Novel approach to solving the inverse equation of point kinetics by the Bernoulli number generalisation method

Figures & data

Table 1. First Bernoulli numbers.

${\mathbf{B}}_1=1/6$	${\mathbf{B}}_5=5/66$	${\mathbf{B}}_9=43867/798$
${\mathbf{B}}_2=1/30$	${\mathrm{B}}_6=691/2730$	${\mathrm{B}}_{10}=174611/330$
${\mathbf{B}}_3=1/42$	${\mathrm{B}}_7=7/6$	${\mathrm{B}}_{11}=854513/138$
${\mathbf{B}}_4=1/30$	${\mathrm{B}}_8=3617/510$	${\mathrm{B}}_{12}=236364091/2730$

Table 2. Maximum difference in pcm for $P(t)=e^{\omega t}$ with $\omega =0.00243,t=1000s$.

T [s]	Euler–Maclaurin	MF	BNGM	BNGM with eps
0.01	6.20 × 10^−08	4.94 × 10^−11	7.50 × 10^−11	7.64 × 10^−11
0.05	3.84 × 10^−05	1.37 × 10^−11	1.86 × 10^−11	2.12 × 10^−11
0.1	6.14 × 10^−04	6.16 × 10^−12	1.55 × 10^−11	1.55 × 10^−11
0.2	9.73 × 10^−03	3.47 × 10^−12	1.05 × 10^−11	1.11 × 10^−11
0.5	3.6 × 10^−01	2.00 × 10^−12	5.40 × 10^−12	5.62 × 10^−12
1	4.66 × 10^0	Not reported	4.06 ×10^−12	4.09 × 10^−12

Table 3. Maximum difference in pcm for $P(t)=e^{\omega t}$ with $\omega =0.01046,t=800s$.

T [s]	Euler–Maclaurin	5-point Lagrange	MF	BNGM	BNGM with eps
0.01	6.20×10^−08	2.37×10^−05	5.65×10^−11	3.44×10^−12	4.54×10^−12
0.05	3.87×10^−05	3.98×10^−04	4.80×10^−11	3.51×10^−12	5.79×10^−12
0.1	6.18×10^−04	1.14×10^−03	1.35×10^−11	3.75×10^−12	2.22×10^−12
0.2	9.8×10^−03	not reported	1.34×10^−11	3.99×10^−12	4.61×10^−12
0.5	3.58×10^−01	1.04×10^−02	1.04×10^−12	3.29×10^−12	3.06×10^−12
1	4.69×10^0	1.83×10^−02	Not reported	3.55×10^−12	3.46×10^−12

Table 4. Maximum difference in pcm for $P(t)=e^{\omega t}$ with $\omega =0.02817,t=600s$.

T [s]	Euler–Maclaurin	5-point Lagrange	MF	BNGM	BNGM with eps
0.01	6.30×10^−08	6.38×10^−05	8.13×10^−11	3.13×10^−12	4.10×10^−12
0.05	3.93×10^−05	1.06×10^−03	8.01×10^−11	3.49×10^−12	5.21×10^−12
0.1	6.28×10^−04	3.05×10^−03	2.13×10^−11	2.19×10^−12	2.59×10^−12
0.2	9.96×*10^−03	not reported	2.10×10^−11	3.13×10^−12	2.13×10^−12
0.5	3.64×10^−03	2.65×*10^−02	9.66×10^−13	2.81×10^−12	3.12×10^−12
1	4.76×10^0	4.36×10^−02	Not reported	1.16×10^−12	2.04×10^−12

Table 5. Maximum difference in pcm for $P(t)=e^{\omega t}$ with $\omega =0.12353,t=300s$.

T [s]	Euler–Maclaurin	5-point Lagrange	MF	BNGM	BNGM with eps
0.01	6.85×10^−08	2.78×10^−04	1.14×10^−10	6.33×10^−12	7.10×10^−12
0.05	4.28×10^−05	4.50×10^−03	2.90×10^−11	2.05×10^−12	2.04×10^−12
0.1	6.83×10^−04	1.24×10^−02	3.02×10^−11	1.88×10^−12	1.98×10^−12
0.2	1.08×10^−02	not reported	8.24×10^−12	1.42×10^−12	1.47×10^−12
0.5	3.9×10^−01	8.29×10^−02	1.48×10^−12	2.67×10^−12	2.50×10^−12
1	5.13×10^0	9.46×10^−02	Not reported	1.99×10^−12	2.10×10^−12

Table 6. Maximum difference in pcm for $P(t)=e^{\omega t}$ with $\omega =1.00847,t=150s$.

T [s]	Euler–Maclaurin	5-point Lagrange	MF	BNGM	BNGM with eps
0.01	1.41×10^−07	2.11×10^−03	4.89×10^−11	3.98×10^−13	3.41×10^−13
0.05	8.80×10^−05	2.59×10^−02	6.25×10^−12	3.41×10^−13	4.54×10^−13
0.1	1.40×10^−03	5.03×10^−02	6.48×10^−12	3.41×10^−13	1.02×10^−13
0.2	2.21×10^−02	not reported	1.71×10^−12	6.82×10^−13	1.02×10^−13
0.5	7.9×10^−01	5.53×10^−01	7.96×10^−13	5.68×10^−13	7.95×10^−13
1	9.76×10^0	4.40×10^0	Not reported	6.82×10^−13	9.09×10^−13

Table 7. Maximum difference in pcm for $P(t)=e^{\omega t}$ with $\omega =11.6442,t=60s$.

T [s]	Euler–Maclaurin	MF	BNGM	BNGM with eps
0.01	1.00×10^−05	5.34×10^−12	2.50×10^−12	2.21×10^−13
0.05	6.23×10^−03	2.16×10^−12	1.82×10^−12	1.13×10^−13
0.1	9.64×10^−02	7.96×10^−12	1.59×10^−12	2.27×10^−13
0.2	1.37×10^0	9.09×10^−12	6.82×10^−13	1.13×10^−13
0.5	3.07×10^1	3.41×10^−13	1.14×10^−13	6.82×10^−13
1	2.06×10^2	Not reported	1.14×10^−13	2.04×10^−12

Table 8. Maximum difference in pcm for $P(t)=e^{\omega t}$ with $\omega =52.80352,t=10s$.

T [s]	Euler–Maclaurin	MF	BNGM	BNGM with eps
0.01	6.77×10^−04	1.36×10^−12	2.27×10^−13	2.27×10^−13
0.05	3.60×10^−01	3.41×10^−13	1.14×10^−13	1.13×10^−13
0.1	4.04×10^0	3.41×10^−13	1.14×10^−13	1.13×10^−13
0.2	3.04×10^1	3.41×10^−13	1.14×10^−13	1.13×10^−13
0.5	2.75×10^2	0	1.14×10^−13	2.27×10^−13
1	1.23×10^3	Not reported	0	5.68×10^−12

Table 9. Maximum difference in pcm for $P(t)=c+dt,\text{break}t=10s$.

T [s]	c	d	MF	BNGM with eps
0.1	1	1	5.12×10^−02	5.42×10^−02
0.5	1	1	1.24×10^0	2.71×10^−02
1	1	1	5.08×10^0	5.43×10^−02
5	1	1	4.46×10^0	1.26×10^−02

Figure 1. Comparison of reactivity in pcm for a form of neutron population density $P(t)=c+dt$.

Table 10. Maximum difference in pcm for $P(t)=c+d{t}^3,\text{break}t=1000s$.

T [s]	c	d	DM	MF	BNGM	BNGM with eps
1	1	(0.0127) ^5/9	3.89×10^−03	1.42×10^−05	7.61×10^−11	7.61×10^−11
1	1	(0.0127) ^4/40	6.20×10^−02	9.57×10^−05	1.35×10^−09	1.35×10^−09
1	1	(0.0127) ^4/4	6.21×10^−01	4.38×10^−04	1.35×10^−08	1.35×10^−08

Figure 2. Comparison of reactivity in pcm for a form of neutron population density $P(t)=c+d{t}^3$.

Table 11. Maximum difference in pcm for $P(t)=c+cosh\text{break}(kt),t=180s$.

T [s]	c	k	MF	BNGM	BNGM with eps
0.1	0	$\pi /180$	1.35×10^−04	5.03×10^−12	1.72×10^−12
0.5	0	$\pi /180$	3.40×10^−03	1.87×10^−12	1.00×10^−12
1	0	$\pi /180$	1.34×10^−02	1.19×10^−12	1.12×10^−12
5	0	$\pi /180$	3.04×10^−01	1.04×10^−12	1.12×10^−12
0.1	100	$\pi /180$	1.43×10^−05	5.21×10^−13	2.43×10^−11
0.5	100	$\pi /180$	3.56×10^−04	3.77×10^−08	1.04×10^−08
1	100	$\pi /180$	1.40×10^−03	1.12×10^−07	1.12×10^−07
5	100	$\pi /180$	3.31×10^−02	1.75×10^−05	1.75×10^−05

Table 12. Maximum difference in pcm for $P(t)=c+sinh\text{break}(kt),t=10000s$.

T [s]	c	k	MF	BNGM	BNGM with eps
0.1	1	1.27×10^−03	7.56×10^−07	8.48×10^−10	8.28×10^−10
0.5	1	1.27×10^−03	1.88×10^−05	3.89×10^−07	3.89×10^−07
1	1	1.27×10^−03	7.47×10^−05	4.06×10^−06	4.05×10^−06
5	1	1.27×10^−03	1.70×10^−03	3.47×10^−04	3.47×10^−04
20	1	1.27×10^−03	1.02×10^−02	1.02×10^−02	1.02×10^−02
0.1	100	1.27×10^−3	2.33×10^−07	2.59×10^−12	6.93×10^−10
0.5	100	1.27×10^−3	5.81×10^−06	2.63×10^−10	3.38×10^−07
1	100	1.27×10^−3	2.31×10^−05	5.58×10^−09	3.76×10^−06
5	100	1.27×10^−3	5.33×10^−04	2.84×10^−06	3.35×10^−04
20	100	1.27×10^−3	6.68×10^−03	1.02×10^−03	1.02×10^−03

Figure 3. Comparison of reactivity in pcm for a form of neutron population density $P(t)=c+cosh(kt)$.

Table 13. Maximum difference in pcm for $P(t)=c+sin\text{break}(kt),t=10s$.

T [s]	c	k	MF	BNGM	BNGM with eps
0.1	1	$\pi /10$	4.22×10^−02	1.14×10^−06	1.14×10^−06
0.5	1	$\pi /10$	1.05×10^0	5.94×10^−04	5.94×10^−04
1	1	$\pi /10$	4.15×10^0	7.62×10^−03	7.62×10^−03
5	1	$\pi /10$	8.55×10^1	1.46×10^0	1.46×10^0

Figure 4. Comparison of reactivity in pcm for a form of neutron population density $P(t)=c+sin(kt)$.