Determination of the leading coefficient in fourth-order Sturm–Liouville operator from boundary measurements

Figures & data

Table 1. The numerical values of obtained for various values of N ∈ {20, 40, 80 }, when ε = 0. The stopping iteration numbers k and the values of the objective function (38) are also included.

N	μ0	μ1	μ2	μ3	k	Obj f
20	0.88	1.04	0.97	1.05	23	2.6 × 10^-7
40	0.90	1.03	0.97	1.08	25	3.7 × 10^-8
80	0.97	1.00	0.99	1.04	41	7.5 × 10^-9

Table 2. The numerical values of obtained for various values of ε ∈ {± 0.1, ± 0.05, 0 }, when N = 40. The stopping iteration numbers k and the values of the objective function (38) are also included.

ε	μ0	μ1	μ2	μ3	k	Obj f
−0.1	0.69	1.11	0.73	0.53	2	3.0 × 10^-2
−0.05	0.70	1.15	0.76	0.54	2	7.5 × 10^-3
0	0.90	1.03	0.97	1.08	25	3.7 × 10^-8
0.05	0.70	1.22	0.95	0.56	5	7.4 × 10^-3
0.1	0.73	1.26	0.86	0.55	2	3.0 × 10^-2

Figure 1. The numerical solution for (a) the deflection u(x), and (b) the flexural rigidity k(x), for various values of ε = {±0.1, ±0.05, 0}, in comparison with the exact solution (u(x),k(x))=(e^x-1-x,x) for example 4.1.

Figure 2. The numerical solution for (a) the deflection u(x), and (b) the flexural rigidity k(x), for various values of ε = {±0.1, ±0.05, 0}, in comparison with the exact solution (u(x),k(x))=(e^x-1-x,1) for example 4.2.