Application of Splines of the Seventh Order Approximation to the
Solution of the Fredholm Integral Equations with Weekly Singularity
I. G. BUROVA, G. O. ALCYBEEV
Department of Computational Mathematics,
St. Petersburg State University,
7-9 Universitetskaya Embankment, St.Petersburg,
RUSSIA
Abstract: - We consider the construction of a numerical solution to the Fredholm integral equation of the
second kind with weekly singularity using polynomial spline approximations of the seventh order of
approximation. The support of the basis spline of the seventh order of approximation occupies seven grid
intervals. In the beginning, in the middle, and at the end of the integration interval, we apply various
modifications of the basis splines of the seventh order of approximation. We use the Gaussian-type quadrature
formulas to calculate the integrals with a weakly singularity. 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 to solve integral equations are given.
Key-Words: - Fredholm integral equation, weekly singularity, polynomial spline approximations.
Received: November 11, 2022. Revised: May 10, 2023. Accepted: June 2, 2023. Published: June 28, 2023.
1 Introduction
As noted in the paper, [1], “Many fields in the area
of applied mathematics rely on the knowledge of
integral equations, as they arise naturally in various
applications in mathematics, engineering, physics,
and technology. They can be used to model a wide
range of physical problems such as heat conduction,
diffusion, continuum mechanics, geophysics,
electricity, magnetism, neutron transport, traffic
theory, and many more. Integral equations provide
solutions to design efficient parametrization
algorithms for algebraic curves, surfaces, and
hypersurfaces. Many initial and boundary value
problems associated with ordinary and partial
differential equations can be reformulated as
integral equations.
Nowadays, many papers are devoted to the
numerical solution of integral equations. We note
several papers on this subject.
[1], presents a numerical iterative method for the
approximate solutions of nonlinear Volterra integral
equations of the second kind, with weakly singular
kernels. The author of the paper derives conditions
so that a unique solution of such equations exists as
the unique fixed point of an integral operator. The
iterative application of that operator to an initial
function yields a sequence of functions converging
to the true solution. Finally, an appropriate
numerical integration scheme (a certain type of
product integration) is used to produce the
approximations of the solution at given nodes.
In the paper, [2], a computational scheme is
presented to solve weakly singular integral
equations of the second kind. The discrete
collocation method in addition to the moving least
squares (MLS) technique established on scattered
points is utilized to estimate the solution of integral
equations. The discrete collocation technique for the
approximate solution of integral equations results
from the numerical integration of all integrals in the
method. The authors utilize an accurate quadrature
formula based on the use of the nonuniform
composite Gauss-Legendre integration rule and
employ it to compute the singular integrals which
appeared in the approach.
In the paper, [3], product integration methods
based on discrete spline quasi-interpolants and
application to weakly singular integral equations are
presented.
In paper, [4], the advanced multistep and hybrid
methods have been used to solve the Volterra
integral equation.
In paper, [5], the kernel was initially
approximated through the Legendre wavelet
functions.
In paper, [6], the forward-jumping methods of
the hybrid type are used for the construction of the
methods with a high order of accuracy.
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In the paper, [7], 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, [8], the method of integral
equations is proposed for some of the electrical
engineering problems (current density, heat
conduction). The presented models lead to a system
of Fredholm integral equations, integro-differential
equations, or Volterra-Fredholm integral equations,
respectively.
[9], presents the thermal desorption functional
differential equations of the neutral type with
integrable weak singularity.
The purpose of the paper, [10], is to investigate
the design of the skin surface electrodes for
functional electrical stimulation using an isotropic
single-layered model of the skin and underlying
tissue. It is shown that the electric potential satisfies
a weakly singular Fredholm integral equation of the
second kind.
In the paper, [11], the Newton-iteration scheme
based on the Galerkin and the multi-Galerkin
operators is constructed. It can be used to solve non-
linear integral equations of the Fredholm-
Hammerstein type for both smooth and weakly
singular algebraic kernels.
In the paper, [12], a new computational method that
is based on a special B-spline is provided.
Previously, the authors constructed a solution to
integral equations with a weak singularity using
splines of the second order of approximation, [13].
In the present paper, the solutions of the integral
equations of the second kind with a weak singularity
are constructed using the splines of the seventh
order of approximation. In addition, a comparison is
made with the results of using splines of the second
order of approximation.
2 Problem Formulation
We discuss the numerical solution of the integral
equation of the second kind
where
We assume that 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
space C is known:
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.
Let be integers, ,
and the spline be such that
. The approximation using basis splines
is built separately on each grid interval. This
approximation is constructed in the form of the sum
of the products of the values of the function at the
grid nodes and the basis splines . Following
the methodology developed by Professor S. G.
Mikhlin, we find the basic functions by solving
the system of approximation relations
With different values of the parameters , we get
basis splines suitable for approximation at the
different parts of the interpolation interval . At
the beginning of the interpolation interval we
use the right basis splines
. In the middle of the
interpolation interval we use the middle basis
splines
. At the end of the interpolation interval
we use the left basis splines
.
2.1 Approximation Theorem
Further, we will use the norm of the vector of the
form:
Consider the approximation with the right basis
splines. Let , in this case, on the
interval the formula for the right spline
takes the form:
where the right basis splines
have the form:
(
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;
Formulas for the left and the middle splines can be
found in paper, [14]. When approximating a
function with the splines of the 7th order of
approximation, the next Theorem is valid.
Theorem. If then we have
the left splines. The following inequalities are valid:
If , then we have the
middle splines. The following approximation
estimate is valid:
.
If , then we have the right
splines. 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:
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
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.
3 Problem Solution
The application of the local splines of the seventh
order of approximation can be applied to calculate
integrals with a weekly singularity. 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
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to each grid interval. Let be the nodes of the set
on the interval :
We represent the integral in the form:
where
The function
can be approximated with the expression:
. Let us denote
Thus we obtain
Now we have to solve the system of equations
We assume that the integrals
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 formulas. When
solving integral equations, local polynomial splines
give an acceptable solution with a small number of
operations and in a short time. If the integral has a
weight function, then we can use the Gaussian
type’s quadrature formulas. In this case, we separate
the singularity into a weight function. It is possible
to use Gaussian-type composite quadrature formulas
on the interval by applying a Gaussian-
type formula with one node on each subinterval.
However, the result will be better if, on each
interval, a Gaussian-type formula with four nodes is
applied.
4 Numerical Examples
Example 1. Consider the following integral
equation from paper, [3]:
,
where is chosen such that the exact solution is
Fig. 1 shows the graph of the right
side of the equation. On the interval we
construct a uniform grid of nodes with the step
Using splines of the seventh order of
approximation, we construct the system of
equations. Note that in numerical calculations of
integrals with a weakly singularity, we can use the
Gaussian-type formulas with three or four nodes
over the interval .
Fig 1: The plot of the function
Calculations are made in the MAPLE system with
Fig. 2, Fig. 3, Fig. 4, and Fig. 5 show
graphs of the solution errors in absolute value for
different numbers of grid nodes. Fig.2 shows the
plot of the errors obtained with the polynomial
splines of the second-order approximation (with 16
nodes), [13]. Note, that the numbers of the grid
nodes are marked along the abscissa axis. Fig.3
shows the plot of the errors obtained with the
polynomial splines of the seventh-order
approximation (with 10 nodes). Fig. 4 shows the
plot of the errors obtained with the polynomial
splines of the seventh-order approximation (with 32
nodes). Fig. 5 shows the plot of the errors obtained
with the polynomial splines of the second-order
approximation (with 32 nodes).
Fig. 2: The plot of the errors obtained with the
polynomial splines of the second-order
approximation (16 nodes).
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Fig. 3: The plot of the obtained errors with the
polynomial splines of the seventh order (10 nodes).
Fig. 4: The plot of the obtained errors with the
polynomial splines of the seventh-order
approximation (32 nodes).
Fig 5: The plot of the obtained errors with the
second-order polynomial splines (32 nodes).
Example 2. Consider the following integral
equation from the paper, [3]:
,
[0, ], where is chosen such that the exact
solution is
The plot of the function is given in Fig.6.
Fig. 6: The plot of the function
We construct a uniform grid of nodes with the step,
Using splines of the seventh order of
approximation, we construct the system of
equations. Fig. 7, Fig. 8, Fig. 9, Fig. 10, and Fig. 11
show graphs of the solution errors in absolute value
for different numbers of grid nodes. Fig. 7 shows
the plot of the errors obtained with the polynomial
splines of the second-order approximation (with 64
nodes). Fig.8. shows the plot of the errors obtained
with the polynomial splines of the second-order
approximation (with 16 nodes). Fig. 9 shows the
plot of the errors obtained with the polynomial
splines of the second-order approximation (with 32
nodes). Fig. 10 and Fig. 11 show the plot of the
errors obtained with the polynomial splines of the
seventh-order approximation (with 32 and 16
nodes). Calculations are made in the MAPLE
system with
Fig. 7: The plot of the obtained errors with the
polynomial splines of the second-order
approximation (64 nodes).
Note that in numerical calculations of integrals
with a weak singularity, we can use Gaussian-type
formulas with three or four nodes over the interval
, or composite Gaussian-type quadrature
formulas. For example, when , the division
into 128 subintervals was used, and in the case of
, the division into 256 subintervals was used.
On every subinterval, one node was taken. The
results are shown in Fig.10 and Fig. 11. Calculations
are made in the MAPLE system with
Fig. 8: The plot of the obtained errors with the
polynomial splines of the second order (16 nodes).
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Fig. 9: The plot of the obtained errors with the
polynomial splines of the second-order
approximation (32 nodes).
Note that in numerical calculations of integrals
with a weak singularity, we can use the Gaussian-
type formulas with three or four nodes over the
interval, , or composite Gaussian-type
quadrature formulas. For example, when ,
the division into 128 subintervals was used, and in
the case of , the division into 256
subintervals was used. On every subinterval, one
node was taken. The results are shown in Fig. 10
and Fig. 11. Calculations are made in the MAPLE
system with Fig. 12 shows a graph of
the solution error when the Gaussian-type formula
with four nodes was applied on every grid interval
and . We get the maxima error in
absolute value which equals to 0.2
Calculations are made in the MAPLE system
with .
Fig. 10: The plot of the obtained errors with the
polynomial splines of the seventh order (32 nodes).
Fig. 11: The plot of the obtained errors with the
polynomial splines of the seventh order (16 nodes).
Fig. 12: The plot of the obtained errors with the
polynomial splines of the seventh order (64 nodes).
Let be the exact solution and the
approximate solution of the integral equation. Table
1 shows the maxima errors in absolute value
( ).
The errors obtained using splines of the second
order of approximation are given in the second
column in Table 1. The errors obtained using splines
of the seventh order of approximation are given in
the third column in Table 1. The last column in
Table 1 contains the result from Paper, [3]. The first
column contains the number of nodes . In Table 1,
the error of the solution was calculated when the
composite quadrature Gauss type’s formula with
four nodes on every grid interval was
applied in the case of splines of the seventh order of
approximation. Note that the values of the solution
error presented in Table 1 correspond to the
theoretical errors of approximation with the
corresponding splines of the second or the seventh
order of approximation.
Table 1. The maxima errors in absolute value
n
The maxima errors in absolute value
Splines of the
2nd order of
approximation
Splines of the
7th order of
approximation
Results
from
paper, [3]
16
0.3
0.4
0.4
32
0.6
0.2
0.4
4 Conclusion
In this paper, we present a numerical scheme to
solve the Fredholm equation of the second kind with
a weak singularity. This scheme can be used for
engineering calculations. If we compare numerical
methods based on local spline approximations of
different orders of accuracy, we note the following.
Numerical methods based on the local spline
approximations of the seventh order of
approximation, give a more accurate result when the
solution is sufficiently smooth. In this case, we can
use a small number of grid nodes. The advantages of
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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.
In the following papers, a generalization will be
made to the case of a solution from several
variables.
Acknowledgment:
The authors are gratefully indebted to St. Petersburg
University for their financial support in the
preparation of this paper (Pure ID 104625746), as
well as to a resource center for providing the
package Maple.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
G.O. Alcybeev executed the numerical experiments.
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 104625746).
Conflict of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
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