Page 175 - IJOCTA-15-2

S. Das / IJOCTA, Vol.15, No.2, pp.368-378 (2025)
Definition 2. 14 For f : R + → R, fractional left Therefore,
Riemann-Liouville integral of order α > 0 is de-
α
fined as t I f(t) = 1 R T (s−t) α−1 f(s)ds, t > 0.
T Γ(α) t ζ(t; z, z) = ((A(t) + F)z, z) 2 ,
L (Ω)
Z ∂z(x, t) ∂z(x, t)
= Σ n c dx
i,j=1 ij (x, t)
Ω ∂x i ∂x j
Definition 3. 14 For f : R + → R, fractional left Z
Caputo derivative of order α ∈ (0, 1) is defined as + c 0 (x, t)z(x, t)z(x, t)dx
α
C D f(t) = 1 R t (t − s) 1−α ′ Ω
f (s)ds.
0 t Γ(1−α) 0 Z
1
+ c(x, ρ(t, z(t)))z(x, t)z(x, t)dx, z ∈ H (Ω)
0
Ω
∂ 2
n
c
Definition 4. 14 For f : R + → R, fractional right ≥ Σ i,j=1 i,j (x, t)∥ z(x, t)∥ 2
L (Ω)
∂x i
Caputo derivative of order α ∈ (0, 1) is defined as 2
α
C D f(t) = −1 R T (s − t) 1−α ′ +c 0 ∥z(x, t)∥ 2
L (Ω)
f (s)ds.
t T Γ(1−α) t Z
1
+ az(x, t)z(x, t)dx, z ∈ H (Ω)
0
Ω
∂
2
Also when T = ∞ it is fractional order α Weyl ≥ Σ n c z(x, t)∥ 2
i,j=1 i,j (x, t)∥
L (Ω)
′
integral of f . ∂x i
2
+c 0 ∥z(x, t)∥ 2
L (Ω)
2
+a∥z(x, t)∥ 2
L (Ω)
Definition 5. Bilinear form ζ(t; z, ϕ) is defined 2
1
1
below in H (Ω) for each t ∈ (0, T) as ≥ λ∥z∥ H (Ω) , λ > 0. (10)
0 0
1
ζ(t; z, ϕ) = ((A(t) + F)z, ϕ) 2 , z, ϕ ∈ H (Ω). 1
L (Ω)
0
(6) It is assumed that ∀z, ϕ ∈ H (Ω) the bilinear
0
1
form t → ζ(t; z, ϕ) is C with respect to t ∈ (0, T)
Then is symmetric, i.e.,
ζ(t; z, ϕ) = ζ(t; ϕ, z) (11)
ζ(t; z, ϕ) = ((A(t) + F)z, ϕ) 2 ,
L (Ω)
Z ∂z(x, t) ∂ϕ(x, t)
= Σ n c dx The existence of unique solution for the sys-
i,j=1 ij (x, t)
Ω ∂x i ∂x j tem (3) − (5) is derived using Lax-Milgram
lemma.
Z 15
+ c 0 (x, t)z(x, t)ϕ(x, t)dx
Ω
Z
1
+ Fz(x, t)ϕ(x, t)dx, z, ϕ ∈ H (Ω) (7)
0
Ω
3.1. Fractional Green’s theorem
Lemma 2. Let z be a solution of the system
∞
Lemma 1. The bilinear form ζ(t; z, ϕ) defined in (3) − (5), then ∀ϕ ∈ C (Q) and ϕ(x, T) = 0 in Ω
1
(6) is coercive on H (Ω), i.e., and ϕ = 0 in Σ, z satisfies the following equation.
0
ζ(t; z, ϕ) ≥ λ∥z∥ 2 , λ > 0. (8)
1
H (Ω)
0 Z T Z
α
C
( D z(x, t) + A(t)z(x, t))ϕ(x, t)dxdt =
0
t
0 Ω
Z
Proof. We know ellipticity is sufficient to prove − z(x, 0) t I 1−α ϕ(x, 0)dΩ
coerciveness. As Ω T
T ∂ϕ(x, t)
Z Z
ζ(t; z, ϕ) = ((A(t) + F)z, ϕ) 2 , − z(x, t) dΓdt
L (Ω)
Z 0 Γ ∂ν
∂z(x, t) ∂ϕ(x, t)
n
= Σ i,j=1 ij (x, t) dx Z T Z ∂z(x, t)
c
Ω ∂x i ∂x j + ϕdΓdt
Z ∂ν
0 Γ
+ c 0 (x, t)z(x, t)ϕ(x, t)dx Z T Z
∗
α
Ω + z(x, t)( t D ϕ(x, t) + A (t)ϕ(x, t))dxdt
Z T
1
+ Fz(x, t)ϕ(x, t)dx, z, ϕ ∈ H (Ω) (9) 0 Ω
0
Ω (12)
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