Page 96 - IJOCTA-15-3

Sayed Saber et.al. / IJOCTA, Vol.15, No.3, pp.464-482 (2025)
Substitute the series and fractional derivative of the solution to this problem is:
approximations into each of the differential equa- K
ˆ
X
tions: υ j (t) = Γ j (k)(t − ι 0 ) kα j , ι ∈ [ι 0 , T],
j=0
α
Σ ∞ a C 0,ι n where Γ j (k) satisfies
n=0 n D (ι ) = −a 1 ˆu(ι j ) + a 2 ˆu(ι j )ˆv(ι j ) + · · · + a 20 ,
ˆ
j
α
Σ ∞ b C 0,ι n α((k + 1)α j + 1) ˆ
n=0 n D (ι ) = −a 8 ˆu(ι j )ˆv(ι j ) − · · · + a 21 ,
ˆ
j
α
n=0 n D (ι ) = a 15 ˆv(ι j ) + · · · − a 19 ˆv(ι j ) ˆw(ι j ),
Σ ∞ c C 0,ι n α(kα j + 1) Γ j (k + 1) = g j (k, Γ 1 , Γ 2 , ..., Γ n ),
j
and with initial conditions Γ j (0) = d j . The
where j = 0, 1, . . . , m. differential transform of g j (k, Γ 1 , Γ 2 , ..., Γ n ) cor-
Based on the complexity of the algebraic sys- responds to g j (t, υ 1 , υ 2 , ..., υ n ) for j = 1, 2, ..., n.
tem, the coefficients a n , b n , and c n can be de- Partition the interval [ι 0 , T] into M subintervals
termined numerically or symbolically. Once the [ι m−1 , ι m ], where m = 1, 2, ..., M, using a step size
coefficients are determined, approximate solu- of h = (T − ι 0 )/M and nodes ι m = ι 0 + mh. The
tions are constructed as: principles of MSGDTM are outlined below.
Initially, the MSGDTM method solves (3)-(4)
n=0 n ι ,
ˆ u(t) ≈ Σ N a n on the first interval [ι 0 , ι 1 ], yielding the approx-
ˆ v(t) ≈ Σ N b n imate solution υ i,1 (t) for t ∈ [ι 0 , ι 1 ]. For m ≥
n=0 n ι ,
2, applying MSGDTM to (3)-(4) on [ι m−1 , ι m ]
ˆ w(t) ≈ Σ N c n uses the continuity condition υ i,m (ι m−1 ) =
n=0 n ι ,
υ i,m−1 (ι m−1 ). Repeating this process results in
where N is the number of terms chosen based on approximations υ i,m (t) for all m = 1, 2, ..., M.
the desired accuracy. The final approximation is defined as:

υ i,1 (t) if ι ∈ [ι 0 , ι 1 ],



 υ i,2 (t) if ι ∈ [ι 1 , ι 2 ],
υ j (t) = .
.
4.2. MSGDTM .


υ i,M (t) if ι ∈ [ι M−1 , ι M ].

Let the differential transforms of ˆu(t), ˆv(t), and
ˆ
ˆ
ˆ
ˆ w(t) be represented as ˆu(k), ˆv(k), and ˆw(k), MSGDTM remains effective for all step sizes h.
respectively. Assume the initial values satisfy For system (1), the MSGDTM procedure yields:
ˆ u(0) = ˆu 0 , ˆv(0) = ˆv 0 , and ˆw(0) = ˆw 0 . The solu- α(αk + 1)
ˆ
ˆ u(k + 1) =
tion process begins with these initial conditions in ˆ
α(α(k + 1) + 1)
the transformed domain at k = 0. Applying the 2 ˆ
ˆ
ˆ
ˆ
recurrence relations derived earlier, we iteratively − b 1 ˆu(k) + b 2 ˆu(k)ˆv(k) + b 3 ˆv (k)
ˆ
ˆ
ˆ
compute ˆu(k + 1), ˆv(k + 1), and ˆw(k + 1) for each 3 ˆ ˆ 2 ˆ
+ b 4 ˆv (k) + b 5 ˆw(k) + b 6 ˆw (k)
k up to the desired approximation order K. The

3 ˆ
approximate solutions in the original domain for + b 7 ˆw (k) + b 20 ,
ˆ u(t), ˆv(t), and ˆw(t) can then be reconstructed via
ˆ
the inverse differential transform. Using the MS- ˆ v(k + 1) = α(αk + 1)
ˆ
GDTM methodology, partition the interval [0, T] α(α(k + 1) + 1)
into M subintervals. To illustrate the approach, − b 8 ˆu(k)ˆv(k) − b 9 ˆu (k) − b 10 ˆu (k)
ˆ
ˆ
2 ˆ
3 ˆ
63 67
consider the system analyzed in: -
ˆ
ˆ
ˆ
+ b 11 ˆv(k)(1 − ˆv(k)) − b 12 ˆw(k)
α
D υ 1 (t) = g 1 (t, υ 1 , υ 2 , ..., υ n ),
1 2 ˆ 3 ˆ
α
D υ 2 (t) = g 2 (t, υ 1 , υ 2 , ..., υ n ), − b 13 ˆw (k) − b 14 ˆw (k) + b 21 ,
2
. . . (3) ˆ w(k + 1) = α(αk + 1)
ˆ
ˆ
α(α(k + 1) + 1)
α
D υ n (t) = g n (t, υ 1 , υ 2 , ..., υ n ).
n
ˆ
3 ˆ
2 ˆ
b 15 ˆv(k) + b 16 ˆv (k) + b 17 ˆv (k)
subject to

ˆ
ˆ
ˆ
υ j (ι 0 ) = d j , i = 1, 2, ..., n, (4) − b 18 ˆw(k) − b 19 ˆv(k) ˆw(k) .
The solution of the problem (3)-(4) is sought over The initial values: ˆu(0) = d 1 , ˆv(0) = d 2 , and
the domain [ι 0 , T]. The kth-order approximation ˆ w(0) = d 3 . Following is the differential transform
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