The Local Nonpolynomial Splines and Solution of Integro-Differential
Equations
I. G. BUROVA
Department of Computational Mathematics
St. Petersburg State University
RUSSIA
Abstract: The application of the local polynomial splines to the solution of integro-differential equations was
regarded in the author’s previous papers. In a recent paper, we introduced the application of the local
nonpolynomial splines to the solution of integro-differential equations. These splines allow us to approximate
functions with a presribed order of approximation. In this paper, we apply the splines to the solution of the
integro-differential equations with a smooth kernel. Applying the trigonometric or exponential spline
approximations of the fifth order of approximation, we obtain an approximate solution of the integro-
differential equation at the set of nodes. The advantages of using such splines include the ability to determine
not only the values of the desired function at the grid nodes, but also the first derivative at the grid nodes. The
obtained values can be connected by lines using the splines. Thus, after interpolation, we can obtain the value
of the solution at any point of the considered interval. Several numerical examples are given.
Key-Words: Local nonpolynomial splines, local trigonometric splines, local exponential splines, integro-
differential equation, the fifth order of approximation
Received: September 9, 2021. Revised: August 16, 2022. Accepted: September 19, 2022. Published: October 24, 2022.
1 Introduction
Many mathematical models are described by linear
or nonlinear integral equations. Integral equations
appear in nonlinear physical phenomenons such as
electomagnetic fluid dynamics, reformulation of
boundary value problem. Integro-differential
equations are encountered in modeling various
processes.
Integro-differential equations (IDEs) have been
used extensively in biological models, economics,
oscillation theory, ocean circulations, control theory
of industrial mathematics and other fields [1], [2].
Paper [3] is devoted to the study of an integro-
differential system of equations modeling the
genetic adaptation of a pathogen by taking into
account both the mutation and selection processes.
Using the variance of the dispersion in the
phenotype trait space as a small parameter the
authors provide a complete picture of the dynamical
behaviour of the solutions of the problem.
The (2+1) dimensional Konopelchenko–Dubrovsky
equation (2D-KDE) is an integro differential
equation which describes a two-layer fluid in
shallow water near ocean shores and stratified
atmosphere (see paper [4]).
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
(see paper [5]).
In study [6], a numerical technique with hybrid
approximation is developed for solving high-order
linear integro-differential equations including
variable delay under the initial conditions. These
types of problems are of applications in
mathematical physics, mechanics, natural sciences,
electronics and computer science.
As noted in paper [7], the wireless sensor network
and industrial internet of things have been a growing
area of research which is being exploited in various
fields such as smart homes, smart industries, smart
transportation, and so on. There is a need for a
mechanism which can easily tackle the problems of
nonlinear delay integro-differential equations for
large-scale applications of the Internet of Things. In
paper [7], the Haar wavelet collocation technique is
developed for the solution of nonlinear delay
integro-differential equations for the wireless sensor
network and the industrial Internet of Things. The
method is applied to the nonlinear delay Volterra,
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delay Fredholm and delay Volterra–Fredholm
integro-differential equations which are based on the
use of Haar wavelets.
Paper [8] noted that it is well known that the
study of many processes of the natural sciences can
be reduced to solving the Volterra
integrodifferential equations. Recent studies on
certain problems such as the HIV virus, bird flu
virus, and diseases associated with mutations of
viruses have become relevant. A solution to such
problems is associated with finding solutions of
VIDEs. There are several classes of methods for
solving IDEs. In contrast to the known methods,
paper [8] developed the finite difference hybrid
method by a combination of power series and the
shifted Legendre polynomial through a block
method.
At present, many authors are trying to construct
more accurate methods for solving integro-
differential equations. In paper [9] efficient
numerical methods are given to solve the linear
Volterra integral equations and Volterra Integro
differential equations of the first and second types
with exponential, singular, regular and convolution
kernels.
In paper [10] the authors introduce a numerical
method for solving the nonlinear Volterra integro-
differential equations. In the first step, the authors
apply the implicit trapezium rule to discretize the
integral in a given equation. Further, the Daftardar-
Gejji and Jafari technique is used to find the
unknown term on the right side.
In study [11], the second order linear Volterra
partial integro-differential equation are solved with
the collocation method based on the Lerch
polynomials.
In paper [12], a 6th order Runge-Kutta with the
seven stages method for finding the numerical
solution of the Volterra integro-differential equation
is considered. Here the integral term in the Volterra
integro-differential equation approximated using the
Lagrange interpolation numerical method is
discussed.
Finite elements, splines and wavelets are often
used to construct computational schemes for solving
integro-differential equations.
In paper [13], the authors present a new mixed
finite element method for a class of parabolic
equations with p-Laplacian and nonlinear memory.
In paper [14], a new collocation method based on
the Pell–Lucas polynomials is presented to solve the
parabolic-type partial Volterra integro-differential
equations.
In paper [15], a new three-point linear rational
finite difference (3LRFD) formula is investigated,
which is combined with the compound trapezoidal
scheme to discretize the differential term and
integral term of second-order linear Fredholm
integro-differential equation (SOLFIDE)
respectively, and then the corresponding 3LRFD-
quadrature approximation equation can be derived
and then generate the large and dense linear system.
In paper [16], the authors consider the Jacobi
collocation method for the numerical solution of the
neutral nonlinear weakly singular Fredholm integro-
differential equations.
Paper [17], focuses on an efficient spline-based
numerical technique for numerically addressing a
second-order Volterra partial integrodifferential
equation. In paper [George] the time derivative is
discretized using a finite difference scheme, while
the space derivative is approximated using the
extended cubic B-spline basis.
Paper [18], aims to present a new method for the
approximate solution of two-dimensional nonlinear
Volterra–Fredholm partial integro-differential
equations with boundary conditions using two-
dimensional Chebyshev wavelets.
In paper [19], the authors approximate the solution
of Fredholm integro-differential equations of the
second kind by using exponential spline function.
The proposed method reduces to the system of
algebraic equations.
Although two-dimensional (2D) parabolic
integro-differential equations (PIDEs) arise in many
physical contexts, there is no general available
software that is able to solve them numerically. To
remedy this situation, in paper [20], the authors
provide a compact implementation for solving 2D
PIDEs using the finite element method (FEM) on
unstructured grids. Piecewise linear finite element
spaces on triangles are used for the space
discretization, whereas the time discretization is
based on the backward-Euler and the Crank–
Nicolson methods. The quadrature rules for
discretizing the Volterra integral term are chosen so
as to be consistent with the time-stepping schemes;
a more efficient version of the implementation that
uses a vectorization technique in the assembly
process is also presented.
In work [21], the Legendre wavelet collocation
method is implemented for the numerical solution of
nonlinear integral and integro-differential equations.
The authors approximate the solution with the
Legendre wavelet.
In this paper we consider the solution of the
linear Volterra–Fredholm integro-differential
equations of the second kind with a continuous
kernel and a continuous right-hand side. When
solving such equations, the application of
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polynomial spline approximations is one of the
possible ways to calculate a solution in the grid
points (see the author's paper [22]). After we have
computed the solution at the grid nodes, we can
compute the solution at additional points between
these nodes using the interpolation with these spline
approximations. Moreover, the resulting line can not
only be continuous, but it can also be quite smooth.
Methods for constructing such approximations were
previously considered in the author’s papers.
The method for calculating the error of
approximation using the non-polynomial splines is
given in the author’s paper [23]. When solving
linear Volterra–Fredholm integro-differential
equations of the second kind, the use of non-
polynomial splines can give a more accurate result.
However, there may be problems with the
calculation of the integral. In this case, it is
necessary to apply the corresponding quadrature
formulas. These quadrature formulas can also be
built using the local non-polynomial splines. The
second section presents formulas for the
trigonometric, exponential and polynomial splines
of the fifth order of approximation. The third section
presents the results of numerical experiments.
2 Construction of Nonpolynomial
Approximations
2.1 Nonpolynomial Approximations of the
Fifth Order
The basic splines of the fifth-order approximation
are found separately on each grid interval. Recall
that when constructing an approximation on a finite
interval , we have to use different types of
fifth-order approximations. We have to distinguish
between the approximations near the left end of the
interval , the right end of the interval ,
and near the middle of the interval . We note
that the approximation with the middle splines gives
a smaller approximation error compared to the
approximations with the left splines or with the right
splines. To do this, we have to solve a system of
approximation relations. Suppose that the functions
form a Chebyshev system, and
the determinant of the system is nonzero. Let the
values of the function be known at the nodes
of the grid
.
Approximation with the local splines of the fifth
order of approximation is built separately on each
grid interval . Denote In the
case of the middle basis splines, the system of
equations looks as follows:
, (1)
.
In the case of the left basis splines, the system of
equations looks as follows:
, (2)
.
In the case of the right basis splines, the system of
equations looks as follows:
, (3)
.
Solving this system of equations, we obtain
formulas for the basis splines.
2.2 Trigonometric Splines of the Fifth Order
of Approximation
In order to simplify the expressions, we will do the
following on the equidistant set of nodes with step
on . Let us introduce a variable Now
we get for : To construct
trigonometric splines of the fifth order of
approximation, we take
(4)
Let supp Now the middle
trigonometric basis splines (according relation (1))
on the interval : can be represented as
followed:
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It is easy to calculate that on the interval ,
the middle trigonometric basis functions satisfy the
inequalities:
We construct an approximation with the middle
splines on the interval according to the
formula:
, (5)
In author [23]’s paper, a technique for constructing
the error of approximation of functions by non-
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polynomial splines is given. In the case of using
local trigonometric splines of the fifth order of
approximation, following this technique, we note
the following. According to this theory, on the
interval , we represent the function
as
Here are arbitrary constants.
It is easy to see that the expression
when
And it is easy to see that the expression
when
It can be obtained that the next inequality is valid:
Having solved the systems of equations (2), (4) we
obtain the formulas of the left trigonometric basis
splines. Having solved the systems of equations (3),
(4) we obtain the formulas of the right trigonometric
basis splines.
We construct the approximation with the left splines
on the interval according to the formula:
. (6)
We construct the approximation with the right
splines on the interval according to the
formula:
. (7)
Note that approximations with the trigonometric
splines have the following properties:
,
when .
2.3 Exponential Splines of the Fifth Order of
Approximation
In the case of applying a system of functions
(8)
we obtain the exponential basis splines.
Let supp Solving a system of
equations, we obtain the left exponential basis
splines (see (2)) of the form:
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.
It is easy to calculate that on the interval ,
the left basis functions satisfy the
inequalities:
We construct an approximation by such splines on
the interval according to the formula:
.
Similarly, we obtain the middle exponential basis
splines of the form:
It is easy to calculate that on the interval ,
the middle basis functions satisfy the inequalities
We construct an approximation by such splines on
the interval according to the formula
,
In author [23]’s paper, a method for constructing an
estimate of the approximation error with the
exponential splines is given. First of all, we note
that the function can be represented in the form:
Here are arbitrary constants. It is easy to see that
the expression
when
It can be obtained that the next inequality is valid:
Having solved the systems of equations (2), (8) we
obtain the formulas of the left exponential basis
splines. Having solved the systems of equations (3),
(8) we obtain the formulas of the right exponential
basis splines.
We construct the approximation with the left
exponential splines on the interval
according to the formula:
,
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We construct the approximation with the right
exponential splines on the interval
according to the formula:
.
Note that approximations with the exponential
splines have the following properties:
,
when .
2.4 Polynomial Splines of the Fifth Order of
Approximation
To construct the polynomial splines of the fifth
order of approximation, we take:
The construction of polynomial basic splines is
considered in detail in the author's articles earlier.
The middle basis splines are used in the middle of
the interval The approximation with the
middle basis splines has the form (see paper [23]):
,
where
It is easy to calculate that on the interval ,
the middle basis functions satisfy the inequalities:
The error of approximation can be written in the
form:
At the end of the interval , we apply the
approximation with the right splines:
where the basis splines are the following:
The error of approximation with the right splines
can be written in the form:
Note that approximations with the polynomial
splines have the following properties:
,
when .
Consider now the approximation by middle splines
on a finite interval
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It follows from formulae (5), (6), (7) that when
approximating with the splines of the fifth order of
approximation on a finite interval , the values
of the function are required at points that go beyond
this finite interval . In particular, when
approximating with the middle splines on the finite
interval , it is necessary to take into account
the values of the function in two additional nodes to
the right and to the left of the boundaries of the
interval
Let us take Denote
.
Thus, the grid of knots was extended to the left of
the interval by two nodes: and to the
right of the interval by two nodes: .
It was assumed that the function values at these
additional nodes are known. To calculate the
maximum error, each grid interval was
divided into 100 parts. At each division point, an
approximation with the cubic splines of the function
was calculated ( in
Maple, ). Table 1 shows the maxima in
absolute value of the actual errors of the
approximation with the middle trigonometric splines
of functions and their first derivative: Table 2 shows
the maxima in absolute value of the actual errors of
the approximation with the middle exponential
splines of functions and their first derivative. Table
3 shows the maxima in absolute value of the actual
errors of the approximation with the middle
polynomial splines of functions and their first
derivative:
Table 1 The actual errors of the approximation of
functions with trigonometric splines
Approximation of
Approximation of
Approximation of
0.0000442
0.0191
0.000290
0.0120
0.000531
0.0305
Table 2 The actual errors of the approximation of
functions with the middle exponential splines
Approximation of
Approximation of
Approximation of
0.0000140
0.000582
0.000429
0.0178
0.000161
0.00927
Table 3 The actual errors of the approximation of
functions with the middle polynomial splines
Approximation of
Approximation of
Approximation of
0.0000142
0.000590
0.000358
0.0148
0.000329
0.0189
Table 4 shows the maxima in absolute value of the
errors of the theoretical approximation of functions
with the trigonometric splines and with the
exponential splines
Table 4. The errors of the theoretical approximation of
functions with the trigonometric splines and with
exponential splines
Approximation of with
trigonometric splines
exponential splines
0.000004871
0.000107
0.0000551
0.000612
0.00130
0.00190
0.000830
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It is easy to see that the results presented in the
Tables confirm the theoretical estimates. In the next
section, we apply exponential, trigonometric and
polynomial splines to solve integro-differential
equations.
3 Problem Solution
In this section, three examples are given to illustrate
the application of the splines in the solving of
integro-differential equations.
For an approximate calculation of the integral
, we use the Newton-Cotes rule. Let
be integer number. As is known, the Newton-
Cotes quadrature rules have the form:
where
When we have
For constructing the numerical method of solving
the integro-ifferential equation with the splines we
can take
where is the approximation of ,
In the case of using the middle splines we get
In the case of using the left splines we get
Thus, using the Newton-Cotes rule when we
have in case of the left splines:
Example 1. First let us solve the Volterra integro-
differential equation
when , , . The exact
solution is the next: .
The calculations were carried out in the Maple
environment. Fig.1. shows the plot of the error of
approximation of the solution of Example 2
obtained with the trigonometric splines when .
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In the figures along with the abscissa axis, the grid
nodes from the interval [0,1] are marked with blue
circles.
Fig.1. The plot of the error of approximation obtained
with the trigonometric splines (Example 1)
Now the goal of the section is to inspect the
numerical technique to approximate the solution of
the linear second-order Fredholm integro-
differential equations (FIDEs) of the form:
with boundary conditions at two points
Example 2. Let us apply our theory to solve the
integral equation
, with boundary conditions
. The exact solution is the next:
.
Fig.2. shows the plot of the error of approximation
of the solution of Example 2 obtained with the
polynomial splines when
Fig.2. The plot of the error of approximation obtained
with the polynomial splines (Example 2)
Fig.3. shows the plot of the error of approximation
of the solution of Example 2 obtained with the
trigonometric splines when
Fig.3. The plot of the error of approximation obtained
with the trigonometric splines (Example 2)
The half-sweep (HS) concept is combined with the
refinement of the successive over-relaxation
(RSOR) iterative method to create the new half-
sweep successive over-relaxation (HSRSOR)
iterative method, which is implemented to get the
numerical solution of a system of linear algebraic
equations (see paper [1]). In paper [1] the
applicability of the half-sweep successive over-
relaxation (HSRSOR) method has been successfully
proven.
Example 3. (This example is taken from the paper
[1]).
Exact solution is ,
.
Method FSRSOR-3LRFD from paper [1] gives the
error of approximation 6.0313E-06 when 32 nodes
were taken.
Fig.4. shows the plot of the error of approximation
of the solution of Example 3 obtained with the
exponential splines when , Digits=20. Fig.5.
shows the plot of the error of approximation of the
solution of Example 3 obtained with the exponential
splines when , Digits=20.
Fig.4. The plot of the error of approximation obtained
with the exponential splines, 32 nodes (Example 3)
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Fig.5. The plot of the error of approximation obtained
with the exponential splines, 16 nodes (Example 3)
Fig.6. shows the plot of the error of approximation
of the solution of Example 3 obtained with the
polynomial splines when , Digits=20
Fig.6. The plot of the error of approximation obtained
with the polynomial splines, 16 nodes (Example 3)
Fig.7. shows the plot of the error of approximation
of the solution of Example 3 obtained with the
trigonometric splines when , Digits=20
Fig.7. The plot of the error of approximation obtained
with the trigonometric splines, 16 nodes (Example 3)
Fig.8. The plot of the error of approximation obtained
with the polynomial splines, 32 nodes
(Example 3)
Fig.9. The plot of the error of approximation obtained
with the trigonometric splines, 32 nodes (Example 3)
The advantages of using such splines include the
ability to determine not only the values of the
desired function at the grid nodes, but also the first
derivative at the grid nodes. The obtained values can
be connected by lines using the splines. Thus, after
interpolation, we can obtain the value of the solution
at any point of the considered interval. Several
numerical examples are given.
From the results presented in the numerical
examples, it follows that before the numerical
solution of the integral equation, the kernel and the
right side of this equation should be analyzed. If the
kernel and the right side are a trigonometric
expression, then it is advisable to use a numerical
method for solving the integral equation based on
trigonometric splines.
If the kernel and the right side are an exponential
expression, then it is advisable to apply a numerical
method for solving the integral equation based on
exponential splines. If the kernel and the right side
are a polynomial expression, then it is advisable to
use a numerical method for solving the integral
equation based on polynomial splines.
The considered examples show, that the use of
non-polynomial splines can give a smaller solution
error even with a small number of grid nodes if the
choice of spline approximation corresponds to the
form of the kernel and the right side of the integral
equation.
4 Conclusion
In this paper, we construct a solution to an integro-
differential equation using non-polynomial splines
on a uniform grid of nodes. To apply quadrature
formulas, the kernel of the integral equation and the
solution are assumed to be sufficiently smooth
functions. In the future, it is planned to construct a
solution to the integro-differential equation on a
non-uniform adaptive grid of nodes.
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Sources of funding for research
presented in a scientific article or
scientific article itself
The authors are gratefully indebted to St. Petersburg
University for financial supporting the preparation
of this paper (Pure ID 92424538, 93852135) as well
as a resource center in St. Petersburg University for
providing the Maple package
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