Splines of the Second and Seventh Order Approximation and the
Stability of the Solution of the Fredholm Integral Equations of the
Second Kind
I. G. BUROVA, G. O. ALCYBEEV, S. A. SCHIPTCOVA
The Department of Computational Mathematics,
St. Petersburg State University,
7-9 Universitetskaya Embankment, St.Petersburg,
RUSSIA
Abstract: - This work is a continuation of a series of works on the use of continuous local polynomial splines
for solving interpolation problems and for solving the Fredholm integral equation of the second kind. Here the
construction of a numerical solution to the Fredholm integral equation of the second kind using local spline
approximations of the second order and the seventh order of approximation is considered. This paper is devoted
to the investigation of the stability of the solution of the integral equation using these local splines.
Approximation constants are given in the theorem about the error of approximation by the considered splines.
Numerical examples of the application of spline approximations of the second and seventh order of
approximation for solving integral equations are given.
Key-Words:- Fredholm integral equation of the second kind, splines of the seventh order of approximation,
splines of the second order of approximation, stability, numerical solution.
Received: May 6, 2022. Revised: August 11, 2023. Accepted: September 12, 2023. Available online: November 16, 2023.
1 Introduction
Integral equations often arise in various
applications. Many problems of astrophysics,
mechanics, viscoelasticity, elasticity, vibrations,
plasticity, hydrodynamics, electrodynamics, nuclear
physics, biomechanics, geology, medicine problems,
and many other problems are formulated in terms of
integral equations. The mathematical model for
many problems arising in different natural science
industries is formulated using differential and
integral equations. The investigation of these
equations is conducted with the help of the
numerical integration theory, [1]. Mathematics and
physics problems are often reduced to solving
integral or integro-differential equations. This is
noted in the following papers. Hypoxy induced
angiogenesis processes can be described by
coupling an integro-differential kinetic equation of
the Fokker-Planck type with a diffusion equation for
the angiogenic factor, [2]. The charged particle
motion for certain configurations of oscillating
magnetic fields can be simulated by a Volterra
integro-differential equation of the second order
with time-periodic coefficients, [3]. In paper, [4],
the Fourier integral transform has been employed to
reduce the problem of determining the stress
component under the contact region of a punch in
solving dual integral equations. In the paper, [5], the
method of integral equations is proposed for some
electrical engineering (current density, radiative heat
transfer, heat conduction) problems. The presented
models lead respectively to a system of Fredholm
integral equations, integro-differential equations, or
Volterra-Fredholm integral equations.
When solving integral equations, splines and
wavelets are often used. The B-spline basis and the
Hartree–Fock integro-differential equations are
reduced to a computationally eigenvalue problem,
[6]. In paper, [7], the Legendre wavelet functions
were used to solve the Fredholm integral equation.
In paper, [8], an efficient modification of the
wavelets method to solve a new class of Fredholm
integral equations of the second kind with a non-
symmetric kernel is introduced. In paper, [9], the
tension spline approximation to obtain the numerical
solution of Volterra–Fredholm integral equation was
developed. In paper, [10], a general spline
maximum entropy method for the approximation of
solutions for solving Fredholm integral equations
was described. In paper, [11], the description of
fuzzy Bezier splines is presented. An iterative
numerical method for approximating the solution of
fuzzy functional integral equations of the Fredholm
type is proposed. In paper, [12], the non-polynomial
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spline functions were used to obtain the numerical
solutions of the Fredholm integral equations of the
second kind. In paper, [13], the wavelet-Galerkin
method for the numerical solution of the Fredholm
linear integral equations and the second-order
integro-differential equations are discussed. A
construction of a quadratic spline-wavelet basis on
the unit interval, such that the wavelets have three
vanishing moments and the shortest support among
such wavelets was proposed, [13]. In paper, [14], a
new collocation technique for the numerical solution
of the Fredholm, Volterra, and mixed Volterra-
Fredholm integral equations of the second kind is
introduced, and a numerical integration formula on
the basis of the linear Legendre multi-wavelets is
also developed. The linear Legendre multi-wavelets
basis for the proposed method is used. In this
technique, the unknown function is approximated by
the truncated linear Legendre multi-wavelets series,
[14].
Good results are obtained by using the
Chebyshev polynomials. Paper, [15], focused on
fuzzy Fredholm integral equations of the second
kind. Using the Chebyshev polynomials due to their
smoothness and reasonable behavior near
boundaries, a new method is proposed to solve the
fuzzy Fredholm integral equation. In paper, [16],
the approximate solution of linear Fredholm integral
equations of the second type on a closed interval is
studied. The Galerkin method enhanced with the
Chebyshev polynomials was used to improve the
approximate solution.
We also note the following papers. In paper,
[17], the authors have used the advanced multistep
and hybrid methods to solve the Volterra integral
equation. In paper, [18], the forward-jumping
methods of the hybrid type are used for the
construction of the methods with a high order of
accuracy. In paper, [19], the Half-Sweep Gauss-
Seidel iteration, which was used to find the
approximate solution of the fuzzy Fredholm integral
equations of the second kind, was applied. In paper,
[20], linear Volterra–Fredholm integral equations of
the second kind were considered in reproducing
kernel space. A new scheme with a high
convergence order for solving the approximate
solutions to oscillation and non-oscillation of exact
solutions was proposed.
In paper, [21], a new technique is offered to
solve three types of linear integral equations of the
2nd kind, including the Volterra-Fredholm integral
equations (as a general case), the Volterra integral
equations, and the Fredholm integral equations (as
special cases). The new technique depends on
approximating the solution to a polynomial of
degree is described.
This work is a continuation of a series of works
on the use of continuous local polynomial splines
for solving interpolation problems and for solving
integral equations, [22], [23]. This paper is devoted
to the investigation of the stability of the solution of
the integral equation using these local splines. As is
known, the solution of integral equations of the
second kind is reduced to finding the frame of the
approximate solution. This means that we find
approximations to the values of the function at the
nodes of the grid (grid function). Usually, the
integral equation is replaced by some difference
scheme with a given order of accuracy. The
approximate values of the function is converged to
the values of the function at the grid nodes if the
two conditions are fulfilled. These conditions are as
follows: an approximation of the equation with a
difference scheme, and the difference scheme is
stable. Let for an approximate solution of the
integral equation be
a difference scheme is constructed
This scheme approximates the original equation
with some order of accuracy. Suppose a linear
normed space of functions defined on the grid is
considered. The operator maps the space to
the space . A difference scheme is said to be
stable on the right-hand side if, for any , the
equation
has a unique solution and
Here is a constant. Next, consider the definition
Thus, the convergence of the approximate
solution to the values of the function at the grid
nodes follows from the approximation of the
original equation and the stability of the solution.
In the works of, [24], conditions for the stability
of the solution of the Fredholm integral equation of
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the second kind using the trapezoidal method were
obtained. Thus, the convergence of the obtained
approximate solution of the Fredholm integral
equation of the second kind to the values of the
function at the grid nodes was proved.
Next, we construct a numerical scheme to solve
the Fredholm integral equation of the second kind.
This paper discusses the stability of the solution
when we use the local splines of the second and
seventh order of approximation. We use these
splines if the kernel and the right side are
sufficiently smooth functions. To construct an
approximate solution at the points between the grid
nodes, we use the interpolation of the same local
splines or the integral equation with an obtained
solution at the nodes.
2 Problem Formulation
Let be a grid of ordered nodes on the interval
: …< . Note that the
approximations with the splines are constructed
separately for each grid interval. Let us
assume that the values of the function are
given at the grid nodes. The approximation using
basis splines is built separately on each grid interval
as the sum of the products of the values of the
function at the grid nodes and the basis splines .
Let ,, be integers, ,
and the spline be such that
. Following the methodology
developed by Professor S.G. Mikhlin, we find the
basis functions by solving the system of
approximation relations
2.1 Polynomial Splines of the Second Order
of Approximation
Let The support of the basis splines
of the second order of approximation occupies two
grid intervals. These splines are convenient to use
on a finite interval, both on a uniform grid of nodes
and on a non-uniform grid of nodes. The
approximation of the function on a finite interval of
interpolation does not have a boundary layer. When
solving the Fredholm integral equation of the
second kind, the minimum number of the grid nodes
is two. We set the support of the basis spline as
follows: . On the interval
we approximate the function by the
following expression:
,
where the basis splines are as
follows:
These splines are the interpolation splines of the
second order of approximation as well as the first
degree. The approximation using these splines is the
continuous approximation.
Note that the minimum number of grid intervals
is one, and the minimum number of grid nodes is
two.
The following statement is valid for the
approximation error.
Let In the case of the splines of
the first degree, it is easy to obtain an estimate of the
approximation error on the interval[22],
[23]
In the case of an uneven grid of nodes, we take the
length of the maximum grid interval as the value of
Further, we will use the norm of the form:
Consider the solution of the Fredholm integral
equation of the second kind
Here, and further, we assume that the kernel
and the right side of the equation are
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continuous. In addition, we assume that the equation
is uniquely solvable and the estimate for the norm of
the inverse operator in space is known:
Suppose
We construct the set of nodes , , on
the interval . We have the relation
On each interval we replace with
approximation .
Now we have the integral equation in the form:
From here we get the system of equation in the form
where
Next, we take instead of and we have to solve
the system of linear algebraic equations:
We assume that the integral
can be computed exactly. Otherwise, we can use
quadrature formulas.
Now, we suppose that , and
. Let be one of those components of the
solution, whose absolute value is the largest.
Therefore, the execution for this component of the
solution is
.
We consider on the interval the
approximation of the function with splines of the
second order of approximation
.
Now we have
where
We assume that . It is easy to
calculate the integrals
Now using the mean value theorem of integral
calculus, we obtain
Similarly, we get
Finally, we obtain the inequality
Thus, we have the estimation
In particular, we can take This implies that
the system is uniquely solvable. The last inequality
means the stability of the solution depends on the
right side of the equation with the constant
.
2.2 Polynomial Splines of the Seventh Order
of Approximation
Now we consider the application of splines of the
seventh order of approximation to solve the
Fredholm integral equations of the second kind.
Different modifications of the splines of the seventh
order of approximation are used at the beginning, in
the middle and at the end of the interpolation
interval The support of the basis spline
occupies seven grid intervals.
First, consider the approximation properties of
polynomial splines of the seventh order of
approximation.
Let be integers, ,
and the spline be such that
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. We find the basis functions by
solving the system of approximation relations
With different values of the parameters , we get
basis splines suitable for th approximation at the
beginning of the interpolation interval (the right
basis splines), in the middle of the interpolation
interval (the middle basis splines), or at the end of
the interpolation interval (the left basis splines).
With and we get the middle
splines. On the interval , we construct the
approximation with the middle splines at a distance
of three grid intervals from the ends of the interval
in the form:
where the middle basis splines
have the
form:
where
Approximations with these basis splines can be
constructed on the grid intervals ,
.
Let us consider the approximation with the left
basis splines. We get the left basis splines when
. In this case, formula (1) on the
interval takes the form:
where the basis splines
have the form
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Approximations with these basis splines can be
applied on the next grid interval ,
.
Let us consider the approximation with the left-
right basis splines. We get the left basis splines
when . In this case, formula (1) on
the interval takes the form
where the basis splines
have the form
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Approximations with these basis splines can be
applied on the next grid intervals ,
.
Consider the approximation with the right basis
splines. Let . In this case, formula (1)
takes the next form on the interval :
where the right basis splines
have the form:
;
Approximations with these basis splines can be
applied on the next grid intervals ,
.
Consider the approximation with the right-left
basis splines. Let , in this case, on the
interval formula (1) takes the form:
where the right-left basis splines
have the
form:
;
;
;
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;
;
;
.
Approximations with these basis splines can be
applied on the next grid intervals ,
.
Consider the approximation with the right-left-
left basis splines. Let . In this case, on
the interval formula (1) takes the form:
where the right-left-left basis splines
have
the form:
;
;
;
;
;
;
.
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Approximations with these basis splines can be
applied on the next grid intervals ,
.
When approximating a function with the splines
of the 7th order of approximation, the next Theorem
is valid.
Theorem. If then the
following inequality is valid:
If then the following
inequality is valid:
If , then the following
approximation estimate is valid:
If , then the following
approximation estimate is valid:
If , then the following
approximation estimate is valid:
If , then the following
approximation estimate is valid:
Proof. In the case of approximating the function
on the interval [] near the left end of the
interval , we use the right basis splines
First, we estimate the approximation error on the
interval [] when the right basis splines are
used. Using the formula of the remainder term of the
interpolation polynomial that solves the Lagrange
interpolation problem, we obtain the relation
There is a product in the
error estimate. Let the ordered grid of nodes be
uniform with step Let us estimate the product of
factors .
Thus, estimating the maximum of the expression
, where
we obtain
.
Similarly, we obtain an approximation estimate on
the grid interval [ with the left and middle
splines.
This completes the proof of the theorem.
Remark 1. The approximation on the interval.
If the interval is divided into 6 grid
intervals, then it is possible to construct an
approximation on the entire interval using the
previously presented approximations on the
intervals , as follows:
for we apply the approximation ;
for we apply the approximation ;
for we apply the approximation ;
for we apply the approximation ;
for we apply the approximation ;
for we apply the approximation .
Remark 2. The stability.
Let , and .
Suppose
The constant in the stability inequality in the
case of splines of the seventh order of
approximation is calculated in the same way as it
was done for the splines of the second order of
approximation. To calculate the constant in the
stability inequality, we need the following values:
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Taking into account the inequalities given above, we
obtain the constants
,
.
Thus, we have the estimations
Suppose that In this case, taking into
account the approximation theorem and the
inequalities given above, we can see that the
approximate solution obtained with the splines of
the seventh order of approximation tends to the
solution to the Fredholm equation.
3 Problem Solution
In this section, we discuss the solution of the
integral equation of the second kind
with the local splines of the seventh order of
approximation.
Let us choose an integer We build ga rid of
nodes {}.
The function
can be approximated with the expression:
. Let us denote
We represent the integral in the form
Using the results from the second section, we can
reduce the integral equation to the solution of a
system of linear algebraic equations. To do this, we
put , , in the equation
And now we have to solve the system of linear
algebraic equations
We can also apply a more detailed approximation
(as shown in sections 2.2.1-2.2.6) on the grid
intervals adjacent to the boundaries of the interval
. We assume that the integral
can be computed exactly.
Otherwise, we can use quadrature formulas. In this
case, it is necessary to take into account the error of
the applied quadrature formula. We can use, for
example, the Simpson's compound formula:
.
Here is even, .
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The formula for the remainder of the Simpson's
compound quadrature rule is well known. In the
case of the interval [, it has the form
.
Proceeding from this formula and taking into
account the approximation theorem for the splines
of the seventh order, it is easy to get a priori
estimation of the number of nodes of the Simpson's
quadrature formula.
Let us calculate the integral from the Runge
function
where
.
We can obtain
We can easily calculate and
Note that this number is less than the
theoretical error of integration (which follows from
the Theorem). We can easily obtain that the
theoretical error of the integration is about
After the approximate solution at the grid nodes is
obtained, we can use the expressions
to solve the problems that arise further, for example,
the construction of a plot of the solution.
Let us consider two examples of the application
of splines of the seventh order of approximation and
splines of the second order of approximation to
solve the Fredholm integral equation of the second
kind.
Example 1. Consider the equation
First, let us construct an ordered grid of nodes
with step ( on the interval
. Using splines of the seventh order of
approximation, we calculate the approximate values
of the solution at these nodes. Next, we construct a
sequence of refining grids on the interval as
follows: we divide each grid interval in half. The
division points are added to the nodes of the
previous grid.
Thus, we get a new grid of nodes. We compare
the values of the approximate solution at some grid
nodes.
Table 1 presents the approximate values of the
solution at some grid nodes when using splines of
the seventh order of approximation when
. Table 2 presents the approximate values of
the solution at the same grid nodes when using
splines of the second order of approximation when
.
Table 1. The Approximate Values of the Solution
when Splines of the Seventh Order of
Approximation were used
Splines of the Seventh Order of
Approximation
0
0
0
0.2
0.24717242
0.24717241
0.4
0.60251062
0.60251061
0.6
1.10193492
1.10193491
0.8
1.79197032
1.79197030
1.0
2.73268142
2.73268140
Table 2. The Approximate Values of the Solution
when Splines of the Second Order of
Approximation were used
Splines of the Second Order of
Approximation
0
0
0
0
0.2
0.24717235
0.24717240
0.24717241
0.4
0.60251051
0.60251059
0.60251061
0.6
1.10193482
1.10193489
1.10193490
0.8
1.79197029
1.79197031
1.79197030
1.0
2.73268156
2.73268145
2.73268140
Having solved the system of linear algebraic
equations, we obtain the framework of an
approximate solution. The obtained values of the
approximate solution at the grid nodes are marked
with circles. Further, using the integral equation and
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the found approximate values of the function, we
connect the found values with a line. The Maple
package was used in the calculations. The plot of the
approximate solution with the splines of the second
order of approximation when is given in
Figure 1. The nodes are marked along the abscissa
axis.
Fig. 1: The plot of the approximate solution when
Example 2. Consider the equation
As is known, in order to obtain an approximate
solution of the integral equation, it is necessary to
apply different methods and calculate with a
different number of grid nodes. Let us start with
calculations using splines of the second order of
approximation. Table 3 presents the approximate
values of the solution at grid nodes when using
splines of the second order of approximation when
. Table 4 presents the approximate values
of the solution at the same grid nodes when using
splines of the second order of approximation when
. Table 5 presents the approximate
values of the solution at the grid nodes when splines
of the second order of approximation when
and splines of the seventh order of
approximation when were used. Table 6
shows the approximate values of the solution when
splines of the seventh order of approximation were
used ( ).
Table 3. The Approximate Values of the Solution
when Splines of the Second Order of
Approximation were used
Splines of the Second Order of
Approximation
0
0.06744487
0.06763105
0.2
0.36908780
0.36939306
0.4
0.65601636
0.65642854
0.6
0.91679162
0.91729428
0.8
1.14101729
1.14159039
1.0
1.31975420
1.32037489
Table 4. The Approximate Values of the Solution
when Splines of the Second Order of
Approximation were used
Splines of the Second Order of
Approximation
0
0.06767764
0.06768929
0.2
0.36946943
0.36948853
0.4
0.65653166
0.65655744
0.6
0.91742003
0.91745147
0.8
1.14173376
1.14176961
1.0
1.32053017
1.32056899
Table 5. The Approximate Values of the Solution
when Splines of the second order of approximation
and the seventh of Approximation were used
Splines
Splines of the Second
Order of Approximation
Splines of the
Seventh Order of
Approximation
0
0.067693111
0.06769317
0.2
0.36949480
0.36949490
0.4
0.65656590
0.65656604
0.6
0.91746179
0.91746195
0.8
1.14178137
1.14178156
1.0
1.32058173
1.32058194
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Table 6. The Approximate Values of the Solution
when Splines of the Seventh Order of
Approximation were used
Splines of the Seventh Order of
Approximation
0
0.06769317
0.067689317
0.2
0.36949490
0.36949490
0.4
0.65656603
0.65656604
0.6
0.91746195
0.91746195
0.8
1.14178156
1.14178156
1.0
1.32058194
1.32058194
The absolute values of the differences between
the solution found on the grid of nodes ( )
and the solution found on the grid of nodes (
) are shown in Figure 2 (blue line); absolute
values of the differences between the solution found
on the grid ( ) and the solution found on the
grid ( ) are shown in Figure 2 (green line).
Fig. 2: The absolute values of the differences
between the solution found on the grid of nodes
( ) and the solution found on the grid of
nodes ( ) (blue line); absolute values of the
differences between the solution found on the grid
( ) and the solution found on the grid (
) (green line).
Figure 3 shows the absolute values of the
differences of the solution found on the grid (
) and the solution found on the grid ( )
when the splines of the seventh order of
approximation were used.
Fig. 3: The absolute values of the differences of the
solution found on the grid ( ) and the solution
found on the grid ( ) when the splines of the
seventh order of approximation were used.
Figure 4 shows the absolute values of the
differences of the solution found on the grid (
) and the solution found on the grid ( )
when the splines of the seventh order of
approximation were used.
Figure 5 shows the plot of the approximate solution
when the splines of the seventh order of
approximation were used and
The calculation results show that splines of the
seventh order of approximation give an approximate
solution at the grid nodes with eight correct digits in
the mantissa when
Fig. 4: The absolute values of the differences of the
solution found on the grid ( ) and the solution
found on the grid ( ) when the splines of the
seventh order of approximation were used.
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Fig. 5: The plot of the approximate solution when
the splines of the seventh order of approximation
were used ( .
Comparing the results in the second and third
columns of Table 5, we see that the spline of the
second order of approximation provides five correct
digits in the mantissa when we use 640 grid nodes.
At the same time, to achieve the same accuracy
using splines of the seventh order of approximation,
it is enough to use 10 grid nodes.
4 Conclusion
In this paper, we consider the stability of the
solution of the integral equations of the second kind
using splines of the second and the seventh order of
approximation. It should be noted that with the same
number of grid nodes, polynomial splines of the
seventh order of approximation provide a smaller
error compared to splines of the second order of
approximation. When using splines of the seventh
order of approximation, a large number of nodes is
not recommended.
In the future, numerical schemes for integro-
differential equations will be constructed.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
-I. G. Burova developed the theoretical part,
-G. O. Alcybeev executed the numerical
experiments,
-S. A. Schiptcova executed some numerical
experiments.
The authors are grateful to Professor V.M. Ryabov
for his valuable comments.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
The authors are gratefully indebted to St. Petersburg
University for their financial support (Pure ID
94029567, ID 104210003)
Conflict of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
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WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.1
I. G. Burova, G. O. Alcybeev, S. A. Schiptcova
E-ISSN: 2224-2880
15
Volume 23, 2024
