The Application of Splines of the Seventh Order Approximation to the
Solution of Integral Fredholm Equations
I. G. BUROVA, G. O. ALCYBEEV
Department of Computational Mathematics,
St. Petersburg State University,
7-9 Universitetskaya Embankment, St.Petersburg,
RUSSIA
Abstract: - There are various numerical methods for solving integral equations. Among the new numerical
methods, methods based on splines and spline wavelets should be noted. Local interpolation splines of a low
order of approximation have proved themselves well in solving differential and integral equations. In this
paper, we consider the construction of a numerical solution to the Fredholm integral equation of the second
kind using spline approximations of the seventh order of approximation. The support of the basis spline of the
seventh order of approximation occupies seven grid intervals. We apply various modifications of the basis
splines of the seventh order of approximation at the beginning, the middle, and at the end of the integration
interval. It is assumed that the solution of the integral equation is sufficiently smooth. The advantages of using
splines of the seventh order of approximation include the use of a small number of grid nodes to achieve the
required error of approximation. Numerical examples of the application of spline approximations of the seventh
order for solving integral equations are given.
Key-Words: Fredholm integral equation of the second kind, splines of the seventh order of approximation
Received: October 17, 2022. Revised: April 23, 2023. Accepted: May 15, 2023. Published: May 25 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 is arising in different industries of
natural science, basically formulated using
differential and integral equations. The investigation
of these equations is conducted with the help of the
numerical integration theory, [1]. Mathematical and
physics problems are often reduced to solving
integral or integro-differential equations.
The (2+1) dimensional Konopelchenko–
Dubrovsky equation (2D-KDE) is an integro-
differential equation which describes two-layer fluid
in shallow water near ocean shores and stratified
atmosphere, [2].
In paper [3], the two-dimensional Volterra
integro-differential equations for viscoelastic rods
and membranes in a bounded smooth domain are
studied.
Hypoxy induced angiogenesis processes can be
described by coupling an integro-differential kinetic
equation of Fokker-Planck type with a diffusion
equation for the angiogenic factor, [4].
The development of two numerical techniques for
solving the convection-diffusion type partial
integro-differential equation (PIDE) with a weakly
singular kernel is studied in the paper, [5].
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,
[6].
As it is known, equations of the second and first
kinds are distinguished among the integral and
integro-differential equations. Usually, the solution
of an integral equation is reduced to the solution of a
system of algebraic equations.
The compound trapezoidal scheme is often used
to solve the first-order linear Fredholm integro-
differential equation and the second-order linear
Fredholm integro-differential equation, [7], [8].
In paper [9] the trapezoidal rule is used to
approximate the integral in linear and nonlinear
fractional Fredholm integrodifferential equations.
The B-spline collocation method is used to solve
the system of singular integro-differential equations,
[10].
On the B-spline basis, the Hartree–Fock integro-
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differential equations are reduced to a
computationally eigenvalue problem, [11].
In paper, [12], the approximation of the solution
of Fredholm integro-differential equations of the
second kind using an exponential spline function
was applied.
In paper, [13], the cubic B-spline collocation
method is used to solve the stochastic integro-
differential equation of fractional order.
In the paper, [14], the advanced multistep and
hybrid methods have been used to solve Volterra
integral equation.
In paper, [15], the kernel as well as the right part
of the equation are initially approximated through
Legendre wavelet functions.
In the paper, [16], the forward-jumping methods of
hybrid type are used for the construction of the
methods with a high order of accuracy.
In the paper, [17], 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, [18], the method of integral equations
is proposed for some problems of electrical
engineering (current density, radiative heat transfer,
heat conduction). Presented models lead to a system
of Fredholm integral equations, integro-differential
equations, or Volterra-Fredholm integral equations,
respectively.
This paper discusses the solution methods based
on the use of local splines of the seventh order of
approximation. We use these splines if the kernel
and the right side are sufficiently smooth functions
and we want to use a small number of grid nodes.
To construct an approximate solution at points
between grid nodes, we use the interpolation of the
same local splines.
2 Problem Formulation
Let be a grid of nodes on the interval ,.
Note that the approximations with the splines are
constructed separately for each grid
interval,.
In this paper, we consider the application of
splines of the seventh order of approximation to
solve integral Fredholm 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 a uniform grid of nodes
on the interval ,:
…< with step
. 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,
7, 1,
1,and
the spline be such that supp
,. Following the methodology
developed by Professor S. G. Mikhlin, we find the
basis functions by solving the system of
approximation relations
,∈
,,
0,1,…,6.1
With different values of the parameters ,, we
get basis splines suitable for 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).
2.1 Approximation with the Middle Basis
Splines
With 3 and 3 we get the middle splines.
When constructing an approximation on a finite
interpolation interval, we use the middle splines,
departing 3 grid intervals from the points and
that are the ends of the interval of integration. At the
beginning and the end of the interval ,, we
apply, respectively, the left and the right splines. For
example, 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:
, ∈ ,.
2
where the middle basis splines
have the
form:
/,
where
,
;
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,
,
;
,
,
;
,
,
;
,
,
;
,
,
;
,
,
.
The plots of the basis splines
,
3,…3, are given in Fig. 1,Fig. 2,
Fig. 3, Fig. 4, Fig. 5, Fig. 6, Fig. 7.
Fig. 1: The plot of the basis spline
when
1,
0.
Fig. 2: The plot of the basis spline
when
1,
0.
Fig. 3: The plot of the basis spline
when
1,
0.
Fig. 4: The plot of the basis spline
when
1,
0.
Fig. 5: The plot of the basis spline
when
1,
0.
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Fig. 6: The plot of the basis spline
when
1,
0.
Fig. 7: The plot of the basis spline
when
1,
0.
2.2 Approximation with the Left Basis
Splines
Let us consider the approximation with the left basis
splines. We get the left basis splines when
5,
1. In this case, formula (1) on the interval
,
takes the form:
, ∈
,
.
3
where the basis splines
have the form
/
,
,
;
,
,
;
,
,
;
,
,
;
,
,
;
,
,
;
,
,
.
2.3 Approximation with the Right Basis
Splines
Consider the approximation with the right basis
splines. Let
0,6, in this case, on the
interval
,
formula (1) takes the form:
,∈
,
,4
where the right basis splines
have the
form:
,
,
;
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,
,
;
,
,
;
,
,
;
,
,
;
,
,
;
,
,
.
2.4 Approximation Theorem
Further, we will use the norm of the vector of
the form: ∥∥,max
∈,||
When approximating a function with the
splines of the 7
th
order of approximation, the
next Theorem is valid.
Theorem. If supp
,
,then the
following inequalities are valid:
||
∈
,
95.842∙
∥
∥
,
7! .
If supp
,
, then the following
approximation estimate is valid:
||
∈
,
12.359∙
∥
∥
,
7! .
If supp
,
, then the following
approximation estimate is valid:
||
∈
,
95.842∙
∥
∥
,
7! .
Proof. In the case of approximating the function
on the interval [
,
] near the left end of the
interval ,, we use the right basis splines:
, ∈
,
.
Let us estimate the approximation error on the
interval [
,
] when the right basis splines were
used. Using the formula of the remainder term of the
interpolation polynomial that solves the Lagrange
interpolation problem, we obtain the relation
7!
…
,
∈
,
.
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.
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3 The Application of the Local Splines
of the Seventh Order of
Approximation to Calculate Integrals
First of all, we note how to apply local splines of the
seventh order of approximation to calculate integrals
over the interval ,. As already noted, the spline
approximation of the function is applied separately
for each grid interval. Applying the estimates given
in the theorem, we should calculate the integral
as followed.
Let be the nodes of the set on the interval ,:
⋯
.
We represent the integral in the form:
.
The function ,,∈,,
can be approximated with the expression:
,. Let us denote
, and
let and , 1,2,3, determine the type of
spline: the left, the right, or the middle spline. On
the intervals ,, 0,1,2, we put
,6. On the intervals ,,
3,…,4, we put 3,
3. On the
intervals ,,3,…,1, we put
5,
1.
Let us denote
. Now we use the following
approximations of the function at the first three
grid intervals of the interval ,:
, ∈ ,, 0,1,2,
We use the following approximations of the
function in the middle of the interval ,:
, ∈ ,,
3,…,4.
We use the following approximations of the
function at the last three grid intervals of the
interval ,:
, ∈ ,,
3,…,1.
We assume that the integral ,
can be computed exactly. Otherwise, we can use the
quadrature formulas. In this case, it is necessary to
take into account the error of the applied quadrature
formula. We can use, for example, Simpson's
compound formula.
4 The Application of the Local Splines
of the Seventh Order of
Approximation which is used to
Calculate a Solution of an Integral
Equation
First, we discuss the solution of the integral
equation of the second kind
≡ ,
,
∈,.
We assume that the kernel , and the right side
of the equation are continuous. In addition, we
assume that the equation is uniquely solvable and
the estimate for the norm of the inverse operator in
the space is known: ∥
∥ .
Let us choose an integer 10. We build a
uniform grid with a step
.
Using the results from the third section, we can
reduce the integral equation to the solution of a
system of linear algebraic equations. To do this, we
put
, 0,...,1, ,
takesthesamevaluesas , 0,...,1 in
the equation
,
,
,
.
And now we have to solve the system of linear
algebraic equations
,
,
,
.
0,...,.
First, let us consider some examples of the
application of splines of the seventh order of
approximation in the example of solving the
Fredholm integral equation of the second kind.
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Example 1. Consider the equation
sin ,
∈ 0,1.
Note that the right side of this equation was
constructed according to the exact solution, which
has the form sin10. The plot of the
function is given in Fig.8.
Fig. 8: The plot of the function .
Using splines of the seventh order of approximation
we construct the system of equations.
Solving the system of equations with the number of
grid nodes 16, we obtain the solution error
that is shown in Fig. 9.The nodes are marked along
the abscissa axis. For comparison, in Fig.10 a plot of
the absolute values of the solution error when using
splines of the second order of approximation is
given (see, [19], [20]). Programs for solving the
integral equations were developed in the Maple
system.
Fig. 9: The application of the splines of the seventh
order of approximation
Fig. 10: The application of the composite quadrature
formula of trapezoids
In Fig. 10, Fig. 12, along the abscissa axis, the
integration interval is marked, and in other figures,
grid nodes are plotted along the abscissa axis.
Example 2. Consider the equation
,
∈ 0,1.
The exact solution of this equation is
sin3. The right side of the integral equation
was constructed according to the exact solution.
Using splines of the seventh order of approximation
we construct the system of equations. Solving the
system of equations with the number of grid nodes
16, we obtain the solution error that is shown
in Fig. 11. Fig. 12 shows the error in solving the
equation using the composite quadrature formula of
trapezoids (16).
Fig. 11: The application of the splines of the seventh
order of approximation
Fig. 12: The application of the composite quadrature
formula of trapezoids
Note that to achieve the order of error of 10 using
the trapezoid formula, the number of grid nodes
512 was required, and the computation time was
125 seconds. Using the splines we can construct
formulas for approximating the derivatives of the
function with the given error.
Example 3. Consider the equation
sin ,
∈ 0,1.
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The exact solution of this equation is
sin. The right side of the integral equation was
constructed according to the exact solution. Solving
the equation using splines of the seventh order of
approximation with the number of grid nodes
16, we obtain the solution error that is shown in
Fig. 13.
Fig. 13: The application of the splines of the seventh
order of approximation 16.
Solving the equation using splines of the seventh
order of approximation with the number of grid
nodes 10, we obtain the solution error that is
shown in Fig. 14.
Fig. 14: The application of the splines of the seventh
order of approximation 10
Example 4. Now consider the integral equation:
exp ,
where ∈0,1.
The exact solution of the integral equation is the
next: exp
. The right side of the integral
equation was constructed according to the exact
solution. Applying spline approximations of the
fifth order to the solution of the equation, [19], we
obtain the error which is shown in Fig. 15. When
applying spline approximations of the seventh order
to the solution of the equation, we obtain the error
which is shown in Fig. 16. A program was
developed in the Maple environment. A uniform
grid of nodes was built on 0,1, consisting of 16
nodes 16.
Fig. 15: The plot of the error obtained using spline
approximations of the fifth order (Digits=20,
16)
Fig. 16: The plot of the error obtained using spline
approximations of the seventh order 16
5 Conclusion
In this paper, we consider the solution of integral
equations of the second kind using splines of the
seventh order of approximation. The results of
solving the same integral equations using splines of
the order of approximation less than 7 are given
also. 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 a lower degree. In the future,
numerical schemes for integro-differential equations
will be constructed. It is supposed to develop
numerical methods for solving integral equations
with a weak singularity based on the use of splines
of the seventh order of approximation.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
G. O. Alcybeev executed the numerical
experiments, and I. G. Burova developed the
theoretical part.
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 in the
preparation of this paper (Pure ID 94029567, ID
104210003), as well as to a resource center for
providing the package Maple.
Conflict of Interest
The authors have no conflict of interest to declare.
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
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