Example Solution

For some further visualizations, we will use the following solution.

This visualizes the left-hand side matrix and the right-hand side vector.

The left graphic is the plot of the solution vector. It clearly shows the two initial sequences belonging to the expansion coefficients in the left and right arms. While the expansion coefficients in the center disk are increasing in absolute value, their influence is decreasing because the factor decreases quickly with for . From the absolute size of the coefficients, we also see that the use of high-precision numbers with digits is appropriate.

The right graphic shows the product of the expansion coefficients with the ansatz functions . We see that the influence of the largest modes in the function value at the center disk boundary is approximately .

We define the corresponding wave functions and their gradients.