## 3 Numerical Verification of Solutions
### Analysis of New Solutions (Difference of Two Orders)  
Figure 5 presents the selected simulation results. Again, the compatibility of the newly derived simplified polynomial formulas using the method proposed in this work with solutions calculated using infinite series is seen. The first to be analyzed are solutions ((10), (111) and (112)), when the denominators of the series has a difference from such zeros \(j_{p,k}^{2}-j_{p+2,n}^{2}\), and then the cases ((13), (14) and (15)) are briefly discussed in which the difference in the denominator is as follows: \(j_{p,k}^{2}-j_{p-2,n}^{2}\). Unlike the examples analyzed in Section 3.1, here, three cases of Bessel function orders \(p=2\), \(p=4\) and \(p=6\) were analyzed. As before, the smallest value \(p=2\) was chosen with the intention that in the formulas Equations (13)-(15) the problem with having to subtract the zeros of the Bessel function of negative orders was eliminated.  
The results obtained from comparisons performed for the series defined by Equation (10) indicate that the final result depends only on the order of the Bessel function \(p\). There is no subtracted zero in the final formula; hence, it has no impact on the form of the course. The above affects the courses of these series, which are then straight lines. The larger the order of the function whose zero was considered, the smaller the results that are observed. Interestingly, when \(l=2\) in the next series of Equation (111), the situation is very similar. The results also did not depend on the subtracted zero and on the graph, and it was again possible to observe the constant course of these functions. The values of these functions obtained using Equation (111) were definitely lower than in the case of Equation (10). This was influenced by the derived polynomial form of this series, which, in this case, for \(l=2\) was the square of the solution for \(l=1\). When \(l=3\), i.e., in the case of Equation (112), a gradual increase in the results became visible as \(n\) increased. The reason for this increase was the appearance of the subtracted zero in the final simplified polynomial solution of Equation (112).  
The numerical results for \(l=1\) were obtained using formula Equation (13), showing that the larger the order of the Bessel function whose zeros were subtracted, the smaller the positive initial value (obtained for \(n=1\)). The values of all analyzed series when \(n=2\) were assumed to take negative values. The final asymptotic value to which these infinite series tended as \(n\) approached infinity was also negative. The final asymptotic values had lower negative values as \(p\) increases. The results obtained for Equation (14) showed that for \(l=2\), the entire variability of the analyzed series almost ceased when \(n\geq 4\) for the subtracted zeros. The larger the \(p\) value, the more flattened the course was and the lower the positive asymptotic value. The plots obtained for \(l=3\) from Equation (15) are like a mirror image of those discussed just above when Equation (14) was analyzed. With the difference here, apart from the negative sign, the results were characterized by significantly lower values. However, here too, the greatest variability of the curves was observed for the first \(n=1,\ 2\) and \(3\) zeros.