INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE
ISSN: 2692-5206, Impact Factor: 12,23
American Academic publishers, volume 05, issue 05,2025
Journal:
https://www.academicpublishers.org/journals/index.php/ijai
page 303
APPLICATION OF THE INITIAL INTEGRATION METHOD TO THE
NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS WITH A SMALL
PARAMETER IN FRONT OF THE HIGHEST DERIVATIVE
Ro‘ziyev Allanazar Yuldosh ugli
2nd-year Master's Student in Applied Mathematics, Termiz State University
Normurodov Chori Begaliyevich
Scientific Advisor: PhD Professor
Abstract:
This study addresses the numerical solution of differential equations that contain a
small parameter in front of the highest-order derivative. Such equations frequently arise in
various physical modeling scenarios, such as heat conduction, diffusion, and fluid dynamics.
These equations exhibit special characteristics, where the presence of a small parameter leads
to the formation of boundary layers and sharp changes in the solution. The initial integration
method is a powerful approach for obtaining stable and highly accurate results in solving
such problems numerically. This research analyzes the application features of the method, its
advantages, and its impact on the sensitive zones of the solution. The results demonstrate that
the method is an effective tool for reliably solving complex problems involving small
parameters
.
Keywords:
Initial integration method, small parameter, higher-order differential equation.
Modeling of Physical and Technical Processes Expressed by Differential
EquationsModeling of physical and technical processes using differential equations is one of
the key directions in applied mathematics. Most real-world models include higher-order
derivatives, and in some cases, a small parameter appears in front of these derivatives. For
example:\varepsilon y^{(n)}(x) + a_{n-1}(x)y^{(n-1)}(x) + \ldots + a_0(x)y(x) = f(x), \quad
0 < \varepsilon \ll 1The solutions to such equations differ significantly from those of standard
equations and exhibit singular behavior. That is, boundary layer zones appear in the
solution—regions where the solution changes rapidly—while the rest of the interval remains
smooth. Traditional numerical approaches often fail to provide sufficient accuracy in these
cases, or they incur very high computational costs. Therefore, the method of initial
integration becomes especially relevant. This method, which takes into account the sensitive
structure of such equations, allows for stable and efficient solutions. The essence of the initial
integration method lies in integrating the equation beforehand, thereby reducing higher-order
derivatives to lower-order ones. As a result, the influence of the small parameter is
diminished, and the solution becomes smoother. Particularly, if the differential equation is
expressed using integrals, this method naturally captures the boundary layer zones in the
solution. In our research, we comprehensively cover the classical analysis of high-order
singularly perturbed equations, the algorithmic application of the initial integration method,
the accuracy and stability of numerical solutions, and comparisons with other methods.
INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE
ISSN: 2692-5206, Impact Factor: 12,23
American Academic publishers, volume 05, issue 05,2025
Journal:
https://www.academicpublishers.org/journals/index.php/ijai
page 304
Additionally, we demonstrate the method’s applicability in physical models (such as heat
flow and diffusion models) through examples. Thus, this work highlights the advantages of
new approaches in mathematical modeling and provides methodologically and practically
significant conclusions in solving singular differential equations.
^c • n ? j. A(r, e) + eC(r, e)-^ + b(r, e)x = P(r, E)el6(t-£) (1) look at the question of integrating
a system of second-order Linear Differential Equations dependent on the parameter, where
x(t,£) is an n-dimensional vector R = Et - slowly changing time e < 1-small real parameter;
6(t, e) is a scalar function, i = V - l; A(t, e), B(t, e), C(T, E)—(N x n) matrices p(r, e) — n -
dimensional vector, let the conditions below Be Satisfied: A(R, E), B(T, E), C(T, E)
matrices,and P(T, E)-vector function given r e [0, l] in the range, let e extend to the
converging series by the degrees of parameter: Co Co C(T, = ^ ESCs (t), p(t, = ^ ESPs (t), (2)
s=0 s=0 CC A(x, e) = ^ ESAs (t), b(t,e) = ^ ESBs (T) S=0 s=0 VTE [0, l], deta0(r) * 0(3) DQ
d(T, E) the-derivative of a function is a slow — changing function, i.e. dO Tt = m() (2) series
coefficients As(t), Bs(t), C(t), Ps(t)s = 0.1.2,... and given a function k(r) [0,L] in the study of
an infinitesimally differentiable system in a section, its character is
fs (r) = Bivs-I - ^ ^ ^ Xixjagvs—1—i—j - ^ ¿0¿iajvs-i-j - i=1 i=1 c=0 k=0 i=1 c=0 s k-1s-1—
i s-1 s-1-I - ^¿0a0vs—I-^ ^ ¿iccvk—1—i—j-0-2^ ^ Xiajv'k-1-i-j- i=1 i=0 c=0 i=0 c=0 s-2 s-2
- ^ Civ's—2—I - ^ AIV'sl-2-i s = 1,2,3,... (14) I=1 i=1 based on the condition of the theorem
these roots are simple, then to each of them one p¿r) corresponding to the specific vector (B0-
WiA0) (i = 0 satisfies the relation and is defined in arbitrary scalar product accuracy, which is
nonzero. In this case, the vector A0 added to The Matrix B0 will not exist. In this case (B0 -
WiA0)Z = a0pb (i = l/n) (LS) has no equation solution. We look at the system of equations
(11), (12). Equation (11) has a nonzero solution only and only when ¿2 = -Wi, (i = 1/n), from
this we define 2N different Ä0(r) : Yao (r) = ±i^Wi (r), (i = %n) (16 Then n different v0 (r)
vector functions from(11) are defined: v0(r) = Pi(r), (i = ln) (l7) where(pi(r), A0(r) are the hos
values of the Matrix B0 (r) with respect to The Matrix. (16) munoit plays an important role in
the modeling of many other natural processes. The underlying complexity is the layer zones
generated by the presence of a small parameter and is manifested in the drastically changing
nature of the solution. This makes it difficult for traditional numerical methods to be used, or
they will not have sufficient accuracy as a result. The study initially explored both
theoretically and practically the application of the integralization method to solving such
equations. Based on analyzes and numerical experiments, the following main conclusions
were drawn: 1. Initially, the integrable method takes into account the sensitive zones of small
parametric equations in a natural way. This plays an important role in capturing the drastic
change in solution, especially in the layer zone.2. This method is highly effective in terms of
stability and accuracy. Reduces the uncertainties associated with a small parameter,
smoothing the points of view in the solution.3. In relation to traditional methods, the
computer resource perspective
References:
1. Vazov V. Asymptotic expansions of solutions of ordinary differential equations, Moscow:
Mir, 1973. 464c. 2. Bogolyubov N.N., Mitropolsky Yu.A. The asymptotic method in the
theory of nonlinear oscillations, Moscow: Fizmatgiz, 1963, 410 p. 3. Feshenko S.F.,
INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE
ISSN: 2692-5206, Impact Factor: 12,23
American Academic publishers, volume 05, issue 05,2025
Journal:
https://www.academicpublishers.org/journals/index.php/ijai
page 305
Shkil N. I., Nikolenko L. D. Asymptotic methods in theory of linear differential
equations. Moscow: Nauk dumka, 1966. 252 p. 4. Vasilyeva A. B., Butuzov V. F.,
Asymptotic decomposition of singularly perturbed equations. Moscow: Nauka, 1973,
272 p.
