguments. I am using Matlab. The problem is in elastic wave propagation and scattering. The solution of the differential equations of potential is the cylindrical Bessel:

$ r^2 \frac{d^2 R}{dr^2} + r \frac{dR}{dr} + (r^2 - \alpha^2)R = f(r)$

The '''inhomogeneous Helmholtz equation''' is the equation:

$ \nabla^2 A(r) + k^2 A(r) = -f(r) \mbox { in } \mathbb R^n$

Each set of equations are solved separately in the nth order. I am having problems while solving the set of linear equations when n is high and argument is small.

for example the set of linear equations can be written as:

T11*a1*J(n,x)+T12*a2*J(n,y)+T13*a3*H(n,x)+T14*a4*H(n,y)=b1(x,y)

T21*a1*J(n,x)+T22*a2*J(n,y)+T23*a3*H(n,x)+T24*a4*H(n,y)=b2(x,y)

T31*a1*J(n,x)+T32*a2*J(n,y)+T33*a3*H(n,x)+T34*a4*H(n,y)=b3(x,y)

T41*a1*J(n,x)+T42*a2*J(n,y)+T43*a3*H(n,x)+T44*a4*H(n,y)=b4(x,y)

such that b1, b2, b3 b4 are known boundary conditions, while A=[a1 a2 a3 a4] is the vector of unknowns and T=[Tij] is a matrix of knowns.

If I can scale or normalize J(n,.) and H(n,.)

in small arguments and big n J(n,.) is a very small ~0 while H(n,.) is very big of the order 1e40. The matrix becomes ill conditioned

Thanks Sorry guys I will give an example of the problem. Let us say: - I am solving a scattered wave problem. I have full knowledge of an incident wave (displacements, stresses) that is the boundary, and I know the properties of the different medias. I am trying to find the displacement and stresses in different media. The scattered wave. In other words I am trying to find the amplitudes in different medias.

The potential is assumed for each media to be:

$\phi=\left(a_{1}J_{n}(K*r)+a_{2}*H_{n}(K*r)\right)*e^{in\theta} ,$

$\psi(t) = \left(a_{3}J_{n}(k*r)+a_{4}*H_{n}(k*r)\right)*e^{in\theta} \, $

I am differentiating the potential to get displacements and stresses in terms of the unknowns.

Comment by Chad

2011-12-03T03:49:50Z

@ S. Carnahan, the use of Bessel ratios prove to be very useful. I was sharing my own experience that the ratio could be used to reduce the overflow. At higher order n and small arguments, very small real and large imaginary. This resulted in numerical instability. In other words the ratio can be used to normalize the potentials as: $<math>\phi=\left(a_{1}J_{n}(K*r)*H_{n}(K*r_{out})+a_{2}*\frac{H_{n}(K*r)}{H_{n}(K*r_{in})}\right)*e^{in\theta},</math>$ $<math>\psi(t)=\left(a_{3}J_{n}(k*r)*H_{n}(k*r_{out})+a_{4}*\frac{H_{n}(k*r)}{H_{n}(K*r_{in})}\right)*e^{in\theta},</math>$

Comment by Chad

2011-12-01T16:06:50Z

Here is what I found out from a research paper: -They normalize the potential for each layer and for each nth iteration The potential will have Hankel function equal to 1 at the inner radius, while Bessel J will be multiplied by Hankel at outer radius. @ Antoine your idea is great but as you said what I'm trying to solve is singular by nature. It needed some good understanding of the physics of the problem.

Comment by Chad

2011-12-01T05:39:50Z

I have added more hopefully it will clarify. In response to Antoine a1 and a2 have small coefficients while a3 and a4 has big coefficients. I expect a3 and a4 to be zero if not approach zero. However this is a program and it needs to be very carefully set up at first. Then i won't have to interfer and let it do the calculation.