Fredholm Integral Equation and Splines of the Fifth Order of
Approximation
I. G. BUROVA
Department. of Computational Mathematics
St. Petersburg State University
7/9 Universitetskaya nab, St.Petersburg, 199034
RUSSIA
Abstract: - This paper considers the numerical solution of the Fredholm integral equation of the second kind
using local polynomial splines of the fifth order of approximation and the fourth order of approximation (cubic
splines). The basis splines in these cases occupy five and four adjacent grid intervals respectively. Different
local spline approximations of the fifth (or fourth) order of approximation are used at the beginning of the
integration interval, in the middle of the integration interval, and at the end of the integration interval. The
construction of the calculation schemes for solving the Fredholm equation of the second kind with these splines
is considered. The results of the numerical experiments on the approximation of functions and on the solution
of the Fredholm integral equations are presented. The results of the solution of the integral equation which uses
the polynomial splines of the fifth order of approximation are compared with ones obtained with cubic splines
and with the application of the Simpson’s method. Note that in order to achieve a given error using the
approximation with quadratic splines, a denser grid of nodes is required than when using the approximation
with the cubic splines or splines of the fifth order of approximation.
Key-Words: - Fredholm integral equation of the second kind, polynomial local splines, approximation, fifth
order of approximation, fourth order of approximation
Received: June 18, 2021. Revised: March 20, 2022. Accepted: April 19, 2022. Published: May 20, 2022.
1 Introduction
When solving a number of applied problems,
researchers have to solve the Fredholm integral
equations. One of the classical forms of
representation of dynamic systems is integral
equations. This representation method is compact
and convenient in the case of linear stationary
systems, when the spectral characteristics of the
input and output of the process associated with the
useful signal and noise are known. The integral
equations contain the complete statement of the
problem together with the initial conditions. Of
particular interest is the representation of
nonstationary dynamical systems by integral
equations. Integral equations are divided into two
main classes: linear and non-linear. In this paper, we
consider the solution of the linear Fredholm
equation of the second kind using local interpolation
splines. The solution of the integral equations of
Fredholm and Volterra, using local interpolation
splines of the second, third and fourth order of
approximation, was considered in the author's
earlier papers. Here we will focus on the use of local
interpolation splines of the fifth order of
approximation.
There are many numerical methods for solving
the Fredholm integral equation of the second kind.
There are various classical methods based on the use
of composite quadrature formulas of average
rectangles, trapezoids, and the Simpson’s method.
The properties of these methods such as
approximation, stability and convergence are well
studied. In some cases, classical methods give a
significant error in the solution. Therefore, many
researchers are trying to construct new approaches
to the numerical solution of integral equations,
which can have a smaller error in the solution. It is
often convenient to construct a solution to the
Fredholm equation based on the use of splines. B-
splines are often used for solving Fredholm
equations.
Among the papers published over the past 3
years on this topic, we note the papers [1]-[11]. In
paper [1], some applications to numerical analysis
especially quadrature formulas, differentiation and
numerical solutions of linear Fredholm integral
equations are given. In paper [2], the isogeometric
Galerkin and collocation methods for solving the
Fredholm integral eigenvalue problem on arbitrary
multipatch domains are introduced. In paper [3], the
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solution of the nonlinear Fredholm integro-
differential equation (NFID) in the complex plane
by periodic quasi-wavelets is approximated. In
paper [4], a numerical solution of important weakly
singular type of Volterra - Fredholm integral
equations WSVFIEs using the collocation type
quasi-affine biorthogonal method (based on special
B-spline tight framelets) is provided. In paper [5], a
computational method for solving nonlinear
Volterra-Fredholm Hammerstein integral equations
is proposed by using compactly supported
semiorthogonal cubic B-spline wavelets as the basis
functions. In paper [6], spline functions were used to
propose a new scheme for solving the linear
Volterra–Fredholm integral equations of the second
kind. In paper [7], a method to solve the integral
equations of the second type with degenerate
kernels and shifts, is constructed. In paper [8], an
efficient modification of the wavelets method to
solve a new class of Fredholm integral equations of
the second kind with non symmetric kernel is
introduced. In paper [9] a new computational
method for solving linear Fredholm integral
equations of the second kind, which is based on the
use of B-spline quasi-affine tight framelet systems
generated by the unitary and oblique extension
principles is presented. In paper [10], a new
collocation technique for numerical solution of
Fredholm, Volterra and mixed Volterra-Fredholm
integral equations of the second kind is introduced.
In paper [11], Farnoosh and Ebrahimi developed a
numerical method based on random sampling for
the solution of Fredholm integral equations of the
second kind, which was called the Monte Carlo
method based on the simulation of a continuous
Markov chain.
The theory of constructing approximations using
local splines was developed in the works of Prof.
Yu.K.Demyanovich and Prof. I.G.Burova. Local
polynomial and nonpolynomial splines of the
second and third order of approximation were
successfully used to construct computational
schemes for solving the integral equations of
Fredholm and Volterra.
Local polynomial splines of the fifth order of
approximation were, in particular, also considered in
detail in the works of the author. The features of
constructing error estimates of approximations with
nonpolynomial splines were discussed in detail in
the author's paper [12].
This paper is structured as follows: Section 2 of
this paper considers the main properties of
polynomial splines of the fifth order of
approximation and the splines of the fourth order of
approximation. The approximation formulas for
different arrangements of the supports of the basis
splines, and formulates approximation theorems are
given in it. Section 3 considers the application of the
splines of the fifth order of approximation to the
solution of the Fredholm integral equation of the
second kind. Also, in the third section we discuss
the construction of calculation schemes. Section 4
presents numerical examples.
As already noted, new numerical methods for
solving the Fredholm integral equation of the
second kind were considered in papers [10, 11]. In
our paper, we will solve the same Fredholm integral
equations that were considered in paper [10, 11], but
using local splines of the fifth order of
approximation instead. In addition, we will compare
our results with the results of applying the classical
method such as method of Simpson.
2 Approximation with the Local
Splines
2.1 Local Approximation with the Splines of
the Fifth Order of Approximation
First, we would like to remind the readers of the
main details of the local approximation with the
splines of the fifth order of approximation.
Let , be real and be integer. Let the values of
the function be known at the nodes of the grid
of nodes
.
Denote . In what follows, we will use the
following types of approximations of the function
on interval . At the beginning of the
interval , we apply the approximation with the
left splines
where , , are the values of the function
in nodes the basis splines are the next:
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The graph of the right basis spline is shown in Fig.1.
Fig. 1: The graph of the right basis spline
Let us denote
On each
separate interval , we can estimate the
approximation error in the assumption that the
function is 5 times continuously
differentiable. We receive the error of
approximation from the remainder term of the
Lagrange interpolation. If the grid of nodes is such
that the nodes are equidistant with a step h, then we
can find the error of approximation for
in the form:
,
If the grid nodes are not equidistant, then we denote
the length of the maximum grid interval with h.
The next option: In the middle of the interval ,
we apply the approximation with the middle splines
,
where
If the grid of nodes is such that the nodes are
equidistant with step h, then we can find the error of
approximation for in the form:
,
The graph of the middle basis spline (when supp
) is shown in Fig.2.
Fig. 2: The graph of the middle basis spline
(supp )
The graph of the middle basis spline (when supp
) is shown in Fig.3.
Fig. 3: The graph of the middle basis spline
(supp )
At the end of the interval , we apply the
approximation with the right splines:
where the basis splines are the following:
We receive the error of approximation from the
remainder term of the Lagrange interpolation. If the
grid of nodes is such that the nodes are equidistant
with step , then we can find the error of
approximation for in the form:
, .
The graph of the left basis spline is shown in Fig.4.
Fig. 4: The graph of the left basis spline
Theorem 1.
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Let .
To approximate the function
with the left and right splines, the
following inequalities are valid:
To approximate the function ,
with the middle splines, the following inequality is
valid:
Proof. It is easy to notice that
is an interpolation
polynomial, and are the
interpolation nodes,
Using the remainder term we get
We can use It can easily be
calculated that the next relation is valid:
It follows that on the uniform grid with step h we
have
.
We recall that we constructed an approximation
with an error of using approximation
identities (relations) separately on each grid interval
We define the supports of the
basis splines as follows: For the middle basis splines
we use supp or we can use
. In the case of the middle
basis splines, we distinguish two types:
supp and .
For the left basis splines we use supp
. For the right basis splines we use
. Note, that the following
relations are valid:
and
when k≠ i.
According to the location of the support of the basis
splines relative to the root-point (the point at which
the basis spline is equal to 1), the four variants of
continuous basis splines can be distinguished: the
left basis splines, the middle basis splines, and the
right basis splines. The interpolation using the right
basis splines is used at the beginning of the
interpolation interval We use the
interpolation using the middle basis splines in the
middle of the interpolation interval We use
the interpolation with the use of the left basis splines
at the end of the interpolation interval So,
when approximating with the splines of the fifth
order of approximation at the finite interval ,
we use the four types of basis splines. We can use
only one type of approximation with the fifth-order
basis splines, but in this case we have to use the
function values that lie outside the bounds of the
finite interval
Let a grid of nodes be built on the
interval When approaching with only the
middle splines on the interval we have to add
values at the grid nodes ,
.
When
approaching with only the left splines on the interval
we have to add values at the grid nodes
.When approaching with only the left
splines on the interval we have to add values
at the grid nodes
,
.
2.2 Approximation of the Functions with the
Cubic Polynomial Splines
Now, we recall the features of the approximation of
the functions with the cubic polynomial splines near
the right end of the interval , near the left end
of the interval , and at the middle of the
interval. Let {} be the set of nodes on the interval
. The middle basis splines that form the
continuous polynomial approximation in the interval
can be written as follows:
,
The approximation with these basis splines can
be written in the form:
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The continuous polynomial approximation
near the left end of the interval uses the right
basis spline
of the form:
The approximation with these basis splines can
be written in the form:
The continuous polynomial approximation
near the right end of the interval uses the left
basis spline
of the form:
The approximation with these basis splines can
be written in the form:
Theorem 2.
Let .
To approximate the function
with the left and right splines, the
following inequalities are valid:
To approximate the function ,
with the middle splines, the following inequality is
valid:
Proof. It is easy to notice that is an interpolation
polynomial, and are the
interpolation nodes,
Using the remainder term we get
We can use It can easily be
calculated that
It follows that on the uniform grid with step h
.
The approximation is constructed separately on each
grid interval . When constructing an
approximation on the interval we need the
values of the function at several neighboring nodes
to the right or left of this interval. Therefore, if the
values of the function are given on the grid of
nodes, which is constructed on a finite interval
, then we are forced to use the approximation
with the right or left splines near points .
When constructing an approximation with only the
right cubic splines on the interval we use the
values of the function at the nodes
. When constructing an
approximation with only the left cubic splines on the
interval we use the values of the function at
the nodes . When constructing an
approximation with only the right splines of the fifth
order of approximation on the interval , we use
the values of the function at the nodes
. When constructing an
approximation with only the left splines of the fifth
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order of approximation on the interval , we use
the values of the function at the nodes
. When constructing an
approximation with only the middle splines of the
fifth order of approximation on the interval ,
we use the values of the function at the nodes
.
The following Tables show the approximation errors
of functions on the interval . The actual errors
were calculated as follows. At each grid interval
, an additional grid of 100 nodes was
constructed. Next, the approximation values of the
function at these nodes were calculated. Next, the
error maxima were calculated using the formula:
.
The next tables of theoretical errors contain the
maximum deviations of the exact solution from the
approximate one on the interval [-1,1] based on the
formulas given in the theorems:
.
In 1901, Runge established that the interpolation
process over equidistant nodes on the interval
does not converge with the increasing
number of nodes even for a smooth arbitrarily
differentiable function
Table 1 presents
the actual errors in absolute values of approximation
with the polynomial cubic splines when
. Table 2 presents the theoretical errors in
absolute values of approximation with the
polynomial splines of the fifth order of
approximation when . Table 3 presents the
actual errors in absolute values of approximation
with the polynomial splines using the
polynomial splines of the fifth order of
approximation when . Table 4 presents the
theoretical errors in absolute values of
approximation with the polynomial cubic splines
when . Analyzing the information
presented in the Tables show that with the same
number of interpolation nodes, the approximation
using the middle splines gives a smaller error. The
results of numerical experiments show that the
actual errors of numerical calculations correspond to
theoretical errors. Errors when using splines of the
fifth order of approximation can be less than when
using splines of the fourth order of approximation, if
the interpolated function is sufficiently smooth.
Table 1. The results of the approximation using the cubic
polynomial splines. Actual errors ( .
Function
Cubic polynomial splines
Left splines
Middle splines
0.01388
0.009097
Function
Cubic polynomial splines
Left splines
Middle splines
0.0003098
0.0001746
0.002341
0.001327
Table 2. The results of the approximation using the
polynomial splines of the fifth order of approximation.
Actual errors ( .
Function
Polynomial splines of the fifth
order of approximation
Left splines
Middle splines
0.03372
0.01244
0.00007233
0.00002840
0.0009116
0.0003579
Table 3. The results of the approximation using the cubic
polynomial splines. Theoretical errors ( .
Function
Cubic polynomial splines
Left splines
Middle splines
0.0625
0.03516
0.0003375
0.0001898
0.002604
0.001465
Table 4. The results of the approximation using the
polynomial splines of the fifth order of approximation.
Theoretical errors ( .
Function
Polynomial splines of the fifth
order of approximation
Left splines
Middle splines
0.09496
0.03715
0.00007351
0.00002876
0.0009453
0.0003698
Note that if the derivatives of the solution grow
rapidly, then the approximation by cubic splines
may turn out to be more profitable than the
approximation by splines of the fifth order of
approximation Let the Runge function be given at
the nodes of a uniform grid with a step of
on the interval The approximation error of
the approximation of the Runge function obtained
with the cubic polynomial splines is given in Fig.5.
The maximum of the error in absolute error is
0.009097. The approximation error in absolute value
(of the approximation of the Runge function,
) obtained with the right polynomial splines of
the fifth order of approximation is given in Fig.6.
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Fig. 5: The approximation error obtained with the
middle cubic polynomial splines
Fig. 6: The approximation error obtained with the
right cubic polynomial splines
The approximation error in absolute value (of the
approximation of the Runge function,
)
obtained with the middle polynomial splines of the
fifth order of approximation is given in
Fig.7 (the maximum of the error in absolute error is
0.012438).
Fig. 7: The approximation error (in absolute value)
obtained with the middle polynomial splines of
the fifth order of approximation
Fig. 7 confirms the theoretical estimate (Theorem 2)
that the approximation with the middle splines give
lesser error than the approximation with the left or
right splines.
3 Applying splines to the Solution of
the Fredholm Equation of the Second
Kind
Consider the Fredholm equation of the second kind
(1)
We construct an approximate solution of the integral
equation by applying the polynomial splines of the
fourth order of approximation as follows. Let be
a grid of nodes on the interval . Divide the
interval into parts,
Let On each grid interval
we apply a formula of the form:
Here are the basis splines that are discussed
above, and are unknowns (values of the solution
of the equation at grid points, ) to be
found. Now we transform the following expression
At the beginning of the interval ,
the values of the parameters should
be taken. At the end of the interval at
, the values of the parameters
should be taken.
Denote by the integral
.
Now we take . The problem of solving the
integral equation is reduced to solving the system of
linear algebraic equations. When we have
the equations
When we have the equations
When we have the equations
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The unknowns in the system of equations are ,
4 Results of the Numerical
Experiments
In this section, we present the results of the
numerical experiments.
Problem 1. Now we take the next Fredholm integral
equation:
The exact solution of the integral equation is
Figs. 8, 9 show the errors of the solution
of Problem 1 with polynomial splines of the fifth
order of approximation when , 16,
. Figs. 10, 11 show the errors of the solution of
Problem 1 with cubic polynomial splines when
, 32 ( ). In the Figures, grid nodes are
marked along the axis at the interval
Fig. 8: The plot of the errors of the solution of
Problem 1 with polynomial splines of the fifth order
of approximation,
Fig. 9: The plot of the errors of the solution of
Problem 1 with polynomial splines of the fifth order
of approximation,
Fig. 10: The plot of the errors of the solution of
Problem 1 with polynomial cubic splines,
Fig. 11: The plot of the errors of the solution of
Problem 1 with polynomial cubic splines,
Now we present the result of applying Simpson’s
rule to solving Problem 1. Figure 12 shows the
graph of the error of problem 1 when the Simpson’s
method was used.
Fig. 12: The graph of the error of problem 1 when
the Simpson’s method was used.
The maximum of the error in absolute value is about
These results show that the use of
splines of the fifth order of approximation
contributes to a significant reduction in the number
of grid nodes. However, it must be remembered that
the use of splines of the fifth order of approximation
assumes that the solution of the equation and the
kernel are five times differentiable functions.
Problem 2. Let the right side of the system of
equations be constructed under the assumption that
the solution of the integral equation is
. We leave the kernel of the equation the
same. Figure 13 shows the graph of the error of
Problem 2 ( .
Fig. 13: The plot of the errors of the solution
when the polynomial cubic splines were used and
,
At the same time, when using splines of the fifth
order of approximation, we obtain the error of the
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solution, the graph of which is shown in Fig.14,
.
Fig. 14: The plot of the errors of the solution when
the polynomial splines of the fifth order of
approximation were used and ,
The errors presented in the last two graphs show
that if the derivatives of the solution grow rapidly,
then it is enough to use traditional methods for
solving integral equations. In this case, it is often
advisable to use an uneven grid of nodes
Problem 3. Now, we again take the Fredholm
integral equation from paper [10]:
The exact solution of the integral equation is
Figs. 15,16 show the errors of the solution
of Problem 1 with polynomial splines of the fifth
order of approximation, .
Fig. 15: The plot of the errors of the solution of
Problem 1 with polynomial splines of the fifth order
of approximation,
Fig. 16: The plot of the errors of the solution of
Problem 1 with polynomial splines of the fifth order
of approximation,
Now consider the solution of the following
problem.
Problem 3.
where the right side of is constructed
according to the known solution
.
First, we apply the calculation scheme constructed
using the cubic polynomial splines. Using the set of
nodes {} with we obtain the approximate
solution in the nodes (see Fig.18). Fig.17 shows
the error of the solution obtained in the nodes. It can
be calculated that the following relation is valid:
Fig. 17: The plot of the error of the approximate
solution of Problem 3 ( ).
Then we apply the numerical method with the
splines of the fifth order of approximation. Fig.19
shows the error of the solution of Problem 3 when
We have
Fig. 18: The plot of the approximate solution of
Problem 3
Fig. 19: The plot of the error of the solution of
Problem 3 when splines of the fifth order of
approximation were used ( ).
Problem 4.
As is known, in the internal Dirichlet problem of
potential theory, it is required to find a function
that is harmonic in domain D and takes
given values on the boundary of this domain D
(for example, see Kollatz [13]). Let the boundary
functions be given by the equations
.
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The solution of the internal Dirichlet problem can be
written as:
,
where the angle theta is calculated by the formula
and function satisfies the Fredholm integral
equation of the second kind.
,
where
If area is an ellipse: cos t
with semi-axes , then the kernel of the
integral equation can be written as:
Let us find the function . To do this, we solve
the integral equation
.
Let us choose . Having solved the integral
equation, we obtain the following values:
=-0.04290; := -0.1066; :=-0.1360;
:=-0.1045; :=-0.04017;
:= 0.2330;
:=0.3427; :=0.2322; :=-0.04290.
Fig. 20: The plot of the solution ( )
Then we choose . Further, the obtained
solution can be represented using splines of the fifth
order of approximation in the following form. The
graph of the solution is shown in Fig. 20. In Figures
20, 21, the values of are plotted along the x-axis,
and the calculated values
at the points are
connected using splines of the fifth order of
approximation.
Now we can calculate the temperature on the axis of
the cylinder: . We can use Simpson’s method
and the trapezium method. After the calculations,
we get ≈0.43.
Fig. 21: The plot of the solution ( )
5 Conclusion
In this paper, we study the numerical solution of the
Fredholm integral equation of the second kind using
polynomial splines of the fifth order of
approximation. Here, a comparison is also made
with the results of applying cubic splines to the
solution of the Fredholm equation. The paper also
gives approximation theorems for polynomial cubic
splines of the fourth order of approximation and
polynomial splines of the fifth order of
approximation. As is known, theorems on the
solution of an integral equation follow from
approximation theorems. However, when using
cubic splines and splines of the fourth order of
approximation, assumptions are required about the
sufficient smoothness of the kernel of the integral
equation, the solution of the integral equation and its
right side. If the smoothness is insufficient, then the
desired error reduction cannot be achieved. In this
case, it is preferable to use splines of the second or
third order of approximation or the classical
methods of trapezoids or middle rectangles.
Thus, we summarize the results obtained. If the
kernel of the integral equation is represented by a
delta function, then we almost immediately obtain a
solution. If the kernel of the integral equation and
the solution of the integral equation have sufficient
smoothness, then the method proposed in this paper
will give a good result when we use a small number
of grid nodes.
In the author’s next studies, new numerical
methods for solving the nonlinear Volterra and
Fredholm equations using spline approximations
will be considered.
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Sources of Funding for Research Presented in a
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The author is highly and gratefully indebted to
St. Petersburg University for financial supporting
the publication of the paper (Pure ID 93852135,
92424538)
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WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2022.21.31
I. G. Burova
E-ISSN: 2224-2880
270
Volume 21, 2022
