Direct problem for fractional differential equation with
the generalized Riemann - Liouville time derivative
order 1
< α <
2
H.H. Turdiev
Bukhara State University
hturdiev@mail.ru
September 24, 2023
H.H. Turdiev (BuxSU)
Short title
September 24, 2023
1 / 16
Paragraphs of Text
We consider the following fractional wave equation in the domain
Ω =
{
(
x
,
t
) : 0
<
x
<
1
,
0
<
t
≤
T
}
D
α,β
0+
,
t
u
(
x
,
t
)
−
u
xx
+
q
(
t
)
u
(
x
,
t
) =
f
(
x
,
t
)
,
(1)
the initial conditions of Cauchy type
I
(2
−
α
)(1
−
β
)
0+
,
t
u
(
x
,
t
)
t
=0
=
φ
1
(
x
)
,
∂
∂
t
I
(2
−
α
)(1
−
β
)
0+
,
t
u
(
x
,
t
)
t
=0
=
φ
2
(
x
)
,
x
∈
[0
,
1]
,
(2)
the boundary conditions
u
(0
,
t
) =
u
(1
,
t
)
,
u
x
(1
,
t
) = 0
,
0
≤
t
≤
T
.
(3)
H.H. Turdiev (BuxSU)
September 24, 2023
2 / 16
Here the generalized Riemann-Liouville (Hilfer) fractional differential
operator
D
α,β
0+
,
t
of the order 1
< α <
2 and type 0
≤
β
≤
1 is defined as
follows
1
,
2
D
α,β
0+
,
t
u
(
·
,
t
) =
I
β
(2
−
α
)
0+
,
t
∂
2
∂
t
2
I
(1
−
β
)(2
−
α
)
0+
,
t
u
(
·
,
t
)
,
I
γ
0+
,
t
u
(
x
,
t
) =
1
Γ(
γ
)
t
Z
0
u
(
x
, τ
)
(
t
−
τ
)
1
−
γ
d
τ, γ
∈
(0
,
1)
is the Riemann–Liouville fractional integral of the function
u
(
x
,
t
) with
respect to
t
3
, Γ(
·
) is the Euler’s Gamma function. The functions
f
(
x
,
t
)
,
φ
1
(
x
)
, φ
2
(
x
)
,
w
(
x
)
,
h
(
t
) are known functions.
1
R Hilfer, Applications of Fractional Calculus in Physics, World Scientific: Singapore,
2000.
2
I. Podlubny, Fractional Differential Equations, of Mathematics in Science and
Engineering, vol. 198, Academic Press, New York, NY, USA, 1999.
3
R . Hilfer, Y. Luchko, Z. Tomovski, Operational method for the solution of
fractional differential equations with generalized Riemann-Liouville fractional derivatives,
Fract. Calc. Appl. Anal., 2009, Vol. 12 No.3, pp. 299-318.
H.H. Turdiev (BuxSU)
Short title
September 24, 2023
3 / 16
In
4
,
5
, by R. Hilfer was introduced a generalized form of the
Riemann-Liouville fractional derivative of order
α
and a type
β
∈
[0
,
1]
,
which coincides with the Riemann-Liouville fractional derivative at
β
= 0
and with Gerasimov-Caputo fractional derivative at
β
= 1, and the case
β
∈
(0
,
1) interpolates these both fractional derivatives.
Assume that throughout this article, given functions
φ
1
, φ
2
,
f
satisfy the
following assumptions:
A1)
{
φ
1
, φ
2
} ∈
C
3
[0
,
1]
,
{
φ
(4)
1
, φ
(4)
2
} ∈
L
2
[0
,
1]
, φ
i
(0) =
φ
i
(1) = 0
,
φ
′′
i
(0) =
φ
′′
i
(1) = 0
,
i
= 1
,
2
;
A2) f
(
x
,
·
)
∈
C
[0
,
T
]
and for t
∈
[0
,
T
]
,
f
(
·
,
t
)
∈
C
3
[0
,
1]
,
f
(
·
,
t
)
(4)
∈
L
2
[0
,
1]
,
f
(0
,
t
) =
f
(1
,
t
) = 0
,
f
xx
(0
,
t
) =
f
xx
(1
,
t
) = 0
;
4
R Hilfer, Applications of Fractional Calculus in Physics, World Scientific: Singapore,
2000.
5
S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives.
Theory and applications, Gordon and Breach Science Publishers,Yveron, 1993.
H.H. Turdiev (BuxSU)
Short title
September 24, 2023
4 / 16
Blocks of Highlighted Text
First, note that for the non-selfadjoint operator
X
′′
(
x
) +
λ
2
X
(
x
) = 0 with
X
(0) =
X
(1)
,
X
′
(1) = 0
X
0
(
x
) = 2
,
X
2
k
(
x
) = 4 cos(2
π
kx
)
,
X
2
k
−
1
(
x
) = 4(1
−
x
) sin(2
π
kx
)
,
k
= 0
,
1
,
2
, . . .
(4)
and
Y
0
(
x
) =
x
,
Y
2
k
(
x
) =
x
cos(2
π
kx
)
,
Y
2
k
−
1
(
x
) = sin(2
π
kx
)
, λ
k
= 2
π
k
, ,
k
= 1
,
2
,
3
, . . . ,
(5)
which are Riesz bases in
L
2
[0; 1]. For more details, the reader can
consult
6
,
7
.
6
T. S. Aleroev, M. Kirane, and S. A. Malik, Determination of a source term for a
time fractional diffusion equation with an integral type over-deter mining condition,
Electronic Journal of Differential Equations, Vol. 270, pp. 1-16, 2013.
7
N. Ionkin, Solution of a boundary-value problem in heat conduction with
non-classical boundary condition, Differential Equations Vol. 13: pp. 204-211, 1977.
H.H. Turdiev (BuxSU)
Short title
September 24, 2023
5 / 16
By applying the Fourier method, the solution
u
(
x
,
t
) of the problem
(1)-(3) can be expanded in a uniformly convergent series in term of
eigenfunctions of the form
u
(
x
,
t
) =
X
0
(
x
)
u
0
(
t
) +
∞
X
n
=1
X
2
n
−
1
(
x
)
u
2
n
−
1
(
t
) +
∞
X
n
=1
X
2
n
(
x
)
u
2
n
(
t
)
.
(6)
In view of (1) for (
u
(
x
,
t
)
,
Y
0
(
x
)) =
u
0
(
t
), we obtain the Cauchy type
problem
D
α,β
0+
,
t
u
0
(
t
) +
q
(
t
)
u
0
(
t
) =
f
0
(
t
)
,
I
(2
−
α
)(1
−
β
)
0+
,
t
u
0
(
t
)
t
=0
=
φ
0
,
1
,
d
dt
I
(2
−
α
)(1
−
β
)
0+
,
t
u
0
(
t
)
t
=0
=
φ
0
,
2
.
(7)
For
u
2
n
−
1
(
t
) = (
u
(
x
,
t
)
,
Y
2
n
−
1
(
x
));
n
≥
1
,
in view of (1) we have
D
α,β
0+
,
t
u
2
n
−
1
(
t
) +
λ
2
n
u
2
n
−
1
+
q
(
t
)
u
2
n
−
1
(
t
) =
f
2
n
−
1
(
t
)
,
I
(2
−
α
)(1
−
β
)
0+
,
t
u
2
n
−
1
(
t
)
t
=0
=
φ
2
n
−
1
,
1
,
d
dt
I
(2
−
α
)(1
−
β
)
0+
,
t
u
2
n
−
1
(
t
)
t
=0
=
φ
2
n
−
1
,
2
.
(8)
H.H. Turdiev (BuxSU)
Short title
September 24, 2023
6 / 16
Also, the Cauchy type problem satisfied by
u
2
n
(
t
) = (
u
2
n
(
x
,
t
)
,
Y
2
n
(
x
)),
n
≥
1, are
D
α,β
0+
,
t
u
2
n
(
t
) +
λ
2
u
2
n
(
t
) + 2
λ
u
2
n
−
1
(
t
) +
q
(
t
)
u
2
n
(
t
) =
f
2
n
(
t
)
,
I
(2
−
α
)(1
−
β
)
0+
,
t
u
2
n
(
t
)
t
=0
=
φ
2
n
,
1
,
d
dt
I
(2
−
α
)(1
−
β
)
0+
,
t
u
2
n
(
t
)
t
=0
=
φ
2
n
,
2
.
(9)
We solve problems (7)-(9).
Based
8
, we have that the initial problems (7)-(9) is equivalent in the
space
C
α,β
γ
[0
,
T
] to the Volterra integral equation of the second kind
u
0
(
t
) =
t
(
β
−
1)(2
−
α
)
Γ(1 + (
β
−
1)(2
−
α
))
φ
0
,
1
+
t
1+(
β
−
1)(2
−
α
)
Γ(
α
+
β
(2
−
α
))
φ
0
,
2
+
+
1
Γ(
α
)
t
Z
0
(
t
−
τ
)
α
−
1
f
0
(
τ
)
d
τ
−
1
Γ(
α
)
t
Z
0
(
t
−
τ
)
α
−
1
q
(
τ
)
u
0
(
τ
)
d
τ.
(10)
8
T. Sandev, Z. Tomovski, Fractional Equations and Models, Springer Nature
Switzerland AG, 2019. pp. 61-114
H.H. Turdiev (BuxSU)
Short title
September 24, 2023
7 / 16
u
2
n
−
1
(
t
) =
t
(
β
−
1)(2
−
α
)
E
α,
1+(
β
−
1)(2
−
α
)
−
λ
2
t
α
φ
2
n
−
1
,
1
+
+
t
1+(
β
−
1)(2
−
α
)
E
α,α
+
β
(2
−
α
)
−
λ
2
t
α
φ
2
n
−
1
,
2
+
+
t
Z
0
(
t
−
τ
)
α
−
1
E
α,α
(
−
λ
2
n
(
t
−
τ
)
α
)
f
2
n
−
1
(
τ
)
d
τ
−
−
t
Z
0
(
t
−
τ
)
α
−
1
E
α,α
(
−
λ
2
n
(
t
−
τ
)
α
)
q
(
τ
)
u
2
n
−
1
(
τ
)
d
τ,
(11)
H.H. Turdiev (BuxSU)
Short title
September 24, 2023
8 / 16
u
2
n
(
t
) =
t
(
β
−
1)(2
−
α
)
E
α,
1+(
β
−
1)(2
−
α
)
−
λ
2
t
α
φ
2
n
,
1
+
+
t
1+(
β
−
1)(2
−
α
)
E
α,α
+
β
(2
−
α
)
−
λ
2
t
α
φ
2
n
,
2
+
+
t
Z
0
(
t
−
τ
)
α
−
1
E
α,α
(
−
λ
2
n
(
t
−
τ
)
α
)
f
2
n
(
τ
)
d
τ
−
−
2
λ
n
t
Z
0
(
t
−
τ
)
α
−
1
E
α,α
(
−
λ
2
n
(
t
−
τ
)
α
)
u
2
n
−
1
(
τ
)
d
τ
−
−
t
Z
0
(
t
−
τ
)
α
−
1
E
α,α
(
−
λ
2
n
(
t
−
τ
)
α
)
q
(
τ
)
u
2
n
(
τ
)
d
τ.
(12)
H.H. Turdiev (BuxSU)
Short title
September 24, 2023
9 / 16
Theorem 1
We have the estimates
t
γ
|
u
0
| ≤
t
γ
+(
β
−
1)(2
−
α
)
|
φ
0
,
1
|
Γ(1 + (
β
−
1)(2
−
α
))
+
t
1+
γ
+(
β
−
1)(2
−
α
)
|
φ
0
,
2
|
Γ(
α
+
β
(2
−
α
))
+
+
∥
f
0
∥
C
γ
[0
,
T
]
t
α
B
(
α,
1
−
γ
)
Γ(
α
+ 1)
!
E
α,γ
∥
q
∥
C
[0
,
T
]
t
γ
1
α
+
γ
−
1
t
;
t
γ
D
α,β
0+
,
t
u
0
(
t
)
≤ ∥
f
0
∥
C
γ
[0
,
T
]
+
+
∥
q
∥
C
[0
,
T
]
t
γ
+(
β
−
1)(2
−
α
)
|
φ
0
,
1
|
Γ(1 + (
β
−
1)(2
−
α
))
+
t
1+
γ
+(
β
−
1)(2
−
α
)
|
φ
0
,
2
|
Γ(
α
+
β
(2
−
α
))
+
+
∥
f
0
∥
C
γ
[0
,
T
]
t
α
B
(
α,
1
−
γ
)
Γ(
α
+ 1)
!
E
α,γ
∥
q
∥
C
[0
,
T
]
t
γ
1
α
+
γ
−
1
t
,
t
∈
[0
,
T
]
,
where
1
> γ >
(1
−
β
)(2
−
α
).
H.H. Turdiev (BuxSU)
Short title
September 24, 2023
10 / 16
Theorem 2
For fixed n
∈
N we have the estimates
t
γ
|
u
2
n
−
1
|≤
t
γ
+(
β
−
1)(2
−
α
)
M
1
|
φ
2
n
−
1
,
1
|
+
t
1+
γ
+(
β
−
1)(2
−
α
)
M
2
|
φ
2
n
−
1
,
2
|
+
+
∥
f
2
n
−
1
∥
C
γ
[0
,
T
]
t
α
B
(
α,
1
−
γ
)
M
3
Γ(
α
+ 1)
!
E
α,γ
∥
q
∥
C
[0
,
T
]
t
γ
1
α
+
γ
−
1
t
;
t
γ
D
α,β
0+
,
t
u
2
n
−
1
(
t
)
≤ ∥
f
2
n
−
1
∥
C
γ
[0
,
T
]
+
+
λ
2
n
+
∥
q
∥
C
[0
,
T
]
t
γ
+(
β
−
1)(2
−
α
)
M
1
|
φ
2
n
−
1
,
1
|
+
t
1+
γ
+(
β
−
1)(2
−
α
)
M
2
|
φ
2
n
−
1
,
2
|
+
+
∥
f
2
n
−
1
∥
C
γ
[0
,
T
]
t
α
B
(
α,
1
−
γ
)
M
3
Γ(
α
+ 1)
!
E
α,γ
∥
q
∥
C
[0
,
T
]
t
γ
1
α
+
γ
−
1
t
,
where
1
> γ >
(1
−
β
)(2
−
α
).
H.H. Turdiev (BuxSU)
Short title
September 24, 2023
11 / 16
t
γ
|
u
2
n
|≤
t
γ
+(
β
−
1)(2
−
α
)
M
1
|
φ
2
n
,
1
|
+
+
t
1+
γ
+(
β
−
1)(2
−
α
)
M
2
|
φ
2
n
,
2
|
+
∥
f
2
n
∥
C
γ
[0
,
T
]
t
α
B
(
α,
1
−
γ
)
M
3
Γ(
α
+ 1)
+
+2
λ
n
∥
u
2
n
−
1
∥
C
α,β
γ
[0
,
T
]
t
α
B
(
α,
1
−
γ
)
Γ(
α
+ 1)
!
×
×
E
α,γ
∥
q
∥
C
[0
,
T
]
t
γ
1
α
+
γ
−
1
t
,
t
∈
[0
,
T
]
.
(13)
t
γ
D
α,β
0+
,
t
u
2
n
(
t
)
≤ ∥
f
2
n
∥
C
γ
[0
,
T
]
+ 2
λ
n
|
u
2
n
−
1
|
C
γ
[0
,
T
]
+
+
λ
2
n
+
∥
q
∥
C
[0
,
T
]
t
γ
+(
β
−
1)(2
−
α
)
M
1
|
φ
2
n
,
1
|
+
+
t
1+
γ
+(
β
−
1)(2
−
α
)
M
2
|
φ
2
n
,
2
|
+
∥
f
2
n
∥
C
γ
[0
,
T
]
t
α
B
(
α,
1
−
γ
)
M
3
Γ(
α
+ 1)
+
H.H. Turdiev (BuxSU)
Short title
September 24, 2023
12 / 16
+2
λ
n
∥
u
2
n
−
1
∥
C
α,β
γ
[0
,
T
]
t
α
B
(
α,
1
−
γ
)
Γ(
α
+ 1)
!
×
×
E
α,γ
∥
q
∥
C
[0
,
T
]
t
γ
1
α
+
γ
−
1
t
,
t
∈
[0
,
T
]
.
H.H. Turdiev (BuxSU)
Short title
September 24, 2023
13 / 16
Lemma
If the conditions A1), A2) are fulfilled then there are equalities
φ
n
,
i
=
1
λ
4
n
φ
(4)
n
,
i
,
i
= 1
,
2
,
f
n
=
1
λ
4
n
f
(4)
n
,
(14)
where
φ
(4)
n
,
i
=
1
R
0
φ
(4)
i
(
x
)
Y
n
(
x
)
dx
,
i
= 1
,
2
,
f
(4)
n
=
1
R
0
f
(4)
(
x
)
Y
n
(
x
)
dx
,
with
the following estimates:
∞
X
n
=1
φ
(4)
n
,
1
2
≤ ∥
φ
(4)
1
∥
L
2
[0
,
l
]
,
∞
X
n
=1
φ
(4)
n
,
2
2
≤ ∥
φ
(4)
2
∥
L
2
[0
,
l
]
,
∞
X
n
=1
f
(4)
n
2
≤ ∥
f
(4)
∥
L
2
[0
,
T
]
.
(15)
H.H. Turdiev (BuxSU)
Short title
September 24, 2023
14 / 16
Using the above results, we obtain the following assertion.
Theorem
. Let q
(
t
)
∈
C
[0
,
T
]
, A1), A2) are satisfied, then there exists a unique
solution of the direct problem (1)-(3) u
(
x
,
t
)
∈
C
2
,α,β
γ
(Ω)
.
where Ω :=
{
(
x
,
t
) : 0
≤
x
≤
1
,
0
≤
t
≤
T
}
.
H.H. Turdiev (BuxSU)
Short title
September 24, 2023
15 / 16
The End
H.H. Turdiev (BuxSU)
Short title
September 24, 2023
16 / 16
Document Outline - First Section
- Second Section
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