modelDE_Usage

For the Bessel functions, examples of generating solely the functions J₀(x) and J₁(x) are shown. This is achieved by choosing the order in the DE blocks and the appropriate boundary conditions near the origin. The third Bessel example generates a combination of J₁(x) and Y₁(x) functions since the zeroed starting point of the integration is away from the origin (x=1). Integrating this example back toward the origin (next example to the right) reveals the divergence of the Y₁(x) function.

The lower-left example is for the first non-polynomial spherical harmonic, Y_1,1(x). Again, note the appropriate boundary conditions at x~-1 to achieve this.

The last two examples involve the lowest solutions to the DE for the quantum harmonic oscillator: psi_0,1(x).

In all such examples, the resulting normalization will be different from that provided by the corresponding GSL functions. The overall normalizations here are controlled via initial values for the functions and their derivatives. Care must also be taken that enough steps are chosen to avoid discretization errors [e.g., slight deviations from expected behavior can be seen near the boundaries for Y_1,1(x) and psi₀(x)].