Is there a way to scale $J_n(\cdot)$ (Bessel of first kind) and $H_n(\cdot)$ (Bessel of third kind or Hankel)? I am having computer problems with higher orders (higher values 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:
$$r^2 \frac{d^2 R}{dr^2} + r \frac{dR}{dr} + (r^2 - \alpha^2)R = f(r)$$
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 $J_n(\cdot)$ and Hankel $H_n(\cdot)$ (as previously mentioned)
The "inhomogeneous Helmholtz equation" is the equation:
$$\Delta A({\bf{r}}) + k^2 A({\bf{r}}) = -f({\bf{r}}) \mbox { in } \mathbb R^n$$
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:
$$ \begin{cases} T_{11} a_1 J_n(x)+T_{12} a_2 J_n(y)+T_{13} a_3 H_n(x)+T_{14} a_4 H_n(y) &= b_1(x,y) \\ T_{21} a_1 J_n(x)+T_{22} a_2 J_n(y)+T_{23} a_3 H_n(x)+T_{24} a_4 H_n(y) &= b_2(x,y) \\ T_{31} a_1 J_n(x)+T_{32} a_2 J_n(y)+T_{33} a_3 H_n(x)+T_{34} a_4 H_n(y) &= b_3(x,y)\\ T_{41} a_1 J_n(x)+T_{42} a_2 J_n(y)+T_{43} a_3 H_n(x)+T_{44} a_4 H_n(y) &=b_4(x,y) \end{cases} $$
such that $b_1, b_2, b_3, b_4$ 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 $J_n(\cdot)$ and $H_n(\cdot)$, in small arguments and large $n$, $J_n(\cdot)$ is a very small while $H_n(\cdot)$ is very large 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:
$$\phi=\left(a_{1}J_{n}(Kr)+a_{2}H_{n}(Kr)\right)e^{in\theta} ,$$
$$\psi_t = \left(a_{3}J_{n}(k r)+a_{4} H_{n}(k r)\right) e^{in\theta} $$
I am differentiating the potential to get displacements and stresses in terms of the unknowns.