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:

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

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

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

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:

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

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

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