Computing Eigenvalues of Discontinuous Sturm-Liouville Problems with Eigenparameter in All Boundary Conditions Using Hermite Approximation

Table 5

For 𝑁=20, 𝑚=8, and 𝜃=1/6, the exact solutions 𝜇_𝑘 are all inside the interval [𝑎₋,𝑎₊] for different values of 𝜀.

𝜇𝑘	Exact 𝜇𝑘	[𝑎−,𝑎+], 𝜀=10^−5	[𝑎−,𝑎+], 𝜀=10^−10
𝜇1	0.579603114810978	[0.504287, 0.644005]	[0.579603, 0.581089]
𝜇2	1.7849838948888357	[1.76837, 1.80095]	[1.78494, 1.78503]
𝜇3	3.238647349751419	[3.18936, 3.28517]	[3.2375, 3.23979]
𝜇4	4.770562335590527	[4.74591, 4.79495]	[4.77054, 4.77059]
𝜇5	6.3257043646722515	[6.30433, 6.34704]	[6.32536, 6.32605]

ℰ₃(ℱ_𝜃,𝑚)=2.61231×10¹¹, ℰ₂(ℱ_𝜃,𝑚)=4.67787×10¹⁰, 𝛼=1, and ℳ_ℱ𝜃,𝑚=5.31641×10⁹.