Collocation-Variational Approaches to the Numerical Solution of Volterra Integral Equations of the First Kind

Abstract

Linear Volterra equations of the first kind are considered. A class of problems with a unique solution is identified, for which collocation-variational solution methods are proposed. According to the proposed algorithms, an approximate solution is found at nodes of a uniform grid (collocation condition), which yields an underdetermined system of linear algebraic equations. The system thus obtained is supplemented with the minimization condition for the objective function, which approximates the squared norm of the approximate solution. As a result, we obtain a quadratic programming problem with a quadratic objective function (squared norm of the approximate solution) and equality constraints (collocation conditions). This problem is solved by applying the Lagrange multiplier method. Fairly simple third-order methods are considered in detail. Numerical results for test problems are presented. Further development of this approach for the numerical solution of other classes of integral equations is discussed.

INTRODUCTION

In this paper, we numerically solve linear Volterra integral equations of the form

1

where and are given functions with sufficiently smooth elements and is the sought function. For

2

and continuous functions , the problem has a unique continuous solution (see, e.g., [1, 2]).

Approaches to the numerical solution of Eq. (1) with condition (2) can be found in [4–6] (collocation and multistep methods), in [7] (block methods), and in [8]. Results concerning this subject and difficulties arising in the development of methods for solving Eq. (1) are described in [9].

For the above-mentioned problems, we propose single-step methods that have been successfully applied to differential-algebraic equations (see [10] and references therein) and are a generalization of [11].

QUADRATURE RULES AND ALGORITHMS

To construct methods for solving the original problem, we need some results from the theory of approximate integration. In what follows, we will use four-point quadrature rules of the third order, which are described in more detail.

On the interval , we introduce a uniform grid and assume that, given a sufficiently smooth function , we know Then

3

4

where the coefficients satisfy third-order conditions, i.e., the quadrature rules (3), (4) are exact for any polynomials of degree at most three.

Omitting elementary calculations, we find that these coefficients are the solution of the SLAE

5

Setting and (free parameters) in (5), we see that the solution of SLAE (5) is

6

7

Now we describe algorithms for the approximate solution of the Volterra integral equation (1), assuming that is given or preliminarily computed. The algorithms are based on the quadrature rules (3) and (4) with coefficients satisfying relations (6) and (7), respectively. To simplify the presentation, we assume that is a multiple of 3 and introduce the notation

In this case, for Eq. (1), we have

8

and

9

The points and will be referred to as collocation points or collocation nodes.

Assuming that is given and using the indicated quadrature rules, we conclude that , and are a solution of the SLAEor, in vector-matrix form,

10

where

The dimension of these systems is , i.e., they are underdetermined.

SLAE (10) is viewed as equality constraints for minimizing the squared norm of the approximate solution , . In this case, we have the constrained minimization problem

11

with equality constraints (10).

If the norm of the function is chosen unsuccessfully, for example, in the space of continuous or continuously differentiable functions, then problem (11) with constraints (10) is sufficiently complicated, so we assume that

(i) is an interpolating polynomial of third degree passing through the points

(ii)

12

Our consideration is restricted to the special case of (12), namely, . To evaluate the definite integral in formula (12), we use a well-known quadrature rule (see, e.g., [12]). Then we have

13

where and, by the norm of a finite-dimensional vector, we mean the Euclidean norm.

The coefficients depend on the choice of a quadrature rule and the formulas for approximate computation of .

The coefficients are uniquely determined by the obvious equality i.e.,

For example, at , the coefficients are and

At , the coefficients are and

At , the coefficients are and

At , the coefficients are and

Thus, since is given, on each integration interval , we have the following quadratic programming problem: find the minimum of the objective function under equality constraints (10).

Since the multiplication of the objective function (13) by an arbitrary nonzero number does not influence finding the argument of the constrained minimum, the given problem is equivalent to the problem

14

with constraints (10).

Since the first to third terms in (14) contain small terms of order , respectively, all (or some) of them can be dropped. For example, retaining only the third and fourth terms or only the last one in (14), we obtain two mathematical programming problems:

1. Find

15

2. Find

16

with equality constraints (10).

Problems (15), (10) and (16), (10) can be solved by applying the Lagrange multiplier method. Since the objective functions (15) and (16) are quadratic and (10) are equality constraints, the solution of these problems is the solution of the corresponding SLAEs. For example, a solution of problems (16) with constraints (10) is a solution of the SLAE of the form

17

where

18

where the vector is given by formula (10).

Proposition. Assume that the following conditions hold for the integral equation (1):

(i) the elements belong to the class ;

(ii) , .

Then , where are solutions of problems (17).

The proof is based on a discrete analogue of the Gronwall–Bellman lemma (see [3, 6]).

Assuming that in (12), we obtain another family of algorithms. For example, if , then (by analogy with problem (14)) we have the following constrained minimization problem for a quadratic function:

1. Find

19

with equality constraints (10).

For , we have the following problem:

2. Find

20

with equality constraints (10).

By analogy with (13), these problem are equivalent to finding a constrained minimum of the functions and , respectively.

As for the case , for , the terms containing or and can be dropped from formula (19). For , we can drop the term containing .

Then we obtain the following family of algorithms: for , find

21

or

22

with equality constraints (10).

For we have the following family of methods: find

23

with equality constraints (10).

Problems (21)–(23) can be solved by applying the Lagrange multiplier method. In this case, the constrained minimum of the functions , , and , is found exactly by solving corresponding SLAEs.

Note that the stability and convergence analysis of methods (19)–(23) is of particular interest. The properties of these algorithms depend not only on the choice of quadrature rules (see formula (6)), i.e., on the parameters and , but also on the choice of approximations of the solution derivatives (see formulas (12) and (13)), i.e., on the parameters . Several versions of such approaches are considered. A preliminary analysis of the algorithms has shown that they are stable.

NUMERICAL COMPUTATIONS

In this section, we present numerical results obtained for test examples by applying algorithm (17) with parameters and . The results are given in tables. We use the notation er =

Example 1 (see [6], p. 149). Consider the integral equationwith the exact solution given by The numerical results obtained for , , and are presented in Table 1.

Table 1.  . Numerical computations of Example 1 for , , and

	0.1	0.05	0.025
er	0.0039	0.0006	0.00009

For the parameter values , , and , the numerical results for this example are given in Table 2.

Table 2.  . Numerical computations of Example 1 for , , and

	0.1	0.05	0.025
er	0.0027	0.0004	0.00006

Example 2 (see [6], p. 517). Consider the integral equationwith the exact solution given by . The numerical results obtained for parameter values , , and are presented in Table 3.

Table 3.  . Computations of Example 2 for parameter values , , and

	0.1	0.05	0.025
er	0.085	0.012	0.0018

For the parameter values , , and , the numerical results for this example are presented in Table 4.

Table 4.  . Numerical results for Example 2 for parameter values , , and

	0.1	0.05	0.025
er	0.02	0.0035	0.00052

The numerical results obtained in these examples agree with the proposition. Additionally, other numerous test examples without stiff components were computed by applying algorithm (17) with various values of the parameters and . These experiments were also found to agree well with the proposition.

CONCLUSIONS

A class of Volterra integral equations of the first kind was distinguished, for the numerical solution of which collocation-variational methods of the third order were proposed. These algorithms are reduced to mathematical (quadratic) programming problems with a quadratic objective function (some analogue of the squared norm of the approximate solution) and with equality constraints (collocation condition). This problem is equivalent to solving a nonsingular SLAE. Numerical computations showed that further development of this approach is promising. In the future, a detailed study of the collocation-variational methods (21)–(23) and higher order methods is also intended for more general problems, in particular, for Volterra integral equations with a degree of instability (see [3]) higher than 1 and for first-kind equations with a kernel involving a weak singularity.

FUNDING

This work was supported by the Russian Science Foundation, project no. 22-11-00173.

CONFLICT OF INTEREST

The author of this work declares that he has no conflicts of interest.

Translated by I. Ruzanova

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