Journal of the Egyptian Mathematical Society (2016) 24 , 666–671

Egyptian Mathematical Society

Journal of the Egyptian Mathematical Society

www.etms-eg.org www.elsevier.com/locate/joems

Original Article

A novel fractional technique for the modified point

kinetics equations

✩

Ahmed E. Aboanber, Abdallah A. Nahla

∗

Department of Mathematics, Faculty of Science, Tanta University, Tanta 31527, Egypt

Received 3 January 2016; revised 19 February 2016; accepted 21 February 2016 Available online 5 April 2016

Keywords

Point kinetics equations; Multi-group of delayed neutrons; Relaxation time effects; Riemann–Liouville defini- tion; Fractional calculus

Abstract A fractional model for the modified point kinetics equations is derived and analyzed. An analytical method is used to solve the fractional model for the modified point kinetics equations. This methodical technique is based on the representation of the neutron density as a power series of the relaxation time as a small parameter. The validity of the fractional model is tested for different cases of step, ramp and sinusoidal reactivity. The results show that the fractional model for the modified point kinetics equations is the best representation of neutron density for subcritical and supercritical reactors.

2010 Mathematics Subject Classifications: 81V35; 82D75; 34A08

Copyright 2016, Egyptian Mathematical Society. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license

( http://creativecommons.org/licenses/by-nc-nd/4.0/ ).

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. Introduction

he neutron density and the precursor concentrations of de- ayed neutrons at the center of a homogeneous nuclear reactor

This work was presented in the International Conference for Math- matics and Applications (ICMA15), 27 – 29 Dec. 2015, Cairo, Egypt. DEA-11) Corresponding author. Tel.: +020 1151667155.

E-mail addresses: ahmed.aboanber@science.tanta.edu.eg (A.E. Aboan- er), a.nahla@science.tanta.edu.eg (A.A. Nahla).

Peer review under responsibility of Egyptian Mathematical Society.

Production and hosting by Elsevier

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1110-256X(16)30003-7 Copyright 2016, Egyptian Mathematical Society. Pronder the CC BY-NC-ND license ( http://creativecommons.org/licenses/by-ttp://dx.doi.org/10.1016/j.joems.2016.02.001

re described by a system of stiff coupled linear and/or non- inear differential equations. An important peculiarity of the re- ctor kinetics is the stiffness of the system. A host of mathemat-cal methods are developed to solve this system as a functionf neutron density with different energy groups of delayed neu- rons. Although, it is still currently requires more effort from thecientists to develops a new mathematical techniques and com- utational scheme to overcome this problem. The continuous

ndication of the neutron density and its rate of change are im-ortant for the safe startup, accurate determination of reactivity ffects and operation of reactors. Recently, the interest of the nu-lear reactor scientists is the development and analysis of differ- nt versions and approximations of the fractional-order point eactor kinetics model for a nuclear reactor ”fractional neutron

oint kinetics equations (FNPKE)”. For example: Espinosa- aredes, Polo-Labarrios, et al. [1–7] ; Ray and Patra [8] ; Nowak

duction and hosting by Elsevier B.V. This is an open access article nc-nd/4.0/ ).

A novel fractional technique for the modified point kinetics equations 667

et al. [9,10] ; and Schramm et al. [11] . Espinosa-Paredes et al.[1] is the first scientific group derived the FNPKE. Aboanberand Nahla commented on the paper [1] through letter to edi-tor [12,13] . In this letter, the corrected form for the fractionalneutron point kinetics equations is developed. In this work wefocus on the derivation of the fractional modified point kineticsequations (FMPKE) and its analytical solution. The developedsolution of the FMPKE is based on the representation of theneutron density as a power series in terms of the relaxation timeas a small parameter, which is less than 10 −3 ( s ) .

The presented paper is organized as follows: Section 2 con-tains the derivation of the fractional modified point kineticsequations. An analytical method based on the representationof the neutron density as a power series of a small parameter ispresented in Section 3 . A comparison between the neutron den-sity of the fractional modified point kinetics equations and thepoint kinetics equations is discussed in Section 4 . The conclu-sion is suggested in Section 5 .

2. Fractional modified point kinetics equations (FMPKE)

The explicit forms taken by the neutron conservation equationand the corresponding equation for the current density in theone-speed case are [14]

1 v

∂

∂t �(r , t) + ∇. J (r , t) + �t �(r , t) = �s �(r , t) + S(r , t) , (1)

1 v

∂

∂t J (r , t) + ∇.

∫ 4 π

ˆ � ˆ �ϕ (r , ˆ �, t ) d ̂ � + �t J (r , t )

= μ̄0 �s J (r , t) + S 1 (r , t) (2)

where ϕ (r , ˆ �, t ) is the time dependent angular flux, as a func-tion of space r and neutron direction of motion

ˆ �, �(r , t) ≡∫ 4 π ϕ (r , ˆ �, t ) d ̂ � is zero moment of the angular flux represents

the neutron density, whereas the first moment of the angularflux J (r , t) ≡ ∫

4 πˆ �ϕ(r , ˆ �, t) d ̂ � represents the neutron current

density, v is the neutron speed, �t is the total cross section, �s

is the scattering cross section, μ̄0 = 〈 ̂ �. ̂

�́〉 is the average scatter-ing angle cosine, S(r , t) ≡ ∫

4 π S(r , ˆ �, t) d ̂ � is the neutron sourceterm and S 1 (r , t) ≡

∫ 4 π

ˆ �S(r , ˆ �, t) d ̂ � is first moment of theneutron source.

Assume that the angular flux is only linearly anisotropic andthe neutron source is isotropic, that is ∇.

∫ 4 π

ˆ � ˆ �ϕ (r , ˆ �, t ) d ̂ � =1 3 ∇�(r , t) and S 1 (r , t) = 0 . The system of Eqs. (1) and (2) takethe following form

1 v

∂�(r , t) ∂t

+ ∇ · J (r , t) + �a �(r , t) = S(r , t) (3)

1 v

∂

∂t J (r , t) + �tr J (r , t) +

1 3 ∇�(r , t) = 0 (4)

where �a = �t − �s is the time dependent absorption cross sec-tion, �t is the total cross section, �s is the scattering cross sec-tion, �tr = �t − μ̄0 �s is the transport cross section.

Eqs. (3) and (4) representing the P 1 approximation (i.e. theone-speed approximation). To simplify this system of equations,Fick’s suggested for Eq. (4) that the time derivative 1

v ∂

∂t J (r , t)can be neglected in comparing with the other terms [14] , whichcontradicts Cattaneo’s law [15] . Let us divide Eq. (4) by �tr

and taking the fractional derivative on the first term as follows

[16,17] :

τ κ ∂ κ

∂t κJ (r , t) + J (r , t) = −D ∇�(r , t) (5)

where D =

1 3�tr

is the neutron diffusion coefficient and τ =1

v �tr =

3 D

v is the relaxation time. Substituting from Eq. (5) into Eq. (3) yields

τ κ ∂ κ

∂t κ

[1 v

∂

∂t �(r , t) + �a �(r , t) − S(r , t)

]+

1 v

∂

∂t �(r , t)

+ �a �(r , t) − S(r , t) − D ∇

2 �(r , t) = 0 (6)

The neutron diffusion equations with delayed neutrons areobtained by adding the source term of reactor kinetics with Igroups of delayed neutrons as follow:

τ κ ∂ κ

∂t κ

[1 v

∂

∂t �(r , t) + (�a − (1 − β) ν� f )�(r , t)

−I ∑

i=1

λi C i (r , t)

]

+

1 v

∂

∂t �(r , t) + [�a − (1 − β) ν� f ]�(r , t)

−I ∑

i=1

λi C i (r , t) − D ∇

2 �(r , t) = 0 (7)

where S(r , t) = (1 − β) ν� f �(r , t) +

I ∑

i=1 λi C i (r , t) , β =

I ∑

i=1 βi is

total fraction of delayed neutrons, β i is the fraction of i -groupof delayed neutrons, ν is the mean number of fission neutrons,�f is the fission cress section, λi is the decay constant of i -groupof delayed neutrons, I is the total number of delayed neutrongroups and C i ( r , t ) is the precursor concentrations of i -group ofdelayed neutrons which satisfy the following equations

∂C i (r , t) ∂t

= βi ν� f �(r , t) − λi C i (r , t) , i = 1 , 2 , . . . , I . (8)

Consider that:

�(r , t) = vp(t ) φ(r ) , C i (r , t ) = c i (t ) φ(r ) (9)

where φ( r ) is the fundamental function, which is obtained fromthe diffusion equation:

∇

2 φ(r ) + B

2 g φ(r ) = 0 (10)

B

2 g is the geometric buckling appropriate for the reactor geome-

try (Table 3.3, page (60), [18] ). Using Eq. (10) in Eqs. (7) and (8) leads to the fractional mod-

ified point kinetics equations with multi-group delayed neutronsas

τ κ d κ

dt κ

[

d p(t) dt

−( ρ

�− μ + α

)p(t) −

I ∑

i=1

λi c i (t)

]

+

d p(t) dt

−( ρ

�− μ

)p(t) −

I ∑

i=1

λi c i (t) = 0 (11)

dc i (t) dt

= μi p(t) − λi c i (t) , i = 1 , 2 , 3 , . . . , I (12)

where p ( t ) is the neutron density, ρ =

ν� f −�a (1+ L 2 B 2 g )

ν� f is the time

dependent reactivity, L

2 =

D

�a is the diffusion length, � =

1 vν� f

is the prompt neutron generation time, α = vDB

2 g , μi =

βi �

andthe initial conditions of this differential equations are p(0) =p 0 , c i (0) =

μi λi

p 0 , i = 1 , 2 , 3 , . . . , I .

668 A.E. Aboanber, A.A. Nahla

tdpd

3

Tt

at

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w

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w

|

r

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O

3

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Tbo

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T

c

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The fractional calculus involves different definitions of he fractional operator as well as the Grünwald–Letnikov erivative, Riemann–Liouville fractional derivative, and Ca- uto derivative. Let us introduce the definition of the fractional erivatives as:

1. The Grünwald–Letnikov definition of fractional derivatives is defined as

GL D

κ f (t) = lim

h → 0 h −κ

n ∑

m =0

γ κm

f (t − mh ) (13)

where γ κm

= (−1) m

(κm

), γ κ

0 = 1 and γ κm

= (1 − κ+1

m

)γ κ

m −1 , m = 1 , 2 , . . . . 2. The Riemann–Liouville definition of fractional derivatives

is defined as

RL D

κ f (t) =

1 �(m − κ)

∂ m

∂t m

∫ t

0

f (ξ )

(t − ξ ) κ+1 −m

dξ,

m − 1 < κ < m (14)

where m is a positive integer and �(m − κ) is the gammafunction whose argument is (m − κ) .

3. The Caputós derivative is defined as

C D

κ f (t) =

1 �(m − κ)

∫ t

0

f (m ) ( ξ )

( t − ξ ) κ+1 −m

dξ, m − 1 < κ < m

(15)

where f (m ) (ξ ) =

∂ m f (ξ )

∂ξm

. Analytical solution of the FMPKE

he solution for the fractional modified point kinetics equa- ions (FMPKE) can be obtained in several ways, in this paper annalytical solution of the FMPKE is proposed. Let us assume hat

(t) =

d p(t) dt

−( ρ

�− μ + α

)p(t) −

I ∑

i=1

λi c i (t) (16)

ith initial value q (0) = −αp 0 . Substituting into Eq. (11) , yields

d p(t) dt

=

(ρ(t) �

− μ

)p(t) +

I ∑

i=1

λi c i (t) − τ κ d κq (t) dt κ

(17)

et us rewrite Eqs. (17) and (12) in matrix form as

d dt

| �(t) 〉 = N | �(t) 〉 − τ κ d κq (t) dt κ

| 1 〉 (18)

here,

�(t) 〉 =

⎛

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

p(t) c 1 (t) c 2 (t)

. . . c I (t)

⎞

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

, N =

⎛

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

ρ

�− μ λ1 λ2 · · · λI

μ1 −λ1 0 · · · 0 μ2 0 −λ2 · · · 0 . . .

. . . . . .

. . . . . .

μI 0 0 · · · −λI

⎞

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

,

| 1 〉 =

⎛

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

1 0 0 . . . 0

⎞

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

.

The vector | �( t ) 〉 can be represented by a power series of theelaxation time τ κ :

�(t) 〉 = | ψ 0 (t) 〉 + τ κ | ψ 1 (t) 〉 + · · · (19)

ince τ ≤ 10 −4 , then we can neglect the terms of order two ( τ 2 κ ).

Substituting into Eq. (18) we get

d dt

| ψ 0 (t) 〉 + τ κ d dt

| ψ 1 (t) 〉

= N | ψ 0 (t) 〉 + τ κN | ψ 1 (t) 〉 − τ κ d κq (t) dt κ

| 1 〉 + · · · (20)

.1. Zero order of relaxation time

he zero order of relaxation time, τ 0 , is

d dt

| ψ 0 (t) 〉 = N | ψ 0 (t) 〉 (21)

his equation represents the point kinetics model which can

e derived using Fick’s approximation. The analytical solution

f this system is reported in most reactor analysis articles. Forompleteness, it is recalled briefly that, with this assumption, the oefficient matrix N is independent on time and all eigenvalues k are real which can be determined as:

et(N − ωI ) = 0 ⇒

(ω − ρ

�

)+

I ∑

i=1

ωμi

ω + λi = 0 (22)

his equation is called inhour equation . The eigenvectors | U j 〉orresponding to different eigenvalues ω j are orthogonal i.e.

V l | U j 〉 = δl, j =

{1 , l= j ;0 , l = j . and

∑ I j=0 | U j 〉〈 V j | = I .

According to the references [20–24] , the analytical formula f the eigenvectors | U j 〉 of coefficient matrix N can be deter-ined from (N − ω j I ) | U j 〉 = 0 as

U j 〉 = σ j

⎛

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

1

μ1

(ω j + λ1 ) μ2

(ω j + λ2 )

. . .

μI

(ω j + λI )

⎞

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

, (23)

nd the eigenvectors 〈 V j | of the adjoint matrix, the transportf coefficient matrix, N

T = N

† can be determined by solve thequation [(N − ω j I ) | V j 〉 ] † = 0 or 〈 V j | (N

T − ω j I ) = 0 as

V j | = σ j

(1

λ1

(ω j + λ1 )

λ2

(ω j + λ2 ) · · · λI

(ω j + λI )

)(24)

y the normalization condition 〈 V j | U j 〉 = 1 then σ j = (1 + I i=1

μi λi (ω j + λi ) 2

) −1 2 , ∀ j.

A novel fractional technique for the modified point kinetics equations 669

t

The exact analytical solution of Eq. (21) is

| ψ 0 (t) 〉 = exp (N t) | ψ 0 (0) 〉 =

I ∑

j=0

e ω j t | U j 〉〈 V j | ψ 0 (0) 〉 (25)

where | ψ 0 (0) 〉 is the initial state vector of reactor. Eq. (25) represents the exact solution of the point kinetics

model or zero order of relaxation time. Generally, in the pointkinetics model, the reactivity ρ is a time dependent. For thiscase, we have an n time intervals, t m +1 − t m

= h and t = t n = nh .Through the time interval [ t m

, t m +1 ] , the reactivity is consideredconstant. Then, the solution of the point kinetics model withtime dependent reactivity takes the form

| ψ 0 (t m +1 ) 〉 = exp (N h ) | ψ 0 (t m

) 〉 =

I ∑

j=0

e ω j h | U j 〉〈 V j | ψ 0 (t m

) 〉 (26)

3.2. First order of relaxation time

The first order of relaxation time, τ κ , is

d dt

| ψ 1 (t) 〉 = N | ψ 1 (t) 〉 − d κq (t) dt κ

| 1 〉 (27)

The final term can be calculated from Eqs. (16) and (21) as

d κq (t) dt κ

= −αd κ p(t)

dt κ(28)

Substituting from Eq. (25) yields

d κ p(t) dt κ

=

d κ

dt κ

I ∑

j=0

e ω j t σ j 〈 V j | ψ 0 (0) 〉 (29)

Let us use the Riemann–Liouville fractional derivatives [19] :

d κ p(t) dt κ

=

I ∑

j=0

t −κE 1 , 1 −κ (ω j t) σ j 〈 V j | ψ 0 (0) 〉 (30)

where E 1 , 1 −κ (ω j t) is the Mittag-Leffler function which is definedas:

E a,b (z ) =

∞ ∑

l=0

z l

�(al + b) , a, b > 0 . (31)

The general analytical solution of Eq. (27) takes the form:

| ψ 1 (t) 〉 = α exp (N t) ∫ t

0 exp (−N ξ )

d κ p(ξ )

dξκdξ | 1 〉 (32)

where | ψ 1 (0) 〉 = 0 . To calculate the integration, substituting from Eq. (34) we

have:

I =

∫ t

0 exp (−N ξ )

d κ p(ξ )

dξκdξ =

I ∑

j=0

σ j I j 〈 V j | ψ 0 (0) 〉 (33)

where

I j =

∫ t

0 exp (−N ξ ) ξ−κE 1 , 1 −κ (ω j ξ ) dξ

=

I ∑

i=0

∫ t

0 e −ω i ξ ξ−κE 1 , 1 −κ (ω j ξ ) dξ | U i 〉〈 V i | =

I ∑

i=0

I j,i | U i 〉〈 V i |

(34)

Substituting from Eq. (31) yields:

I j,i =

∫ t

0 e −ω i ξ ξ−κE 1 , 1 −κ (ω j ξ ) dξ =

∞ ∑

l=0

∫ t

0

e −ω i ξ ξ−κ (ω j ξ ) l

�(l − κ + 1) dξ

=

∞ ∑

l=0

ω

l j

∫ t

0

e −ω i ξ ξ l−κ

�(l − κ + 1) dξ =

∞ ∑

l=0

ω

l j γ (l − κ + 1 , ω i t)

ω

l−κ+1 i �(l − κ + 1)

(35)

where γ (l − κ + 1 , ω i t) is lower incomplete gamma functionwhich is defined as:

γ (s, z ) = z s �(s ) e −z ∞ ∑

l=0

z l

�(s + l + 1) (36)

Substituting from Eqs. (35) and (34) into Eq. (33) we have:

I =

I ∑

j=0

σ j 〈 V j | ψ 0 (0) 〉 I ∑

i=0

∞ ∑

l=0

ω

l j γ (l − κ + 1 , ω i t)

ω

l−κ+1 i �(l − κ + 1)

| U i 〉〈 V i | (37)

Substituting into Eq. (32) yields:

| ψ 1 (t) 〉 = α

I ∑

i=0

σi e ω i t I ∑

j=0

σ j

∞ ∑

l=0

× ω

l j γ (l − κ + 1 , ω i t)

ω

l−κ+1 i �(l − κ + 1)

| U i 〉〈 V j | ψ 0 (0) 〉 (38)

Let us divide the time t to n intervals as t m +1 − t m

= h and = t n = nh . Through the small time interval [ t m

, t m +1 ] , the Eq.(38) becomes:

| ψ 1 (t m +1 ) 〉 α

I ∑

i=0

σi e ω i h I ∑

j=0

σ j

L ∑

l=0

× ω

l j γ (l − κ + 1 , ω i h )

ω

l−κ+1 i �(l − κ + 1)

| U i 〉〈 V j | ψ 0 (t m

) 〉 (39)

Substituting from Eqs. (26) and (39) into Eq. (19) we get:

| �(t m +1 ) 〉

I ∑

i=0

e ω i h | U i 〉〈 V i | ψ 0 (t m

) 〉 + τ k α

I ∑

i=0

σi e ω i h I ∑

j=0

σ j

L ∑

l=0

× ω

l j γ (l − κ + 1 , ω i h )

ω

l−κ+1 i �(l − κ + 1)

| U i 〉〈 V j | ψ 0 (t m

) 〉

4. Results and discussion

The behavior of the fractional modified point kinetics equa-tions with multi-group of delayed neutrons are presented forpressurized water reactor (PWR) [14,18] . The delayed neu-tron parameters of this reactor are: τ = 10 −4 (s ) , λ1 = 0 . 0127 ,λ2 = 0 . 0317 , λ3 = 0 . 115 , λ4 = 0 . 311 , λ5 = 1 . 4 , λ6 = 3 . 87 ( s −1 ) ,

β1 = 0 . 000266 , β2 = 0 . 001491 , β3 = 0 . 001316 , β4 = 0 . 002849 ,β5 = 0 . 000896 , β6 = 0 . 000182 , β = 0 . 007 , � = 2 . 0 × 10 −5 ( s )and α = 2220 . 0 ( s −1 ) . The initial conditions are p(0) = p 0 =1 . 0 , c i (0) =

βi p 0 �λi

, and

d p(t) dt | t=0 =

dc i (t) dt | t=0 = 0 , i = 1 , 2 , 3 , . . . , 6 .

The neutron density is calculated for three types of reactivity:step, ramp and sinusoidal reactivities.

Figs. (1) –( 4 ) show the neutron density for a step reactivityρ = −0 . 007 = −1($) , ρ = −0 . 003 = −0 . 429($) , ρ = 0 . 003 =0 . 429($) and ρ = 0 . 007 = 1($) respectively. As shown in these

670 A.E. Aboanber, A.A. Nahla

Fig. 1 Neutron density for negative step reactivity ρ = −0 . 007 .

Fig. 2 Neutron density for negative step reactivity ρ = −0 . 003 .

Fig. 3 Neutron density for positive step reactivity ρ = +0 . 003 .

Fig. 4 Neutron density for positive step reactivity ρ = +0 . 007 .

Fig. 5 Neutron density for ramp reactivity ρ = βt.

fit

1

(

t

p

fft

ci

t

td

m

o

lt

ws

ra

gures, the results of fractional modified point kinetics equa- ions (FMPKE) with fractional order κ = 0 . 25 , 0 . 5 , 0 . 75 and.0 are compared with the results of the point kinetics equationsPKE). The important fact in the presented Figs. (1) –(3) , is thathere is a relaxation in the neutron density of the FMPKE com-ared with the PKE at the beginning. Figs. (1) and (2) shows

or negative reactivity the FMPKE presents sub diffusive ef- ects, i.e., greater resistance to movement of the neutron respect o PKE due to the decreasing of the neutron density with in-reasing the fractional order, in addition to the negative term

ncluding the relaxation time and the fraction rate of change ofhe neutron density. On the other side, when the positive reac-ivity is applied, apparently showing a super diffusive behavior ue to the increasing of the neutron density with time for PKEodel, in addition to the relaxation time and the fraction rate

f change for FMPKE model as show in Figs. (3) and (4) . At aarge positive reactivity, Fig. (4) shows a good representation for he relaxation in the neutron density of the FMPKE and PKE,hich confirm that the FMPKE is the best representation for

upercritical reactor. The effects of the ramp, ρ = βt = 1 . 0($ /s ) , and sinusoidal

eactivity, ρ = βsin (

2 π5 t

), on the FMPKE are also included

nd shown in Figs. (5 ) and ( 6 ), respectively. The effect of the

A novel fractional technique for the modified point kinetics equations 671

Fig. 6 Neutron density for sinusoidal reactivity ρ = βsin ( 2 πt 5 ) .

relaxation time on the neutron density for time varying reactiv-ity using FMPKE is also satisfied and clear.

5. Conclusion

Fractional neutron point kinetic model for the FMPKE hasbeen analyzed for the dynamic behavior of the neutron mo-tion in which the relaxation time associated with a variationin the neutron flux involves a fractional order acting as expo-nent of the relaxation time, to obtain the best operation of anuclear reactor dynamics. The numerical stability of the resultsfor neutron dynamic behavior for subcritical reactivity, super-critical step and time varying reactivity and for different valuesof fractional order are shown and compared with the classicneutron point kinetic equations (PKE). The fractional modelretains the main dynamic characteristics of the neutron motionin which the relaxation time associated with a rapid variationin the neutron flux contains a fractional order, acting as expo-nent of the relaxation time, to obtain the best representation ofa nuclear reactor dynamics.

Acknowledgments

The authors would like to express their sincere thanks and grat-itude to the editor and referee(s) for their helpful comments toimprove the quality and readability of this work.

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