Solution of Integral Equations Using Local Splines of the Second Order
I.G.BUROVA, G. O. ALCYBEEV
Dept. of Computational Mathematics
St. Petersburg State University
St. Petersburg, RUSSIA
Abstract— Splines are an important mathematical tool in Applied and Theoretical Mechanics. Several
Problems in Mechanics are modeled with Differential Equations the solution of which demands Finite Elements
and Splines. In this paper, we consider the construction of computational schemes for the numerical solution of
integral equations of the second kind with a weak singularity. To construct the numerical schemes, local
polynomial quadratic spline approximations and second-order nonpolynomial spline approximations are used. The
results of the numerical experiments are given. This methodology has many applications in problems in Applied
and Theoretical Mechanics
Keywords—Splines, integral equation of the second kind, weak singularity, local spline, polynomial spline,
nonpolynomial spline
Received: January 13, 2022. Revised: November 14, 2022. Accepted: December 13, 2022. Published: December 31, 2022.
1. Introduction
A lot of mathematical formulations of physical phenomena
contain integral and/or integro-differential equations. These
equations occur in many applications such as in the
transport of air and ground water pollutants, oil reservoir
flow, in the modeling of semiconductors etc. Currently
many papers have been devoted to the numerical solution of
integral equations with a weak singularity. Let's mention
some papers published recently.
In paper [1] an iterative scheme to approach the solution
of nonlinear integro-differential Fredholm equation with a
weakly singular kernel using the product integration method
is developed.
Cubic trigonometric B-spline functions are used in the
paper [2] to solve the convection-diffusion type partial
integro-differential equation (PIDE) with a weakly singular
kernel. Cubic trigonometric B-spline (CTBS) functions are
used for interpolation in both methods. The first method is
the CTBS based collocation method which reduces the
PIDE to an algebraic tridiagonal system of linear equations.
The other method is the CTBS based differential quadrature
method which converts the PIDE to a system of ODEs by
computing spatial derivatives as weighted sum of function
values.
A new orthogonal basis for the space of cubic splines has
been used in paper [3] for obtaining the numerical solutions
of a partial integro-differential equation with a weakly
singular kernel.
In paper [4] a meshless method in local setting and
Laplace transform are coupled to approximate partial
integro-differential equations (PIDEs).
In paper [5] a numerical scheme is developed to solve
the Volterra partial integro-differential equation of the
second order having a weakly singular kernel. The scheme
uses cubic trigonometric B-spline functions to determine the
weighting coefficients in the differential quadrature
approximation of the second order spatial derivative.
In the paper [6] the trigonometric cubic B-spline
collocation method is extended to the solution of a second
order partial integro-differential equation with a weakly
singular kernel.
Splines are often used to solve various problems:
interpolation; solving the Cauchy problem; Image
compression; and enlargement (see, for example, [8]-[11]).
In paper [11], splines were used to solve Volterra integral
equations of the second kind with a smooth kernel. This
methodology has many applications in problems in Applied
and Theoretical Mechanics
In section 2 of this paper, for the approximate calculation
of integrals with a weak singularity, we use polynomial and
nonpolynomial splines of the second order of
approximation. In section 3 the numerical examples are
given.
2. The construction of the method
In this paper, we consider the numerical solution of the
Fredholm integral equation of the second kind with a weak
singularity
We assume that the kernel has the form:
where is a weight function,
,
We assume that the function is bounded
function on In this paper, to construct calculation
formulas, we use quadratic basis splines and a Gaussian-
type quadrature formula.
Let an ordered grid of nodes be constructed on the
interval : We represent the
integral
as the sum of integrals over the
grid segments:
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2022.17.31
I. G. Burova, G. O. Alcybeev
E-ISSN: 2224-3429
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Volume 17, 2022
Applying local interpolation splines of the second order of
approximation (see [8]), we obtain , where
Recall that we have proved the approximation error theorem
(see paper [7]).
Denote the norm:
Theorem. If , are the nodes,
, and then the next estimation is
valid
Denote . Let us construct a quadrature
formula of the Gaussian type with two nodes:
It is necessary to calculate the nodes and the coefficients
, The nodes and coefficients of the quadrature
formula of the Gaussian type with the weight can be
found in the traditional way. Let the function be such
that , where are the
nodes of the quadrature formula of the Gaussian type.
For the convenience of these calculations, we will write
the polynomial in the form:
.
First, we calculate the moments :
Next , we solve the system of the equations and find the
unknowns :
.
Now, we can solve the quadratic equation
and find its roots . These roots are the nodes of the
quadrature formula. Now we determine the coefficients
of the quadrature formula by solving the system of
equations:
,
.
Recall the theorem on the remainder term of a quadrature
formula of the Gaussian type. In the case of two nodes, the
remainder term ,
takes the form:
, .
When calculating sequentially, we get a chain of equalities:
+
.
Now we are constructing a system of linear algebraic
equations by setting Solving the
system of equations, we obtain the solution values at
the grid nodes , If necessary, we can
connect the found values using the piecewise linear basis
splines.
Remark. If functions and form a Chebyshev
system, then the basis functions and can be
determined by solving the system of equations
+ () ,
+ ,
Let us assume that the determinant of the system is not
equal to zero. Let us study the case when and
, where is a continues function. Let us
construct a nonpolynomial approximation of the function
on each grid interval in the form:
where
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Depending on the choice of the function , different
error estimates are obtained. Approximation errors of the
solution obtained with the non-polynomial splines are
discussed in papers [10]-[11].
3. The Results of the Numerical
Experiments
Example 1. Let us start with solving the integral equation
with a weak singularity
Here
, the function is constructed using
the functions and the exact solution
We construct the set of equidistant nodes with the step
We develop a program in Maple with Digits=10. First
consider the use of the approximation with the polynomial
splines.
Fig. 1 shows the plot of the exact and approximate solutions
of the integral equation when
Fig.1. The plot of the exact and approximate solutions of the integral
equation when
Fig. 2 shows the plot of the error of the solution of the
integral equation when
Fig.2. The plot of the errors of the solutions of the integral equation when
Figs. 3-5 show the plots of the errors of the solutions of the
integral equation. Fig. 3 shows the plots of the errors of the
solutions of the integral equation when . Fig. 4
shows the plots of the errors of the solutions of the integral
equation when . Fig. 5 shows the plots of the errors
of the solutions of the integral equation when .
Fig.3. The plot of the errors of the solutions of the integral equation when
Fig.4. The plot of the errors of the solutions of the integral equation when
Fig.5. The plots of the errors of the solutions of the integral equation when
It can be seen that with an increase in the number of nodes,
the solution error decreases. Now consider the use of
approximation with the nonpolynomial (exponential)
splines. We take Fig. 6 shows the plot of
the exact and approximate solutions of the integral equation
when Fig. 7 shows the plot of the error of the
solution of the integral equation when
Fig.6. The plot of the exact and approximate solutions of the integral
equation when (exponential splines)
Fig.7. The plot of the errors of the solutions of the integral equation when
(exponential splines)
The results of numerical experiments show that well-chosen
basis functions can significantly reduce the error of the
solution.
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
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Example 2. Let us continue with solving the next
integral equation with a weak singularity
Here
, the function is constructed using
the functions and the exact solution
The plot of the error of the solution obtained with the
exponential splines when is given in Fig.8.
Fig.8. The plot of the errors of the solutions of the integral equation when
(Example 2, exponential splines)
The plot of the error of the solution obtained with the
polynomial splines when is given in Fig.9.
Fig.9. The plot of the errors of the solutions of the integral equation when
(Example 2, polynomial splines)
Example 3. Let us continue with solving the next
integral equation with a weak singularity
Here
, the function is constructed using
the functions and the exact solution
The plot of the error of the solution obtained with the
exponential splines when is given in Fig.10.
Fig.10. The plot of the errors of the solutions of the integral equation when
(Example 3, exponential splines)
The plot of the error of the solution obtained with the
polynomial splines when is given in Fig.11.
Fig.11. The plot of the errors of the solutions of the integral equation when
(Example 3, polynomial splines)
4. Conclusion
Splines are an important mathematical tool in Applied
and Theoretical Mechanics. Several Problems in Mechanics
are modeled with Differential Equations the solution of
which demands Finite Elements and Splines. In this paper,
we considered the construction of computational schemes for
the numerical solution of integral equations of the second
kind with a weak singularity. To construct the numerical
schemes, local polynomial quadratic spline approximations
and second-order nonpolynomial spline approximations are
used. The results of the numerical experiments are given.
ACKNOWLEDGMENT
The authors are gratefully indebted to St. Petersburg
University for their financial support in the preparation of
this paper (Pure ID 94029567), as well as to a resource
center for providing the package Maple.
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