Volume 4, issue 6, 2025
280
SOLVING FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS AND THEIR
APPLICATIONS IN HEAT CONDUCTION AND BIOLOGICAL MODELING
Saodat Kurbonkulova Burkhon kizi
3rd-year student, Chirchik State Pedagogical University
saodatqurbonqulova691@gmail.com
Annotation:
This article analyzes the method of solving fractional integro-differential equations
using the Neumann series. The convergence of the solution for equations based on the Caputo
fractional derivative is proven by applying the Banach fixed-point theorem. The efficiency of the
method is demonstrated through numerical examples and is applied to problems of heat
conduction and biological population dynamics. The results are of significant importance in
modern materials science and biology, and contribute to the advancement of fractional
mathematics within the scientific community of Uzbekistan.
Keywords:
fractional integro-differential equations, Neumann series, Caputo derivative,
convergence, heat conduction, biological modeling.
Introduction.
In recent years, fractional differential and integro-differential equations have taken
on an important role in mathematics and applied sciences [1, 2]. They are used to model
memory-effect processes, such as anomalous diffusion, viscoelastic materials, biological
population dynamics, and signal transmission [3, 4]. In Uzbekistan’s mathematical school,
research on fractional equations is evolving, particularly through the works of Kadirkulov and
Khudaybergenov
[5].
Compared to classical differential equations, fractional equations provide a more accurate
description of the behavior of complex systems. However, obtaining analytical solutions for such
equations is challenging; hence, iterative methods, especially the Neumann series, are widely
employed [6]. The Neumann series is an effective tool for solving integral equations, and its
convergence is justified through the Banach fixed-point theorem [7].
This article examines the method of solving fractional integro-differential equations via the
Neumann series. The convergence of the solution is analyzed, and the method is applied to heat
conduction and biological population dynamics problems. The aim of this paper is to
demonstrate the effectiveness of the method, verify it through numerical simulations, and
contribute to the development of fractional mathematics in the scientific context of Uzbekistan.
Definition 1.
The Caputo fractional derivative of order α\alphaα is defined as follows:
'
0
1
( )
(
)
( )
(1
)
t
D u t
t s u s ds
Г
a
a
a
-
=
-
-
where
1
0
( )
t z
Г z
e t dt
-
-
=
is the gamma function [1].
Definition 2.
An integral equation of the form
Volume 4, issue 6, 2025
281
0
( )
( )
( , )
( )
t
u t
f t
K t s D u s ds
a
=
+
,
[ ]
0,
t
T
in which the upper limit of integration depends on a free variable, is called a Volterra integral
equation of the second kind. Here,
( )
u t
is the unknown function to be determined,
( )
f t
is a
given
function,
( , )
K t s
is
the
kernel
function ,where
[ ]
( )
0,
f t C T
,
[ ] [ ]
( , )
( 0,
0, )
K t s C
T
T
.
Definition 3.
The Neumann series is an iterative method for solving integral equations and is
defined as:
0
( )
( )
u t
f t
=
,
1
0
( )
( )
( , )
( )
t
n
n
u t
f t
K t s D u s ds
a
+
=
+
,
0,1,2,.......
n
=
If the convergence conditions are satisfied, the solution can be expressed as:
( ) lim
( )
n
n
u t
u t
®
=
[7].
Theorem 1. (Banach Fixed-Point Theorem).
If the operator
0
:
( )
( , ) ( )
t
A u
f t
K t s D s ds
a
®
+
is a contraction, that is,
Au Av q u v
-
-
,
1
q
<
hen the equation has a unique solution,
and the Neumann series converges to this solution [8].
Now, let us consider the application of the Neumann series to fractional integro-differential
equations
and
analyze
the
convergence
of
the
solution.
Let us examine the following equation:
0
( )
( )
( , )
( )
t
u t
f t
K t s D u s ds
a
=
+
,
[ ]
0,
t
T
here,
[ ]
( )
0,
f t C T
,
[ ] [ ]
( , )
( 0,
0, )
K t s C
T
T
and
D
a
is the Caputo fractional derivative.
The operator is defined as follows:
0
( )
( )
( , )
( )
t
Au t
f t
K t s D u s ds
a
=
+
The Neumann series iterations are given by:
0
( )
( )
u t
f t
=
,
1
0
( )
( )
( , )
( )
t
n
n
u t
f t
K t s D u s ds
a
+
=
+
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282
To demonstrate that the operator
A
is contractive, we verify the following condition:
0
( , )
( ( )
( ))
t
Au Av
K t s D u s v s ds
a
-
=
-
-
Regarding the properties of the Caputo derivative:
1
(
)
(2
)
T
D u v
u v
Г
a
a
a
-
-
-
-
If
[ ]
0
0,
0
sup
( , )
t
t
T
K t s ds K
then
1
0
(2
)
T
Au Av K
u v q u v
Г
a
a
-
-
- =
-
-
,
where
1
0
1
(2
)
T
q K
Г
a
a
-
=
<
-
.Therefore, the operator
A
is a contraction, and by Banach's
fixed-point theorem, the Neumann series converges to a unique solution.
Example.
Let us consider the equation for the parameters
2
( )
f t
t
=
,
( , )
s t
K t s
e
-
=
,
0,5
a
=
,
1
T
=
.
2
0,5
0
( )
( )
t
s t
u t
t
e D u s ds
-
= +
tamiz First, let us examine the first two iterations of the Neumann series..
2
0
( )
u t
t
=
,
2
0,5
2
1
0
( )
( )
t
s t
u t
t
e D s ds
-
= +
,
3
0,5
2
0,5
0
1
4
( )
(
) 2
(0,5)
3
s
s
D s
s
d
Г
t
t t
p
-
=
-
=
3
2
1
0
4
( )
3
t
s t
s
u t
t
e
ds
p
-
= +
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283
The convergence of the iterated kernels mentioned above can be visualized by generating their
graphs using Python.
import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import quad
from scipy.special import gamma
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284
Now,
let
us
examine
the
practical
application
of
the
Neumann
series.
Fractional integro-differential equations are applied in two important fields:
heat conduction
and
biological modeling
.
The fractional heat equation is expressed as follows:
2
2
( , )
u
D u t x
k
x
a
¶
=
¶
,
0
1
a
< <
here
( , )
u x t
represents the temperature distribution,
k
- is the thermal conductivity coefficient.
This equation is transformed into an integral form and the Neumann series is applied to obtain
the solution:
0
0
( , )
(
, )
( , )
t
u t x u
G t s x D u s x ds
a
=
+
-
bu yerda
( , )
G t x
-Grin funksiyasi.
The following figure illustrates
0,7
a
=
how anomalous diffusion behaves differently from
classical diffusion.
T = 1; alpha = 0.5; n = 100
t = np.linspace(0, T, n)
f = lambda t: t**2
K = lambda t, s: np.exp(s - t)
Gamma = gamma(1.5)
u0 = f(t)
u1 = np.zeros(n)
for i in range(n):
integrand = lambda s: K(t[i], s) * (4 * np.sqrt(s**3) / (3 * np.sqrt(np.pi))) # D^0.5 t^2
u1[i] = f(t[i]) + quad(integrand, 0, t[i])[0]
plt.plot(t, u0, 'b-', label='u_0(t) = t^2', linewidth=2)
plt.plot(t, u1, 'r--', label='u_1(t)', linewidth=2)
plt.grid(True)
plt.xlabel('t')
plt.ylabel('u(t)')
plt.title('Rasm 1: Volterra tenglamasi uchun Neumann qorisining iteratsiyalari')
plt.legend()
plt.savefig('rasm1.png', dpi=300)
plt.savefig('rasm1.pdf')
plt.show()
Volume 4, issue 6, 2025
285
Conclusion.
In this article, the method of solving fractional integro-differential equations using
the Neumann series was analyzed. The convergence of the solution was proven based on
Banach’s theorem and validated through numerical examples. It was demonstrated that the
method can be effectively applied in heat conduction and biological modeling. The results
contribute to the advancement of fractional mathematics within the scientific environment of
Uzbekistan. Future research may focus on extending the application of the Neumann series to
import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import quad
from scipy.special import gamma
T = 1 # Vaqt oralig'i
alpha = 0.7 # Fraksiyali tartib
n = 100 # Nuqtalar soni
t = np.linspace(0, T, n)
k = 1 # Diffuziya koeffitsienti (o'zgartirish mumkin)
u0 = lambda x: np.ones_like(x) # Boshlang'ich shart: u0(x) = 1
G = lambda t_s, x: np.exp(-(t_s)) # Yadro funksiyasi: e^-(t-s)
Gamma = gamma(1 - alpha) # Gamma(0.3) uchun koefitsient
u = [u0(t)] # u0(t) = 1
for k_iter in range(10): # 10 iteratsiyagacha hisoblash
u_new = np.zeros(n)
for i in range(n):
integrand = lambda s: G(t[i] - s, 0) * (u[-1][np.argmin(np.abs(t - s))] / Gamma * (t[i] - s)**(-alpha))
plt.plot(t, u[0], 'b-', label='u_0(t) = 1', linewidth=2) # Boshlang'ich taxmin
plt.plot(t, u[1], 'r--', label='u_1(t)', linewidth=2) # Birinchi iteratsiya
plt.plot(t, u[2], 'g-.', label='u_2(t)', linewidth=2) # Ikkinchi iteratsiya
plt.plot(t, u[-1], 'k-', label='Umumiy yechim (sonli yaqinlash)', linewidth=2)
plt.grid(True)
plt.xlabel('t (vaqt)')
plt.ylabel('u(t) (harorat)')
plt.title(Neumann qorisining 3 iteratsiyasi va umumiy yechim')
plt.legend()
plt.ylim(0, 2)
plt.savefig('issiqlik_grafik.png', dpi=300)
plt.savefig('issiqlik_grafik.pdf')
plt.show()
Volume 4, issue 6, 2025
286
more complex systems, such as multi-dimensional fractional equations or real-time
computational algorithms.
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2.
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