International scientific journal
“Interpretation and researches”
Volume 1 issue 1 (47) | ISSN: 2181-4163 | Impact Factor: 8.2
93
APPLICATION OF GAMMA FUNCTIONS FOR SOLVING ORDINARY
DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER
Khakbergenova Nargiza Fazilovna
Karakalpak state university after named Berdakh
Fractional-order equations play an important role in modeling processes that
cannot be described by standard integer-order differential equations. They are widely
used in physics, engineering, biology, and economics [1-2]. One approach to solving
such equations is to use the Gamma function, which makes it possible to solve
fractional differential equations efficiently. The article discusses the theoretical
foundations of the use of Gamma functions for solving fractional equations, as well
as examples of practical use of this method.
Fractional-order differential equations are a generalization of classical integer-
order differential equations. They describe the dynamics of systems where processes
change gradually and do not obey the classical laws of differentiation. Fractional-
order equations describe phenomena such as the diffusion and damping of vibrations,
which are often found in nature and technology. One of the methods for solving such
equations is to use the Gamma function, which is an important mathematical tool for
working with fractional derivatives [3].
Gamma function defined as
( )
( )
1
0
,
0,
z
t
Г z
t
e dt
R z
−
−
=
it is a generalization of the factorial for complex numbers. For positive integers
n
, the Gamma function takes the form,
( ) (
)
1 !
n
n
=
−
for non-integer values ,
x
( )
x
−
, it is a continuous function, which makes it
extremely useful when working with fractional orders in differential equations.
The gamma function has several important properties [4] that make it useful for
solving fractional differential equations:
Recurrent relation
(
)
( )
1
,
x
x
x
+ =
Relation with the Betta function,
( )
( ) ( )
(
)
,
.
x
y
B x y
x
y
=
+
The gamma function is widely used in the theory of fractional derivatives, for
example, in equations that describe processes with memory and inertia.
In the context of fractional derivatives, the Gamma function plays a key role in
defining fractional integrals and derivatives.
International scientific journal
“Interpretation and researches”
Volume 1 issue 1 (47) | ISSN: 2181-4163 | Impact Factor: 8.2
94
The fractional derivative
( )
f t
of an order function
(
)
0
1
can be defined
in terms of the Riemann-Liouville integral,
( )
(
)
(
)
( )
0
1
,
1
t
d
D f t
t
f
d
Г
dt
−
=
−
−
This approach makes it possible to relate the solution of fractional-order
equations to the Gamma function, which leads to analytical solutions for various
types of equations.
We give examples of solving ordinary differential equations of fractional order
using Gamma functions
1.
Solvethe m equation
3
2
3
2
.
d y
x
dx
=
Multiply both parts by
1
2
1
2
d
dx
then we get
1
2
2
1
2
2
d y
x
dx
dx
=
(1)
we simplify the right-hand side of equation (1) by using [2 ],
(
)
2
(
1)
(
)
,
2 1
n
n
Г n
f
x
x
Г n
−
+
=
− +
(2)
here
1
,
2
=
1
2
n
=
, then
( )
1
1
1 1
2
2
2 2
1
1
1
1
2
2
,
1
1
1
1
2
2
f
x
x
−
+
+
=
=
− +
(3)
Now from (3) we find the Gamma functions, then
1
1
(2 )!
(
)
(
1)
,
2
2
4
!
n
n
n
n
+
= + =
1
2
(
1)
2
4
2
+ =
=
substituting in (3) we obtain the equation
,
2
y
=
integrating it we have
2
1
2
,
4
y
x
C x
C
=
+
+
where
1
2
,
сonst.
С С
−
International scientific journal
“Interpretation and researches”
Volume 1 issue 1 (47) | ISSN: 2181-4163 | Impact Factor: 8.2
95
2.
Solvethe m equation
1
3
2
2
2
1
2
8
.
3
d y
x
x
dx
=
+
by acting on both parts with the
differentiation operator, we get
1
3
2
2
2
1
2
8
,
3
dy
d
x
x
dx
dx
=
+
(4)
Simplifying the right-hand side of (4) by the formula (2), we obtainm
3
2
8
3
8
8
3
dy
x
x
dx
=
+
(5)
(5)
Iuse an equation with separable variables, solving we have
5
2
2
16
,
5
2
x
y
x
C
=
+
+
where
.
С const
−
3. Solve the equation
1
2
1
2
.
d y
x
dx
=
acting on both sides of the equation with the differentiation operator, we have:
1
1
2
4
1
2
dy
d
x
dx
x
=
transforming the right-hand side by formula (2), we have
1
4
5
4
,
3
4
dy
x
dx
−
=
integrating we get a solution in the form
3
4
5
4
4
,
.
3
3
4
y
x
C С
сonst
=
+
−
The gamma function plays a key role in solving fractional differential equations.
It is used to calculate fractional derivatives, normalize solutions, and simplify
expressions for more complex physical processes.
References:
1. Podlubny, I. (1999). Fractional Differential Equations. Academic Press.
International scientific journal
“Interpretation and researches”
Volume 1 issue 1 (47) | ISSN: 2181-4163 | Impact Factor: 8.2
96
2. Kilbas, A. A., Srivastava, H. M., & Trujillo, J. J. (2006). Theory and
Applications of Fractional Differential Equations. Elsevier.
3. Gradshteyn, I. S., & Ryzhik, I. M. (2007). Table of Integrals, Series, and
Products. Academic Press.
4. Bateman, H., & Erdey, M. (1954). Higher Transcendental Functions. Volume
I. McGraw-Hill.
