Page 14 - IJOCTA-15-1

P. Kumar / IJOCTA, Vol.15, No.1, pp.1-13 (2025)

m i−⌈γ⌉ m n−1 m
X X X
X X (γ) n−j−γ
γ C (γ) i−n−γ E n ∆ t = Q 0 + l P n F n (t)
D (Ω m (t)) = a i ∆ i,n t and n,j
t
i=⌈γ⌉ n=0 n=1 j=0 n=0
m
n 2i−2n
(γ) (−1) 2 (2k + 1)(2i − n)!(i − n)! X
∆ i,n = . − g 0 E n F n (t) .
T ∗k−n Γ(i − n + 1 − γ)Γ(2i − 2n + 2)
n=0
(49) (54)
Now the Eqs. (51)-(54) are collocated at m nodes
Now we apply the Chebyshev spectral colloca- t p , p = 0, 1, ..., m − 1 as follows:
tion scheme to derive the solution of the proposed
model (24) as follows:
m n−1 m
X X (γ) n−j−γ γ X
I n ∆ t = β I n F n (t p )
m m n,j p
n=1 j=0 n=0
X X
I m (t) = I n F n (t), P m (t) = P n F n (t),
m m
X X
n=0 n=0 + λ γ B n F n (t p ) P n F n (t p )
m m
X X n=0 n=0
B m (t) = B n F n (t), E m (t) = E n F n (t). m m
X γ γ γ X
n=0 n=0 − I n F n (t p ) − (ν + α + δ ) I n F n (t p ) ,
(50) n=0 n=0
(55)
Using Eqs. (24), (50) and the formula (49), we
get
m n−1 m
X X (γ) n−j−γ γ γ X
P n ∆ n,j p = Λ − δ P n F n (t p )
t
m n−1 m n=1 j=0 n=0
X X (γ) n−j−γ γ X
I n ∆ t = β I n F n (t) m
n,j γ X
n=1 j=0 n=0 − α I n F n (t p ) ,
m m n=0
X
X
+ λ γ B n F n (t) P n F n (t) (56)
n=0 n=0
m m
X γ γ γ X m n−1
− I n F n (t) − (ν + α + δ ) I n F n (t) , X X (γ) n−j−γ
B n ∆ t =
n=0 n=0 n,j p
(51) n=1 j=0
m m
X 1 X
γ
s B n F n (t p ) 1 − B n F n (t p )
m n−1 m K
X X (γ) n−j−γ γ γ X n=0 n=0
P n ∆ t = Λ − δ P n F n (t) m m
n,j X X
n=1 j=0 n=0 + s γ I n F n (t p ) − s γ B n F n (t p )
1 0
m
n=0 n=0
X
− α γ I n F n (t) , m m
X X
γ
n=0 + g B n F n (t p ) E n F n (t p ) ,
(52) n=0 n=0
(57)
m n−1
X X (γ) n−j−γ
B n ∆ t = and
n,j
n=1 j=0
m m
X 1 X m n−1 m
s γ B n F n (t) 1 − B n F n (t) X X (γ) n−j−γ X
t
K E n ∆ n,j p = Q 0 + l P n F n (t p )
n=0 n=0 (53)
m m n=1 j=0 n=0
γ X γ X m
+ s 1 I n F n (t) − s 0 B n F n (t) X
− g 0 E n F n (t p ) .
n=0 n=0
m m n=0
X X (58)
γ
+ g B n F n (t) E n F n (t) ,
n=0 n=0
By substituting from Eq. (50), the initial condi-
and tions can be defined by
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