Volume 04 Issue 06-2024
76
American Journal Of Applied Science And Technology
(ISSN
–
2771-2745)
VOLUME
04
ISSUE
06
Pages:
76-81
OCLC
–
1121105677
Publisher:
Oscar Publishing Services
Servi
ABSTRACT
The article, "Application of Differential Equations in Various Fields of Science," explores the use of differential
equations for modeling economic and natural phenomena. It examines two main models of economic dynamics: the
Evans model for the market of a single product, and the Solow model for economic growth.
The author emphasizes the importance of proving the existence of solutions to differential equations in order to verify
the accuracy of mathematical models. They also discuss the role of electronic computers in developing the theory of
differential equations and its connection with other branches of mathematics such as functional analysis, algebra, and
probability theory.
Furthermore, the article highlights the significance of various solution methods for differential equations, including
the Fourier method, Ritz method, Galerkin method, and perturbation theory.Special attention is paid to the theory of
partial differential equations, the theory of differential operators, and problems arising in physics, mechanics, and
technology. Differential equations are the theoretical foundation of almost all scientific and technological models and
a key tool for understanding various processes in science, such as in physics, chemistry, and biology.
Examples of processes described by differential equations include normal reproduction, explosive growth, and the
logistic curve. Cases of using differential equations to model deterministic, finite-dimensional, and differentiable
phenomena, as well as the impact of catch quotas on population dynamics, are discussed.
In conclusion, the significance of differential equations for research and their role in stimulating the development of
new mathematical areas is emphasized.
KEYWORDS
Research Article
APPLICATION OF DIFFERENTIAL EQUATIONS IN VARIOUS FIELDS OF
SCIENCE
Submission Date:
June 20, 2024,
Accepted Date:
June 25, 2024,
Published Date:
June 30, 2024
Crossref doi:
https://doi.org/10.37547/ajast/Volume04Issue06-15
Kosimova Marjona
M
aster’s degree student of National university of Uzbekistan in mathematics major
, Uzbekistan
Journal
Website:
https://theusajournals.
com/index.php/ajast
Copyright:
Original
content from this work
may be used under the
terms of the creative
commons
attributes
4.0 licence.
Volume 04 Issue 06-2024
77
American Journal Of Applied Science And Technology
(ISSN
–
2771-2745)
VOLUME
04
ISSUE
06
Pages:
76-81
OCLC
–
1121105677
Publisher:
Oscar Publishing Services
Servi
Differential equations, mathematical modeling, economic dynamics, the Evans model, the Solow model, the theorems
on the existence of solutions, evolutionary processes, the Fourier method, the Ritz method, the Galerkin method,
partial differential equations, the averaging theory, physics, chemistry, biology, the logistic curve, catch quotas, the
theory of differential operators, the explosion equation, and normal reproduction.
INTRODUCTION
Mathematical modeling of economic and natural
processes leads to the need to solve equations that, in
addition to independent variables and the desired
functions dependent on them, also contain derivatives
or differentials of unknown functions. Such equations
are called differential.
Differential equations are widely used in models of
economic dynamics, in which they study not only the
dependence of variables on time, but also on their
relationship in time. Such models are: the Evans model
- establishing a balanced price in the market for one
product; as well as a dynamic model of economic
growth, known as the “basic Solow model”.
In the Evans model, the market for a single product is
considered, time is considered continuous. Let
( ), ( ),
( )
d t
s t
p t
- demand, supply and price
according to this product at a time.
Suppose that
supply and demand are linear functions of price, that
is,
( )
d p
a
bp
= −
,
,
0
a b
–
demand decreases
with increasing prices, and,
( )
s p
p
= +
,
,
0
–
supply increases with increasing prices.
The ratio
a
is natural, that is, at zero price,
demand exceeds supply.
The Solow model considers the economy as a whole
(without structural units). This model adequately
reflects the most important macroeconomic aspects of
the production process.
It is important to note that to verify the correctness of
the mathematical model, the existence theorems of
solutions to the corresponding differential equations
are very important, since the mathematical model is
not always adequate to a specific phenomenon and the
existence of a solution to a real problem (physical,
chemical, biological) does not imply the existence of a
solution to the corresponding mathematical problem.
Currently, the use of modern electronic computers
plays an important role in the development of the
theory of differential equations. The study of
differential equations often makes it easier to conduct
a computational experiment to identify certain
properties of their solutions, which can then be
theoretically justified and will serve as the foundation
for further theoretical research.
So, the first feature of the theory of differential
equations is its close connection with applications. In
other words, we can say that the theory of differential
equations was born from applications.
The second feature of the theory of differential
equations is its connection with other branches of
mathematics, such as functional analysis, algebra, and
probability theory.
In the study of specific differential equations that arise
in the process of solving physical problems, methods
are often created that have great commonality and are
applied without a rigorous mathematical justification
to a wide range of mathematical problems. Such
methods are, for example, the Fourier method, the Ritz
method, the Galerkin method, perturbation theory
methods, and others. The effectiveness of the
Volume 04 Issue 06-2024
78
American Journal Of Applied Science And Technology
(ISSN
–
2771-2745)
VOLUME
04
ISSUE
06
Pages:
76-81
OCLC
–
1121105677
Publisher:
Oscar Publishing Services
Servi
application of these methods was one of the reasons
for the attempts of their rigorous mathematical
justification. This led to the creation of new
mathematical theories, new areas of research.
Currently, the theory of partial differential equations is
a rich, highly branched theory. The theory of boundary
value problems for elliptic operators is constructed on
the basis of a recently created new apparatus - the
theory of pseudo-differential operators, the index
problem is solved, and mixed problems for hyperbolic
equations are studied.
In recent decades, a new branch of the theory of partial
differential equations has arisen and is developing
intensively - the theory of averaging of differential
operators. This theory arose under the influence of the
problems of physics, continuum mechanics and
technology, in particular, related to the study of
composites (highly heterogeneous materials widely
used at present in engineering), porous media, and
perforated materials.
Thus, differential equations are currently a complex
collection of facts, ideas, and methods that are very
useful for applications and stimulating theoretical
research in all other branches of mathematics. Many
sections of the theory of differential equations have
grown so much that they have become independent
sciences. It can be said that most of the paths
connecting abstract mathematical theories and natural
science applications go through differential equations.
Differential equations are the theoretical basis of
almost all models used in science and technology. Such
processes are reflected in physics, chemistry, biology
and almost all areas of science. Almost all problems of
physics lead to the need to solve differential equations.
This is due to the fact that virtually all physical laws that
describe physical phenomena are differential
equations for certain functions that describe these
phenomena. Such physical laws are a theoretical
generalization of numerous experiments and
characterize the evolution of the sought quantities in
the general case, both in space and in time. The
solution of differential equations is a key task for many
areas of human activity, and also plays an important
role in the knowledge of the world.
The theory of ordinary differential equations is one of
the main tools of mathematical natural science. This
theory allows us to study all kinds of evolutionary
processes with the properties of determinism, finite-
dimensionality and differentiability. Before giving
exact mathematical definitions, consider a few
examples.
1. Examples of evolutionary processes
. A process is
called deterministic if its entire future course and its
entire past are uniquely determined by the current
state. Many of the various states of the process are
called phase space. So, for example, classical
mechanics considers the movement of systems whose
future and past are uniquely determined by the initial
positions and initial velocities of all points of the
system. The phase space of a mechanical system is a
set whose element is a set of positions and velocities
of all points of a given system. Particle motion in
quantum mechanics is not described by a deterministic
process. Heat distribution is a semi-deterministic
process: the future is determined by the present, but
the past is not. A process is called finite-dimensional if
its phase space is finite-dimensional, that is, if the
number of parameters needed to describe its state is
finite. So, for example, Newtonian mechanics of
systems from a finite number of material points or
solids belongs to this class. The dimension of the phase
space of a system of material points is 6n, and that of a
system of n solids is
––
. Fluid motions studied in
Volume 04 Issue 06-2024
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American Journal Of Applied Science And Technology
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VOLUME
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Pages:
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OCLC
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Publisher:
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hydrodynamics, processes of string and membrane
vibrations, wave propagation in optics and acoustics
are examples of processes that cannot be described
using finite-dimensional phase space.
A process is called differentiable if its phase space has
the structure of a differentiable manifold, and a change
in state over time is described by differentiable
functions. So, for example, the coordinates and speeds
of the points of a mechanical system change over time
in a differentiable way.
The movements studied in shock theory do not
possess the differentiability property.
Thus, the motion of a system in classical mechanics can
be described using ordinary differential equations,
while quantum mechanics, the theory of thermal
conductivity, hydrodynamics, the theory of elasticity,
optics, acoustics, and impact theory require other
means.
Two more examples of deterministic finite-dimensional
and differentiable processes: the process of
radioactive decay and the process of reproduction of
bacteria with a sufficient amount of nutrient. In both
cases, the phase space is one-dimensional: the state of
the process is determined by the amount of substance
or the number of bacteria. In both cases, the process is
described by an ordinary differential equation.
We note that the form of the differential equation of
the process, as well as the very fact of the determinism,
finite-dimensionality, and differentiability of a
particular
process
can
only
be
established
experimentally, therefore, only with a certain degree
of accuracy. In the future, we will not emphasize this
circumstance every time and will talk about real
processes as if they exactly coincided with our
idealized mathematical models.
2. An evolution equation with one-dimensional phase
space.
Consider the equation
( ),
x
v x
x
=
. (1)
This equation describes an evolutionary process with
one-dimensional phase space. The right side defines
the vector field of the phase velocity: a vector is
applied at the point. Such an equation, the right side of
which is independent of, is called autonomous. The
evolution rate of an autonomous system, that is, a
system that does not interact with others, is
determined only by the state of this system: the laws
of nature do not depend on time.
The solution to this equation is given by the formula
0
0
( )
x
x
d
t
t
v
− =
(2)
3. Example: the equation of normal reproduction
.
Suppose that the size of the biological population (for
example, the number of bacteria in a Petri dish or fish
in a pond) is equal and that the growth rate is
proportional to the number of individuals present. The
breeding equation (This assumption is approximately
satisfied, while there is a lot of food.)
Our assumption is expressed by the differential
equation of normal reproduction
,
0
x
kx k
=
(3)
According to the meaning of the problem
0
x
, so
that the direction field is specified in the half-plane; It
is clear from the form of the direction field that it
x
grows with growth
t
, but it is not clear whether the
infinite values
x
will be reached in a finite time (the
vertical asymptote of the integral curve
t
) or does the
Volume 04 Issue 06-2024
80
American Journal Of Applied Science And Technology
(ISSN
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VOLUME
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Pages:
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OCLC
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1121105677
Publisher:
Oscar Publishing Services
Servi
solution remain finite for all? Along with the future,
the past is also unclear: will the integral curve tend to
the axis
0
x
=
while striving for a finite negative limit
t
or an infinite?
Fortunately, the breeding equation is solved explicitly.
0
0
(
)
0
0
0
0
,
(
) ln
,
x
k t t
x
d
x
t t
k t t
x e
x
k
x
−
− =
−
=
=
(4)
This theorem was discovered by Barrow
precisely when solving the simplest differential
equations, now called equations with separable
variables. Therefore, the solutions of the normal
multiplication equation grow exponentially at
t
→ +
and exponentially decrease at
t
→ −
;
neither infinite nor zero values
x
are reached at finite
t
. To double the population according to the
equation of normal reproduction, it is therefore
always necessary the same time, regardless of its
number.
4. Example: explosion equation
. Now suppose that the
growth rate is proportional not to the number of
individuals, but to the number of pairs:
2
x
kx
=
.
(5)
In this case, with large growths it is much faster than
normal, and with small growths it is much slower (this
situation is more likely to occur in physicochemical
problems, where the reaction rate is proportional to
the concentrations of both reagents; however, it is
now so difficult for some whales to find a mate that
whale breeding obeys equation (5), moreover, a little).
The field of directions does not seem to differ much
from that for ordinary reproduction, but calculations
show that the integral curves behave completely
differently. Assume for simplicity that
1
k
=
. Using
Barrow's formula, we find a solution
2
dx
t
C
x
=
+
, that is
1
x
t
C
= −
−
at
t
C
, that is at. Integral
curves - half hyperbole. Hyperbola has a vertical
asymptote.
So, if population growth is proportional to the number
of pairs, then the number of population becomes
infinitely large in a finite time. Physically, this
conclusion corresponds to the explosive nature of the
process. (Of course, at if too close to
C
, the
idealization adopted in the description of the process
by the differential equation is not applicable, so that
the real number of people in a finite time does not
reach infinite values.).
5. Example: logistic curve.
The usual breeding equation
is suitable only as long as the number of individuals is
not too large. With an increase in the number of
individuals, competition due to food leads to a
decrease in the rate of growth. The simplest
assumption is that the coefficient
k
depends on
x
how a linear inhomogeneous function (if not too large,
any smooth function can be approximated by a linear
inhomogeneous function) :
k
a
bx
= −
. Thus we
come to the competition equation taking into account
competition
(
)
x
a
bx x
=
−
. The coefficients
a
and
b
can be turned into a unit by choosing the scales
t
and
x
. We get the so-called
logistic equation
.
(1
)
x
x x
= −
.
The vector field of the phase velocity
v
and the
field of directions on the plane
( , )
t x
. We see that
Volume 04 Issue 06-2024
81
American Journal Of Applied Science And Technology
(ISSN
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2771-2745)
VOLUME
04
ISSUE
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Pages:
76-81
OCLC
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1121105677
Publisher:
Oscar Publishing Services
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1)
the process has two equilibrium positions:
0
x
=
and
1
x
=
;
2)
between points 0 and 1, the field is directed from 0
to 1, and at
––
to point 1. Thus, the equilibrium
position 0 is unstable (once the population appears
begins to grow), and the equilibrium position 1 is
stable (a smaller population grows, and a larger
population decreases). Whatever the initial state
0
x
, over time, the process reaches a steady
state of equilibrium
1
x
=
. From these
considerations, it is not clear, however, whether
this exit occurs in a finite or in infinite time, i.e., do
the integral curves that begin in the region
0
1
x
have common points with a straight line
1
x
=
? It can be shown that there are no such
common points and that these integral curves
asymptotically tend to the line
1
x
=
at
t
→ +
and to the line
0
x
=
for
t
→ −
. These curves
are called logistic curves. Thus, the logistic curve
has two horizontal asymptotes (
0
x
=
and
1
x
=
) and describes the transition from one state (0) to
another (1) in infinite time.
6. Example: catch quotas.
So far, we have considered a
free population developing according to its internal
laws. Suppose now that we catch part of the
population (say, we fish in a pond or in the ocean).
Assume that the catch rate is constant. We arrive at a
differential catch equation
(1
)
x
x x
c
= −
−
. The
value
c
characterizes the catch rate and is called the
quota. We see that at a not too high catch rate (
1
0
4
c
), there are two equilibrium positions.
REFERENCES
1.
V.I. Arnold. Ordinary differential equations.
Moscow. Publishing House of the Center. 2018.344
s.
2.
V.V. Stepanov. The course of differential equations.
Moscow. Publishing house LCI. 2016.512 s.
