Estimation of parameter functions in ordinary differential equations with a stage structure: a linear case

Figures & data

Figure 1. For given data with $I=2,$$t_1$ can be uniquely determined in case (a), whereas $t_1$ cannot be determined in case (b).

Figure 2. An example of the case in which five data points satisfy the $C$-test condition.

Figure 3. The first and last intervals both contain two data points, whereas the others, only one data point (T1).

Table 1. Simulated data in data set 1.

j	1	2	3	4	5	6	7	8	9	10
$p_j$	0.10	0.20	0.25	0.30	0.45	0.55	0.60	0.85	0.75	0.90
$q_j$	5.46	13.87	21.86	34.03	110.53	195.22	238.65	356.29	329.98	363.17

Figure 4. Given information in data-set 1.

Table 2. Minimization results for the functional (18) for data-set 1.

$j$	0	1	2	3	4	5	6	7
$t_j$	0.10000	0.22691	0.36204	0.46342	0.55737	0.60754	0.77273	0.90000
${\mathit{\theta }}_j$	9.31914	8.84667	7.15449	5.46094	3.76750	2.07546	0.38248

Figure 5. The dashed line indicates the exact solution of (2020 $$\begin{array}{c}\hfill \frac{du}{dt}=ru(1-\frac{u}{K})\text{for}t\in (p_1,p_{10})\text{and}u(p_1)=u_0,\end{array}$$20 ), the solid line is the derived solution and dots represent data points.

Table 3. Simulated data in data-set 2.

j	1	2	3	4	5	6	7	8	9	10
$p_j$	-0.90	-0.75	-0.55	-0.45	-0.20	0.00	0.25	0.30	0.60	0.85
$q_j$	0.7843	0.8688	0.9461	0.9701	0.9973	1.00	1.0052	1.0090	1.0747	1.2272

Figure 6. Given information in data-set 2 (dots represent data points).

Table 4. Minimization results for the functional (18) for data set 2.

$j$	0	1	2	3	4	5	6	7
$t_j$	-0.90000	-0.75750	-0.59574	-0.34737	-0.00000	0.28494	0.59400	0.85000
${\mathit{\theta }}_j$	0.69334	0.47768	0.25085	0.013339	0.020822	0.20346	0.53083

Figure 7. The dashed line indicates the exact solution of (2121 $$\begin{array}{c}\hfill \frac{du}{dt}=t^{2}u\text{in}(p_1,p_{10})\text{and}u(p_1)=exp(p_1^3/3).\end{array}$$21 ), the solid line is the derived solution and dots represent data points.