Local Interpolation Splines and Solution of Integro-Differential
Equations of Mechanic’s Problems
I. G. BUROVA
Department of Computational Mathematics,
St. Petersburg State University,
RUSSIA
Abstract: - Integro-differential equations are encountered when solving various problems of mechanics.
Although Integro-Differential equations are encountered frequently in mathematical analysis of mechanical
problems, very few of these equations will ever give us analytic solutions in a closed form. So that construction
of numerical methods is the only way to find the approximate solution. This paper discusses the calculation
schemes for solving integro-differential equations using local polynomial spline approximations of the
Lagrangian type of the fourth and fifth orders of approximation. The features of solving integro-differential
equations with the first derivative and the Fredholm and Volterra integrals of the second kind are discussed.
Using the proposed spline approximations, formulas for numerical differentiation are obtained. These formulas
are used to approximate the first derivative of a function. The numerical experiments are presented.
Key-Words: - Fredholm integro-differential equations, Volterra-Fredholm integro-differential equations, local
polynomial splines, problems of mechanics, numerical solution
Received: May 25, 2021. Revised: April 28, 2022. Accepted: June 14, 2022. Published: July 28, 2022.
1 Introduction
The history of the development of the theory of
integro-differential equations (i.e. integral equations
relating an unknown function and its derivatives)
began with the work of Volterra [1]. The
investigation of the theory of elasticity was the
beginning of Volterra's theory of integro-differential
equations. In 1909, Volterra published two papers in
which he suggested that the deformation is a linear
functional of pressure. In this case, the system of
linear integro-differential equations is the main one,
by solving which it is possible to determine the
deformation from a known force and pressure.
Integro-differential equations are encountered in
solving various problems of mechanics. Of the
problems that lead to the solution of integro-
differential equations, we can cite: the Proctor
problem on the equilibrium of an elastic beam, the
Volterra problem of torsional vibrations, the Prandtl
problem for calculating an aircraft wing. Integro-
differential equations with hinged boundary
conditions are used to study the vibrations of
suspension bridges.
For an approximate solution of integro-
differential equations, one can use various
representations of functions in the form of series, in
particular, power series. It should also be noted that
the simplest approach to solving the integro
differential equation is to replace the definite
integral with an approximating summation of a
finite number of suitably weighted discrete values of
approximate solution of an unknown function.
Integro-differential equations arise when solving
various problems of mechanics. As it is generally
known, obtaining an analytical solution for some
integro-differential equations is not possible. In this
regard, various numerical methods have been
developed for finding approximate solutions to such
equations. For example, for an approximate solution
of integro-differential equations, one can use
various representations of functions in the form of
series, in particular, power series. The improvement
of numerical methods for solving such problems is a
very important area of computational mathematics.
Let us list a few works that have recently been
published. Numerical solutions of the Fredholm
integro-differential equations of the second kind
have been considered in many papers (see, for
example, [2]-[10]. In paper [2] four numerical
methods are compared, namely, the Laplace
decomposition method (LDM), the Wavelet–
Galerkin method (WGM), the Laplace
decomposition method with the Pade approximant
(LD–PA) and the homotopy perturbation method
(HPM). In paper [3] superconvergent versions for
the numerical solution of a class of linear Fredholm
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integro-differential equations of the second kind are
discussed. In paper [4] the Hermite wavelet method
(HWM) is applied to approximate the solution of the
integro-differential equations. In paper [5] the B-
splines-least-square method and weight function of
B-splines, were proposed for solving integro-
differential equations.
The following methods are noted in these papers:
the Legendre method, Bernoulli polynomials [6],
pseudospectral methods, piecewise linear
approximation, polynomial approximation, rational
approximation [7], exponential spline [8],
differential transformation [9], and Schauder bases
[10]. The existence, uniqueness, and stability of
solutions for a class of systems of non-linear
complex Integro-differential equations on complex
planes were investigated in [11]. The Abel integral
equation of the second kind was investigated in
paper [12].
Local polynomial splines have good
approximation properties and are easy to use. The
application splines of the fifth and fourth order of
approximation to the construction of the solution of
Fredholm integral equation was considered in the
author's paper [13]. In this paper, we explore the
application of the local polynomial splines to the
construction of the solution of integro-differential
equations with the first derivative in more detail.
Using the proposed spline approximations, formulas
for numerical differentiation are obtained. These
formulas are used to approximate the first derivative
of a function. In Section 2 we consider the
polynomial cubic splines of the fourth order of
approximation. In Section 3 we consider the
polynomial splines of the fifth order of
approximation and the use of them for solving the
Fredholm and Volterra integro-differential equations
of the second kind. A comparison of the results of
applying splines of the fourth and fifth orders of
approximation with the results of applying the
methods are considered in paper [2].
2 Local Splines of the Fourth order of
Approximation and Applications
Let , be real and be integer. Let the values of
the function be known at the nodes of the grid
. Denote .
Recall that the approximation by local polynomial
splines is built separately on each grid interval
. This approximation has the form of the
product of the function values at the grid nodes and
the basis functions. Basis functions are determined
by solving a system of equations. Prof. S.G. Mikhlin
called this system of equations a system of
approximation relations. Note that the basic splines
are calculated in advance once, and then they are
used in solving various problems, including
interpolation, solving differential and integral
problems by variational methods, constructing grid
schemes, etc. When applied on a finite interval
the left, the right and the middle splines have
to be applied. This is due to the fact that it is
necessary to use grid nodes only on the given
interval
Each basis function has support consisting of s
grid intervals. The theory of approximation by
minimal interpolation splines was built in Yu.K.
Demyanovich and I.G.Burova’s works.
Approximation theorems by interpolating
polynomial splines were obtained earlier by the
authors. Assume that a uniform grid of nodes is built
and the length of the interval is equal to .
Features of the use of the polynomial cubic
splines of the fourth order of approximation, and
polynomial splines of the fifth order of
approximation are noted in the author's paper [13].
Now we recall the approximation properties of these
splines. First consider the use of polynomial cubic
splines. Approximation is constructed separately on
each grid interval as a sum of
products of function values at grid nodes and basis
splines. Approximations differ at the beginning, in
the middle and at the end of the interval .
The approximation with the right polynomial
splines is used at the beginning of the interval
and can be written in the form:
,
where
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Note that the derivative of the function satisfies the
relation:
The formulae for the first derivative of the basis
splines on a uniform grid of nodes with step takes
the form:
Let us denote
.
Table 1 shows the maximum errors in the
approximation of functions and also the maximum
errors in the approximation of their first derivative
when the right splines were used on a uniform grid
with a grid step . The grid of knots were
extended 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. The maximum errors in absolute values
in the approximation of functions and of their first
derivative
The graph of the error of the approximation of
Runge function
with the right cubic splines is
shown in Fig.1. The graph of the error of the
approximation of the first derivative of Runge
function
with the right cubic basis splines is
shown in Fig.2.
Fig. 1: The graph of the error of the approximation
of Runge function
with the right cubic
splines
Fig. 2: The graph of the error of the approximation
of the first derivative of Runge function
with
the right cubic splines
Note that the formula for approximating the
function by right splines implies the formula for
approximating the first derivative on a uniform grid
of nodes with step
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The continuous polynomial approximation
near the right end of the interval uses the left
basis spline
of the form:
Note that the derivative of the function satisfies
the relation:
Let The formulae for the first
derivative of the basis splines on a uniform grid of
nodes with step take the form:
The graph of the error of the approximation of
Runge function
with the left cubic splines is
shown in Fig.3. The graph of the error of the
approximation of the first derivative of Runge
function
with the left cubic splines is shown
in Fig.4.
Fig. 3: The graph of the error of the approximation
of Runge function
with the left cubic splines
Fig. 4: The graph of the error of the approximation
of the first derivative of Runge function
with
the left cubic splines
The errors in the approximation errors of the Runge
function (Figs. 1, 3) and the derivative of the Runge
function (Figs. 2, 4) confirm the theoretical results
presented in the Theorem and Tables 1, 2. In
addition, we should remember that we should not
approximate the Runge function with the Lagrange
interpolation polynomials on a uniform grid on the
interval [-1,1]. The problem is that the norm of the
error in approximating the Runge function by
interpolation polynomials tends to grow infinitely as
the degree of the interpolation polynomial increases.
In our case, when we apply spline approximations,
we obtain a completely satisfactory result. Here,
when the uniform grid is refined, the approximation
error decreases. This follows from the theoretical
results formulated in the theorems.
In addition, looking at Figures 1-4, we can see that
the use of an appropriate non-uniform grid can give
a solution with a smaller error.
In earlier works of the author, also the middle
splines were considered. With application of the
middle splines, we get a smaller approximation
error. In this paper, we will not dwell on
approximations by the middle splines.
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Let us denote
Table 2 shows the maximum errors in the
approximation of functions and their first derivative
with the left splines on a uniform grid with a grid
step . The grid of knots has been extended
to the right by two nodes: .
Table 2. The maximum errors in absolute values
in the approximation of functions and of their first
derivative
0
Note that the formula for approximating the
function by right splines implies the formula for
approximating the first derivative on a uniform grid
of nodes with step :
The following Theorem is true.
Theorem 1. Let .
To approximate the
function with the left and right
splines, the following inequalities are valid:
The proof can be found in paper [13].
Consequence. Let the values of the function be
given at the grid nodes with step . For an
approximate calculation of the first derivative of a
function, the following equalities are valid.
Let us dwell on the case of the presence in the
equation of the derivative of both the first and
second orders. We will replace these derivatives
both with the help of known numerical
differentiation formulas and with the help of
formulas obtained using cubic splines.
Problem 1. Consider the integro-differential
equation
with the and the exact solution is the
next: .
The function under the integral sign is
approximated by the cubic splines. For the
approximation we use the formulae from the
Consequence to Theorem 1.
The error of the solution of Problem 1 obtained with
cubic splines at 9 grid nodes ( ) is shown in
Figs. 5, 6. Fig. 5 shows us the solution when
, Fig. 6 shows us the solution when
.
Fig. 5: The plot of the error of the solution of
Problem 1 obtained when .
Fig. 6: The plot of the error of the solution of
Problem 1 obtained when .
3 Local Splines of the Fifth order of
Approximation and Applications
Next, we are interested in comparing the results of
when we apply the fifth-order polynomial splines
[13]. Denote . In what follows, we will
use the following types of approximations of the
function on interval . At the beginning
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of the interval , we apply the approximation
with the right splines
where , , are the values of the function
in nodes the basis splines are the next:
In the middle of the interval , we apply the
approximation with the middle splines:
,
where
At the end of the interval , we apply the
approximation with the left splines:
where the basis splines are the following:
Table 3 shows the maximum errors in the
approximation of functions and also the maximum
errors in the approximation of their first derivative
when the right splines were used on a uniform grid
with a grid step . The grid of knots has
been extended to the right by three nodes:
.
Table 3. The maximum errors in absolute values in
the approximation of functions and of their first
derivative
The graph of the error of the approximation of the
Runge function
with the right splines of the
fifth order of approximations shown in Fig.7.
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Fig. 7: The graph of the error of the approximation
of the first derivative of the Runge function
with the right splines of the fifth order of
approximation
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:
Consequence. Let the values of the function be
given at the grid nodes with step . For an
approximate calculation of the first derivative of a
function, the following equalities are valid.
Next, we present the results of solving several
integro-differential equations.
Problem 2. Consider the following equation:
The kernel of this equation is the next: ,
and the exact solution is the following:
. To solve this equation, we use polynomial
splines of the fifth order of approximation. Let us
take 8 nodes. The graph of the solution error is
shown in Fig. 8.
Fig. 8: The plot of the solution error of Problem 2.
Problem 3. Now, consider the following equation
from paper [2].
where with the exact solution
. The interval [0,1] was divided into 32
subintervals. The grid nodes are renumbered from 0
to 32.Fig. 9 shows the plot of the solution errors that
was obtained using polynomial splines of the fifth
order of approximation. Fig. 10 shows the plot of
the solution errors that were obtained using cubic
splines of the fourth order of approximation.
Fig. 9: The plot of the solution error of Problem 3
(splines of the fifth order of approximation)
Digits=18
As noted earlier, in paper [2] the next numerical
methods were compared, namely, the Laplace
decomposition method (LDM), the Wavelet-
Galerkin method (WGM). Figures 11-15 show plots
of the error of solution obtained with the cubic
splines and WGM-method, with the cubic splines
and LDM-method, with the splines of the fifth order
of approximation and the LDM-method, with the
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splines of the fifth order of approximation and the
WGM-method.
Fig. 10: The plot of the solution error of Problem 3
(cubic splines of the fourth order of approximation)
Digits=18
Fig. 11: The plot of the solution error of Problem 3
obtained with the LDM-method (blue) (paper [2])
Fig. 12: The plot of the solution error of Problem 3
obtained with the cubic splines (red) and the LDM-
method (blue)
Table 4 presents the errors in solving this equation
obtained using polynomial splines of the fourth
order of approximation (column 2), and using
polynomial splines of the fifth order of
approximation (column 3). Column 1 lists the node
numbers.
Table 4. The errors in solving Problem 3 are
obtained when using polynomial splines of the
fourth order of approximation.
Number of
Cubic spline of
Splines of the
node
the 4th order of
approximation
5th order of
approximation
0
0
0
1
0.5086
0.204
2
0.1323
0.3001
3
0.2452
0.3148
4
0.3908
0.3882
5
0.5707
0.4033
6
0.7870
0.4805
7
0.1042
0.5011
8
0.1339
0.5844
9
0.1682
0.6128
10
0.2073
0.704
11
0.2519
0.7438
12
0.3025
0.8473
13
0.3597
0.9005
14
0.4242
0.10
15
0.4969
0.1091
16
0.5789
0.1231
17
0.6711
0.1327
18
0.7750
0.1494
19
0.8922
0.1623
20
0.1024
0.1826
21
0.1173
0.1998
22
0.1341
0.2249
23
0.1532
0.2479
24
0.1749
0.2594
25
0.1998
0.2796
26
0.2284
0.3104
27
0.2612
0.3510
28
0.2974
0.3923
29
0.3328
0.4446
30
0.3601
0.4953
31
0.3778
0.5222
32
0.4220
0.1134
Fig. 13: The plot of the solution error of Problem 3
obtained with the cubic splines (red) and the WGM
-method (green points)
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Fig. 14: The plot of the solution error of Problem 3
obtained with the splines of the fifth order of
approximation and the WGM -method
Fig. 15: The plot of the solution error of Problem 3
obtained with the splines of the fifth order of
approximation (red) and the LDM-method (blue)
In Figures 11-15, the solution at the grid nodes,
obtained using splines of the fourth and fifth
approximation orders, is marked with red circles,
the solutions obtained by other methods of paper [2]
are marked by green and blue circles. Thus, it can be
seen that for this equation, the error of the solution
with the splines turned out to be no worse than when
using the WGM-method, and the LDM-method.
Problem 4. Consider the following integro-
differential equation:
with the initial condition .
The solution of this integro-differential equation
cannot be represented explicitly. To solve this
nonlinear integro-differential equation, we will first
use splines of the fourth order of approximation,
then splines of the fifth order of approximation. To
approximate the first derivative, we will use
numerical differentiation formulas obtained on the
basis of the corresponding splines. The interval [0,1]
was divided into 32 subintervals. The grid nodes are
renumbered from 0 to 32. Table 5 presents the
solutions of this equation obtained using polynomial
splines of the fourth order of approximation
(column 2), and using polynomial splines of the
fifth order of approximation (column 3).
Table 5. The solutions of Problem 4
Number of
node
Cubic spline of
the 4th order of
Splines of the
5th order of
approximation
approximation
0
0
0
1
-0.3125
-0.3125
2
-0.6250
-0.6250
3
-0.9374
-0.9374
4
-0.1250
-0.1250
5
0.1562
0.1562
6
0.1874
0.1874
7
0.2186
0.2186
8
0.2497
0.2497
9
0.2807
0.2807
10
0.3117
0.3117
11
0.3426
0.3426
12
0.3734
0.3734
13
0.4040
0.4040
14
0.4345
0.4345
15
0.4647
0.4647
16
0.4948
0.4948
17
0.5247
0.5247
18
0.5542
0.5542
19
0.5835
0.5835
20
0.6124
0.6124
21
0.6410
0.6410
22
0.6692
0.6692
23
0.6969
0.6969
24
0.7242
0.7242
25
0.7509
0.7509
26
0.7771
0.7771
27
0.8027
0.8027
28
0.8277
0.8277
29
0.8520
0.8520
30
0.8756
0.8756
31
0.8984
0.8984
32
0.9205
0.9205
Column 1 of Table 5 lists the node numbers. It
should be noted that in the paper [2] the results of
numerical experiments are obtained based on the
application of four different methods (the Laplace
decomposition method (LDM), the Wavelet–
Galerkin method (WGM), the Laplace
decomposition method with the Pade approximant
and the homotopy perturbation method (HPM)). The
results obtained using the method based on the use
of splines coincide with the results of applying the
methods of paper [2].
4 Conclusion
As is known, in the numerical solution of equations
and systems of equations, several different solution
methods are usually used. This is necessary for a
verification of the result. If the kernel, coefficients
and the right side of the equation are sufficiently
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smooth, then the proposed method can give a
solution with a smaller error.
This paper shows the results of applying local
interpolation splines of the fourth and fifth order of
approximation for solving integro-differential
equations with Fredholm and Volterra integrals of
the second kind. The main focus was on the
equations with the first derivative.
Comparisons of the results of applying local
splines with the use of other methods for solving
integro-differential equations are shown. It is shown
that in some cases the application of the approach to
solving integral equations based on splines gives a
smaller error for the same number of nodes. In
addition, the approach based on spline
approximations is quite simple to implement and
gives a reliable result.
We emphasize once again that the advantage of
the spline approach is the simple implementation of
the algorithm. As a result of applying this approach,
we have to solve a system of equations (linear or
nonlinear). As a result, we obtain an approximation
to the solution of the original integro-differential
equation in the form of grid function values at the
grid nodes.
To obtain an approximate solution at points
between the grid nodes, it is convenient to use the
same spline approximations. The result is a
continuous line.
To obtain a twice continuously differentiable
approximate solution, a special method considered
by the author earlier can be used. In this case, it is
necessary to solve a system of linear algebraic
equations additionally. The matrix of this system of
equations will have a tape form.
In the future, other types of integro-differential
equations and systems of equations will be
considered.
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Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
The author is highly and gratefully indebted to St.
Petersburg University for financial supporting the
publication of the paper (Pure ID 92424538)
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