In this section, we consider the classical integer order neutron point kinetic equations for m-delayed groups as follows [24,53]:

dn t dt

t l n t c S ti ii

m( ) ( ) ( ) ( )= −� ��

�

�� + +

= ∑ρ β λ 1

(4.1)

dc t dt l n t c t i m i i

i i ( ) ( ) ( ), , , ,= − =β λ 1 2 … (4.2)

where: n(t) is the time-dependent neutron density c ti( ) is the ith precursor density ρ(t) is the time-dependent reactivity function βi is the ith delayed fraction β β=

l is the neutron generation time λi is the ith group decay constant S(t) is the neutron source function

The classical neutron point kinetic equation is considered in matrix form as follows [60]:

dx t dt Ax t B t x t S t ( ) ( ) ( ) ( ) ( )= + + (4.3)

where:

x t

n t c t c t

c tm

( )

( ) ( ) ( )

( )

=

•

–

³ ³ ³ ³ ³ ³ ³ ³

—

˜

µ µ µ µ µ µ µ µ

with initial condition x x( )0 0=

Here, we define A as

A

l

l

l

l

=

−

−

−

−

�

�

� � � � � � � � � � �

�β λ λ λ

β λ

β λ

β λ

0 0

0 0

0 0

�

� � � � � � � � � � �

B(t) can be expressed as

B t

t l

( )

( )

=

�

�

� � � � � � �

�

�

� � � � � � �

ρ 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 1

and S t( ) is defined as

S t

q t

( )

( )

=

•

–

³ ³ ³ ³ ³ ³ ³ ³

—

˜

µ µ µ µ µ µ µ µ

0 0

where: q(t) is the time-dependent neutron source term

In this section, we will apply the MDTM to obtain the solution for classical neutron point kinetic equation (Equation 4.3).