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THE SYSTEM OF WAVE EQUATIONS INVOLVED BY THE RIMAN-
LIOUVILLE FRACTIONAL-ORDER OPERATOR INTO THE
CANONICAL FORM.
Bakhronova .S.B
Asian International University, teacher of the "General Technical Sciences"
department
Abstract.
In this article, the conversion of the system of fractional-order wave
oscillation equations with initial conditions into the canonical form is studied, and
the achieved results are presented.
Key words :
Riemann-Liouville fractional derivative, initial condition, integral
equation,
Today, it is necessary to stimulate scientific research and innovative activities,
to create effective mechanisms for the implementation of scientific and innovative
achievements in practice, to establish specialized research and experimental
laboratories, high technology centers and technological parks at universities and
research institutes. important tasks have been defined. In particular, a lot of
significant work is being done in the field of mathematics. As a clear example of
this, the decision of the President of the Republic of Uzbekistan dated 05.07.2020
on measures to improve the quality of education in the field of mathematics and
research development No. We can cite the Decree of the President of the Republic
of Uzbekistan No. PF-5847 on approval of the concept of development of the
transport system until 2030.
In recent years, the field of fractional calculus has attracted the interest of
researchers in several fields such as mathematics, physics, chemistry, engineering,
and economics and social sciences. Today, fractional calculus is a new branch of
mathematical research that deals with the investigation and application of whole-
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order derivatives and integrals. Advanced systematic research was done and studied
by Liuville, Riemann, Leibniz, Caputo and other scientists in the 19th century.
The fractional derivative of the exponential function obtained by Liouville in
1832 and the fractional derivative of the power function obtained by Riemann in
1847 were studied [1]. In other words, there are several definitions for the derivative
of a fraction, and all of them are mathematically correct.
The initial, initial boundary value problem for fractional order differential
equations has been studied by many researchers.
Let's look at the system of differential equations with a fractional part:
(𝐼 ⋅ 𝐷
𝑡
𝛼
+ 𝐴
𝜕
𝜕𝑥
+ 𝐵
𝜕
𝜕𝑦
) 𝑈(𝑥, 𝑦, 𝑡) = 𝑓(𝑥, 𝑦, 𝑡), (1)
Here
𝑈 = (𝜌, 𝑢, 𝑣)
∗
,
∗
- the transposition sign,
𝐷
𝑡
𝛼
and
0 < 𝛼 < 1
the Riemann-
Liouville fractional derivative
of order ,
𝐴 = (
0
𝜌
0
𝑐
𝑜
2
0
1
𝜌
𝑜
0
0
0
0
0
) ,
𝐵 = (
0
0 𝜌
0
𝑐
𝑜
2
0
0
0
1
𝜌
𝑜
0
0
) ,
𝑓 = (
𝑓
1
𝑓
2
𝑓
3
) (𝑥, 𝑦, 𝑡).
Definition 1
so
𝑛 − 1 < 𝛼 < 𝑛
the Riemann-Liouville
fractional integral
and derivative of order a are defined as follows:
𝐼
𝑡
𝑛−𝛼
𝑢(𝑥, 𝑡) =
1
Γ(n − α)
∫(t − τ)
n−α−1
t
0
u(x, τ)dτ,
and
𝐷
𝑡
𝛼
𝑢(𝑥, 𝑡) = (
𝑑
𝑑 𝑡
)
𝑛
(𝐼
𝑡
𝑛−𝛼
𝑢(𝑥, 𝑡))(𝑥, 𝑡), 𝛼 ∈ 𝐶, 𝑅𝑒(𝛼) > 0, 𝑛 < [𝑅𝑒(𝛼)] + 1.
Now , with respect to the variables
𝑡
and
𝑥
, we bring the system (1) into the
canonical form. Let's make an equation for this
|𝐴 − 𝜆𝐼| = 0, (2)
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where is
𝐼 − 3 × 3
the dimensional unit matrix.
𝜆
The last equation with respect
to has the following solutions:
𝜆
1
= 𝑐
0
𝜆
2
= −𝑐
0
𝜆
3
= 0.
Now we choose a non-degenerate matrix
T(
𝑥
3
,t) so that the equality is as
follows
𝑇
−1
𝐴𝑇 = Λ (3)
Here
Λ
is a diagonal matrix whose diagonal A contains the eigenvalues of the
matrix.
𝛬 = 𝑑𝑖𝑎𝑔 (𝑐
0
, −𝑐
0
, 0)
Formula (3) assumes equality:
𝐴𝑇 = TΛ.
So, T is a column with number I of the matrix. Mathematical calculations
show that can choose as matrices (not unique)
𝑇
and
𝑇
−1
and satisfying the above
conditions.
𝑇 = (
𝜌
0
𝑐
0
𝜌
0
𝑐
0
0
1
−1
0
0
0
1
),
and
𝑇
−1
=
1
−2𝜌
0
𝑐
0
(
−1 −𝜌
0
𝑐
0
0
−1
𝜌
0
𝑐
0
0
0
0
−2𝜌
0
𝑐
0
).
𝑈
we express the vector function with the following equation
𝑈 = 𝑇𝑉.
this expression in equation (1)- σ hi:
𝐼𝑇𝐷
𝑡
𝛼
𝑉(𝑥, 𝑦, 𝑡) + 𝐴𝑇
𝜕
𝜕𝑥
𝑉(𝑥, 𝑦, 𝑡) + 𝐵𝑇
𝜕
𝜕𝑦
𝑉(𝑥, 𝑦, 𝑡) = 𝑓(𝑥, 𝑦, 𝑡).
We multiply it by
𝑇
−1
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𝑇
−1
𝐼𝑇𝐷
𝑡
𝛼
𝑉(𝑥, 𝑦, 𝑡) + 𝑇
−1
𝐴𝑇
𝜕
𝜕𝑥
𝑉(𝑥, 𝑦, 𝑡) +
+𝑇
−1
𝐵𝑇
𝜕
𝜕𝑦
𝑉(𝑥, 𝑦, 𝑡) = 𝑇
−1
𝑓(𝑥, 𝑦, 𝑡).
Then we get the following result:
𝐸𝐷
𝑡
𝛼
𝑉(𝑥, 𝑦, 𝑡) + 𝛬
𝜕
𝜕𝑥
𝑉(𝑥, 𝑦, 𝑡) + 𝐵
1
𝜕
𝜕𝑦
𝑉(𝑥, 𝑦, 𝑡) = 𝐹(𝑥, 𝑦, 𝑡), (4)
here
𝑇
−1
𝐼𝑇 = 𝐸 = (
1
0 0
0
1 0
0
0 1
) ,
𝛬 = (
𝑐
0
0
0
0
−𝑐
0
0
0
0
0
),
𝐵
1
=
(
0
0
1
2
𝑐
0
0
0
1
2
𝑐
0
𝑐
0
𝑐
0
1 )
, 𝐹(𝑥, 𝑦, 𝑡) =
(
1
2𝜌
0
𝑐
0
𝑓
1
(𝑥, 𝑦, 𝑡) +
1
2
𝑓
2
(𝑥, 𝑦, 𝑡)
1
2𝜌
0
𝑐
0
𝑓
1
(𝑥, 𝑦, 𝑡) −
1
2
𝑓
2
(𝑥, 𝑦, 𝑡)
𝑓(𝑥, 𝑦, 𝑡)
)
.
Results achieved
The Riemann-Liuviil fractional operator was involved
(𝐼 ⋅ 𝐷
𝑡
𝛼
+ 𝐴
𝜕
𝜕𝑥
+ 𝐵
𝜕
𝜕𝑦
) 𝑈(𝑥, 𝑦, 𝑡) = 𝑓(𝑥, 𝑦, 𝑡)
as a result of normalizing the system of differential equations, we got the
following result:
𝐸𝐷
𝑡
𝛼
𝑉(𝑥, 𝑦, 𝑡) + 𝛬
𝜕
𝜕𝑥
𝑉(𝑥, 𝑦, 𝑡) + 𝐵
1
𝜕
𝜕𝑦
𝑉(𝑥, 𝑦, 𝑡) = 𝐹(𝑥, 𝑦, 𝑡)
Summary
Thus, in this article, the canonicalization of the system of fractional differential
equations was studied.
Studying the system of fractional differential equations provides the following
opportunities:
1) Research of tectonic movements in underground layers in geology;
2) Solving thermodynamic problems;
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3) Building models in thermoelasticity issues.
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