An Approximation Solution of Linear Differential Equation using
Kantorovich Methods
WASAN AJEEL AHMOOD
Department of Al-Quran Science,
Al-Iraqia University,
Faculty of Education for Women, Baghdad,
IRAQ
MARWA MOHAMED ISMAEEL
Department of Arabic Language,
Al-Iraqia University,
Faculty of Education for Women, Baghdad,
IRAQ
Abstract: - In our work, we constructed a numerical approximations method to deal with approximations of a
linear differential equation. We explained the general framework of the projection method which helps to
clarify the basic ideas of the Kantorovich methods. We applied the iterative projection methods and presented a
theorem to show the convergence of the constructed solutions to the exact solution. Also, most of the
expressions encountered earlier can be used to define functions. Here are some illustrations. A great deal of
information can be learned about a functioning relationship by studying its graph. A fundamental objective of
section 4, is to acquaint with the graphs of some important functions and develop basic graphing procedures.
Key-Words:- Differential Equation, Kantorovich methods, Graphing lines in the rectangular coordinate system.
Received: May 5, 2022. Revised: January 6, 2023. Accepted: February 4, 2023. Published: March 16, 2023.
1 Introduction
Kantorovich's background was completely in
mathematics but showed a great sense of the
primary economy in which it applied sports
technologies. He was one of the first to use linear
programming as a tool in the economy, and this
appeared in the way a sports post regulates
production and planning mentioned in the above
quotes and is considered a historical document,
containing facts about the discovery of linear
programming. The sports formula for production
problems was presented in optimal planning here for
the first time, and effective methods of solution and
economic analysis were proposed, [2]. The author
has introduced many new concepts in the study of
sports programming such as giving the necessary
and sufficient conditions based on supporting the
various aircraft at the solution point in the
production area, the concept of primitive permanent
methods, interpretation in the economics of
complications, and the method of generating the
column used in linear programming. One of his
most important main businesses in the economy was
the best use of economic resources he wrote in
1942, but it was not published until 1959.
Kantorovich has applied improvement techniques to
a wide range of problems in the economy, as well as
a theory of dealing with the economies of
technological innovations.
One way to solve these problems is implemented as
a set of ODPEVP-kantbp programs, [4], [5], based
on the Kantorovich method that provides reducing
the initial problem to a set of regular differential
equations, [1], with more use of the limited element
method, [3], with Lagrange border tangle.
In [6], the authors presented the associated version
of previous work on the non-linear Shrudnegger
equation, with a focus on building the approximate
solutions of these equations through the way the
differential transformation solves the associated
Schrödinger equations.
In [7], the author obtained the approximate solution
to the written differential equation through an
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DOI: 10.37394/232011.2023.18.2
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Marwa Mohamed Ismaeel
E-ISSN: 2224-3429
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Volume 18, 2023
example of a test that explains the effectiveness of
the method and gave efficiency and the accuracy of
the proposed method.
In [13], the authors have proposed an arithmetic
plan to solve the problem of the subjective value of
the elliptical differential equation in the two -
dimensional field with the terms of the Dirichlet
border. The solution is searched in the form of
Kantorovich's expansion on the basic functions of
an independent variable with the second variable
dealing with a teacher.
In [8] the authors studied an approximate solution,
which is the Newton-Kitturovich method of non-
linear, non-linear equations. These methods are used
to modify the NLTD-VIe for the sequence of an
integrated two-dimensional linear equation.
In [9], the authors used the method of contrast in
Kantorovich to solve the finding analysis of the
CSCS KIRCHOFF-LOVE.
The Kantorovich variational method is an effective
technique to solve the bending problem of
Kirchhoff-LOVE panels with CSCS edges and
under a uniform transverse load, this method gives
solutions to deviations that are one series of endless
terms with solutions close, quickly, and quickly
retreat. It converges with fine solutions with the use
of three Infinite chains.
In [10], the authors have built an assembly and
Kantorovich methods to deal with the numerical
estimates of the various team of Fredholm and
introduced a new repetition method to avoid
isolating the mass operator matrix, and theories to
show the exact solution to numerical examples.
In [11], the authors used Modified Bernstein-
Kantorovich operators to solve the first kind of
linear Fredholm and Volterra integral equations.
In [12], the author gave a survey of applications of
the ideas and methods of Leonid Kantorovich
(1912-1986) to the classical problems of
approximation theory.
2 Kantorovich Theorem
If is a map from a Banach space X into
itself, the point . Then
has a solution .
Proof
Let F satisfy the assumption (A) and assume that,
for every , the differential DF(x) has an
inverse such that:
(1)
Let x(.) be the solution to the Cauchy problem
(2)
The local existence and uniqueness of this solution
follow from ODE has a Lipschitz continuous right-
hand side,
(3)
hence,
(4)
by the above equation, when t=1, we have
F(x(1))=0.
Hence
(5)
is a zero of F.
3 Methodology of The Kantorovich
Method
The Kirchhoff-Love rectangular plate is supported
uniformly, A, simply supported on the edges y = (±
b) ⁄2, and installed on the opposite edges x = (± a) ⁄2,
as shown in Figure 1.
The painting is subject to a transverse load
distributed uniformly from the intensity. The field of
the entire panel, r definition by -A = 2 ≤ x ≤ a = 2; -
B = 2 ≤ y ≤ b = 2, origin.
The Cartesian coordinates system is chosen in the
center of the panels, and the axes of the coordinates
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X and Y are chosen to be parallel to the edges of the
painting as shown in Figure 1. This choice of
Cartesian coordinates takes a maximum feature of
symmetry in the plate and the symmetry of loading.
By the Kantorovich method, the Q(x;y) is
represented as one series of infinite terms that
include changing -chapter functions of X and Y
level coordinates. Q (x; y) deliberate to search for an
accurate solution to the flexion problem in the
Kirchoff-Love panel. As follows:
(6)
Fig. 1: loaded rectangular Kirchhoff-Love plate with
clamped edges and simply supported edges.
In the Kantorovich method, one of the functions is
determined using the terms of restriction and
download limits along the coordinates x or y
coordinate directions.
When boundary conditions y=0 at y = ±b=2 for
gn(y); n is odd number, we suggested
that a suitable shape function in the y direction is:
(7)
put the above equation in the single series of infinite
terms since satisfies all the
boundary conditions
(8)
Let Π is the total potential energy functional, U be
the fold and V is the potential energy of the
transverse loads applied:
(9)
Where
(10)
and D is the hardness of the painting:
(11)
and E is the Youngs flexibility coefficient, h is the
thickness of the plate, is the Laplacian
in x, y coordinates
(12)
Thus,
(13)
Now, by simplifying the above equation with an
evaluation of integration to simplify the total
potential energy:
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(14)
We get the differential equations for the functional
above equation.
(15)
The 4th order ordinary differential equation:
(16)
Where n is odd no. The general solution of the
above equation with the four constants of
integration is given by:
(17)
The general solution with x = 0;
becomes is:
(18)
Where
. Thus, Q is fully determined:
(19)
4 Graphing Lines in the Rectangular
Coordinate System
A great deal of information can be learned about a
functional relationship by studying its graph. A
fundamental objective of this course is to acquaint
you with the graphs of some important functions, as
well as to develop basic graphing procedures. First,
we need to review the structure of a rectangular
coordinate system.
Definition of A Graph of an Equation
The graph of an equation in the variables x and y
consists of all the points in a rectangular system
whose coordinates satisfy the given equation.
An infinite number of ordered pairs satisfy the
equation y=x+2, and all are located on the same
straight line. Table 1 presents the values of some
ordered pairs of numbers that satisfy the equation
y=x+2. These have been plotted in a rectangular
system and connected by a straight line. The
arrowheads in Figure 2 suggest that the line
continues endlessly in both directions.
Table 1. The linear equation
x
y=x+2
-3
-1
-2
0
-1
1
0
2
1
3
2
4
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Fig. 2: Graph of an equation.
The graph of y=x+2 is a straight line and the
equation is called a linear equation. Since two points
determine a line, a convenient way to graph a line is
to locate two intercepts. The x-intercept for y=x+2
is -2, the abscissa of the point where the line crosses
the x-axis. The y-intercept is 2, the ordinate of the
point where the line crosses the y-axis.
Example 1:
Graph the linear equation y=2x-3 using the
intercepts.
Solution:
-To find the x-intercept, let y=0.
(20)
When the line crosses the x-axis, the y-value is 0,
and the
point (3/2,0).
-To find the y-intercept, let x=0.
.
(21)
When the line crosses the y-axis, the x-value is 0,
the point (0,-3). The graph of the linear equation is
presented in Figure 3.
Fig. 3: Graph the linear equation.
Definition of slope
If two points (x1, y1) and (x2, y2) are on a line L,
then the slope m of line L is defined by
.
(22)
Example 2:
Find the slope of the L determined by the points
(-3,4) and (1,-6).
By using (x1, y1)=(-3,4), (x2, y2)=(1,-6). Then
.
(23)
It makes no difference which of the two points is
called (x1, y1) or (x2, y2) since the ratio will still be
the same.
Example 3:
Graph the line with the slope
that passes through
the point (-2,-5).
Solution:
Think of
as the
. Now start at (-2,-5)
and move 3 units up and 2 units to the right. This
locates the point (-2,1). Draw a straight line through
these two points.
Shown in Figure 4: reading from left to right, a
rising line has a positive slope and a falling line has
a negative slope.
Alternately, start at (-2,-5) and move 3 units down
and 2 units to the left to locate (-4,-8).
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Fig. 4: rising and falling line has a negative slope.
Definition of the slope-intercept form of A-
line
Y=mx+b.
(24)
Where m is the slope and b is the y-intercept and the
equation also defines.
A function, thus we consider this equation as a
linear function with the domain consisting of all the
real numbers.
Example 4:
Graph the linear function f defined by y=f(x)=x-1 by
using the slope and y-intercept. Display f(4)=3
geometrically, that is, show the point P
corresponding to f(4)=3.
Solution:
The y-intercept is -1. Locate (0,-1) and use m=2 to
reach (1,1), another point on the line. At point P,
x=4 and f(4)=3. The domain and range of f
consisting of all real numbers are illustrated in
Figure 5.
Fig. 5: The domain and range of f consist of all real
numbers.
Definition of the point-slope form of a line:
y-y1=m(x-x1),
(25)
where m is the slope and (x1, y1) is a point on the
line, a general linear equation with constants and not
both 0.
The slope is
.
(26)
Use this slope and either point, such as (2,-3).
y-y1=m(x-x1)= y+3 =2(x-2). Point-slope form
(27)
Use the solution for the above equation and solve
for y.
y =mx+b, y+3=2x-4. Slope-intercept form
(28)
Rewrite the solution for the above equation
Ax+By=C: y=2x-7 or 2x-y=7
note that all three forms are different ways of
expressing the same equation for the given line
through (2,-3) and (3,-1). Show that x=3 and y=-1
satisfy each form. Table 2 presents the summary of
the algebraic forms of a line.
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Table 2. Summary of the algebraic forms of a line
Slope-intercept
form Point-slope
form General linear
equation
Y=mx+b
Line with slope
m and y-
intercept b.
y-y1=m(x-x1)
Line with
slope m and
through the
point (x1, y1)
Ax+By=C
Line with slope
and y-
intercept
, if
For the general linear equation Ax+By=C, note the
following:
If B=0 and , then
, the equation of a
vertical line.
If A=0 and , then
, the equation of a
horizontal line.
5 Conclusion
In this paper, the Kantorovich method is offered
to solve the ordinary differential equation with the
local existence and uniqueness of this solution
follow from. In the Kantorovich method, one of the
functions is determined using the restraint and
loading boundary conditions along either x or y
coordinate directions. The theorem of existence
and uniqueness of approximate solution is
introduced based on the general theorems of
Kantorovich. Then, the example is given to
show the efficiency of the method.
The existence and uniqueness of the approximate
solution are demonstrated, and an illustrative
example is provided to show the precision and
authenticity of the method. In section 4, we gave
some examples that refer to the rectangular
coordinate system as the Cartesian coordinate
system, or simply the Cartesian plane. The results
will help to determine the emphasis needed for this
section.
References:
[1] L.V. Kantorovich, V.I. Krylov, Approximate
Methods of Higher Analysis (Wiley, New York,
1964).
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(1912-1986), in J Eatwell, M Milgate and P
Newman (eds.), The New Palgrave: A dictionary of
economics 3 (London, 1987), pp. 14-15.
[3] G. Strang, G.J. Fix, An Analysis of the Finite
Element Method (Prentice-Hall, Englewood Cliffs,
New York, 1973).
[4] O. Chuluunbaatar et al, Comput. Phys. Commun.
180, 1358–1375 (2009).
[5] O. Chuluunbaatar et al, Comput. Phys. Commun.
179, 685–693 (2008).
[6] Borhanifara A. and Reza A. Numerical study of
nonlinear Schrdinger and coupled Schrdinger
equations by differential transformation method,
Optics Communications, Vol. 283, Issue 10, Pages
2026-2031, 2010.
[7] Nadhem E., Approximate solution of linear
differential equations, Mathematical and Computer
Modelling, Volume 58, Issues 78, Pages 1502-
1509, 2013.
[8] Hameed H et al. An approximate solution of two
dimensional nonlinear Volterra integral equation
using Newton-kantorovich method. Malaysian
Journal of Science 2016.
[9] Charles C.I. and Benjamin O. M., Kantorovich
variational method for the flexural analysis of
CSCS Kirchhoff-Love plates, Mathematical Models
in Engineering, Vol. 4, Issue 1, p. 29-41, 2018.
[10] Boutheina T. et al., An approximation solution of
linear Fredholm integro-differential equation using
Collocation and Kantorovich methods, Journal of
Applied Mathematics and Computing, 2021.
[11] Tikhomirov V. Leonid Kantorovich and
Approxination Theory, Siberian Mathematical
Journal, vol. 63, pages173-178, 2022.
[12] Gusev et al. Solution of Boundary-Value Problems
using Kantorovich Method, EPJ Web of
Conferences, vol. 108. DOI: 10.1051/ C Owned by
the authors, published by EDP Sciences, 2016.
[13] Suzan C. et al, Numerical Solution of the Fredholm
and Volterra Integral Equations by Using Modified
Bernstein Kantorovich Operators. Mathematics, 9,
1193, 2021. https://doi.org/10.3390/math9111193.
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
Wasan Ajeel: Theorem, examples, and
methodology.
Marwa Mohamed: Investigation, writing with
Figures and Tables.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received
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Conflict of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
