1. Introduction

The theory of integral equations is interesting, not only in itself but also in the absolute importance of its results in the analysis of numerical methods of solution of various equations. Besides the concept of existence and uniqueness with respect to differential equations, the theory is concerned, in particular, with questions of regularity and stability of their solutions.

In the last two and a half decades, there has been a tremendous increase in the applications of first kind integral equations of Volterra type. Among the areas where the applications of such class of equations abound in the natural sciences is the problem of restoration of incoming signals at the entrance of measuring instruments and observatory systems which, in view of their physical nature, allow some distortions of the observed and registered data [1].

An integral equation is a functional equation in which the unknown function appears under one or several integral signs. In an integral equation of Volterra type for instance, the integrals containing the unknown function are characterized by a variable upper limit of integration. To be more precise, an integral equation of the form

(1)

is called a linear Volterra integral equation of the first kind, or that of the form

(2)

is called a linear Volterra integral equation of the second kind. Here, are real numbers, is a parameter, is an unknown function, while are given functions which are square integrable on and in the domain respectively. The function is called the free term, while the function is called the kernel.

One distinguishing feature associated with this class of equations is the uncertainty surrounding their nature and this is reflected in the difficulties encountered when attempting to find their solutions using one or the other of the existing numerical methods [2]. On one hand, Volterra integral equations of the first kind appear to be special cases of Fredholm integral equations of the first kind and are consequently classified among the lists of ill-conditioned problems solvable by the classical regularization means. On the other hand, given a set of limitations (for example, given that their kernels and right-hand sides are “good” and “smooth”) Volterra integral equations of the first kind belong to the class of well-posed problems and hence can be solved through any direct method based on the discretization of the unknown solutions. Both approaches seem to be a little on the extreme. The use of discretization process transforms Volterra integral equations of the first kind into another problem (often systems of linear equations and may be solved by the well-known singular value decomposition) that can lead to solutions which may or may not deviate strongly from the (minimum norm) solution of the original equation [3]. A complete regularization (with its accompanying difficulties and unresolved questions) turns out to be a too complicated approach as it is observed that the methods for Volterra equations of the first kind are not nearly as ill-posed as methods for Fredholm integral equations of the first kind.

In this study, we investigate the regularization methods with a view to identifying the properties and types that can be used in solving Volterra type equations of the first kind, but without destroying the Volterra structure. The methods in view are perhaps the discrete regularization methods. The discrete approximation methods provide another approach to regularize the original problem. In this case, the regularization parameter is the discretization parameter (or step size) and coordination between this parameter and the amount of noise in the problem is required to obtain good approximations in the presence of noise.

2. The Concept of a Regularizing Operator

We are already aware of the fact that the problem of solving Volterra integral equations of the first kind in some spaces, for example, in the triple (C,V,C) is, to a certain extent, ill-posed. Even in those spaces where the problems are well-conditioned, for example, in the space (C,V,C⁽¹⁾), we still encounter significant difficulties, leading to some slight instabilities in the results during actualization. Consequently, to improve on the stability of the solutions which implies the accuracy of the desired results, we go out for the most stable methods. Among such methods are the regularization methods.

Methods for construction of approximate solutions of ill-posed problems that are stable with respect to small perturbations of the initial data are referred to as regularization methods. The approach is based on what is called the fundamental concept of a regularizing operator [4]. In what follows, we examine the concept in detail.

Suppose the operator A in the equation

(3)

where Y and F are some metric spaces and A is a continuous operator mapping Y into F, is such that its inverse is also continuous in the set and the set Y of possible solutions, is not compact.

If the right–hand member of the equation is an element that differs from the exact right–hand member by no more than , that is, if , it is obvious that the approximate solutions of equation (3) cannot be defined as the exact solution of this equation with approximate right–hand member , that is, according to the formula .

The numerical parameter characterizes the error in the right–hand member of equation (3). Therefore, it is natural to define with the aid of an operator depending on a parameter having a value chosen in accordance with the error in the initial data . Specifically, as , that is, as the right–hand member of equation (3) approaches (in the metric of the F) the exact value , the approximate solution must approach (in the metric of the space) the exact solution that we are seeking for in the equation .

Now, assuming that the elements and are connected by , we define the following concepts:

Definition 2.1. An operator is said to be a regularizing operator for the equation in a neighbourhood of if:

i) there exists a positive number such that the operator is defined for energy in and every such that

ii) for every , there exists a , such that the inequality implies the inequality , where

Definition 2.2. An operator depending on a parameter α is called a regularizing operator for the equation Ay = f in a neighbourhood of if:

i) there exists a positive number such that the operator is defined for every and every for which

ii) there exists a function of such that, for every , there exists a number such that the inclusion and the inequality imply where

Again, there is no assumption of the uniqueness of the operator . It is easy to point out here that the function also depends on .

If , we know from studies by Tikhonov [4] that we can take for an approximate solution of equation (3) with approximately known right–hand member the element obtained with the aid of the regularizing operator where in accordance with the error in the initial data . This solution is called a regularized solution of equation (3). The numerical parameter α is called the regularization parameter. Obviously, every regularizing operator defines a stable method of approximate construction of the solution of equations (3) provided the choice of α is consistent with the accuracy of the initial data . If we know that we can, by definition of a regularizing operator, choose the value of the regularization parameter in such a way that, as , the regularized solution approaches (in the metric of Y) the desired exact solution , that is, . This justifies taking as an approximate solution of equation (1) the regularized solution. Thus, the problem of finding an approximate solution of equation (1) that is stable under small changes in the right–hand member reduces to:

i) finding regularizing operators;

ii) determining the regularization parameter α from the supplementary information pertaining to the problem, for example, the size of the error in the right–hand member .

This method of constructing approximate solutions is called the regularization method.

Out of all the operators from F and Y that depend on the parameter α and which are defined for every f∈F and every positive α, we need to single out the operators that are continuous with respect to f. In doing this, we can give sufficient conditions for belonging to the set of regularizing operators of equation (3). This is an immediate consequence of the following theorem.

Theorem 2.1. Let A denote an operator from Y into F and let denote an operator from F into Y that is defined for every element f∈F and every positive α that is continuous with respect to f. If for every element y∈Y, then the operator is a regularizing operator for the equation

Proof. It will be sufficient to show that the operator possesses property (ii) of Definition 2.2.

Let and denote fixed elements of Y and F respectively such that . Let denote a fixed positive number. Then, for every element such that , we have

(4)

Since the operator is continuous with respect to f at the “point” , it follows that, for sufficiently small positive , the inequality

(5)

implies the inequality

(6)

where .

Since it holds that

(7)

and with

It follows from inequalities (4), (6), and (7) that, for every and ,

(8)

Since there exist for every , a such that, for and , inequalities (5) and (8) imply

This completes the proof of Theorem 1.

3. Regularization Methods

3.1. Tikhonov’s Regularization Method

Consider the Volterra integral equation of the first kind

(9)

where

To ensure the stability of the solution of equation (9), we introduce the condition of minimum smoothing functional given as

(10)

where, as it is so – called, the stabilizing functional, Ω[y], is usually written in the form [3]

(11)

What is more, α > 0 is called the regularization parameter, defines the order of the regularization (zero order if and first order if ). Expanding condition (10) using condition (11) and bearing in mind the Volterra structure of equation (9) yields the following equation of the second kind:

(12)

where

(13)

(14)

and

(15)

Expressions (13) and (14) may be written in another form as

(16)

In place of equations (12) - (16), we may use the relation

(17)

where

(18)

(19)

and

(20)

are valid for Fredholm first kind equation and setting in the later, and

(21)

In all the cases considered above, the original Volterra integral equations of the first kind are replaced with equations of the second kind (precisely, integro-differential equation of the second kind) of Fredholm type. Consequently, the Tikhonov’s regularization method results in the loss of the Volterra structure but assists in the preliminary transformation of equation (10) (using, for example, the finite sum and the finite difference methods) into a system of linear algebraic equations with a full (positive definite) matrix (not triangular). This calls for a reasonable loss of time when solving it using any computational device. In spite of all these shortcomings, the use of Tikhonov’s regularization method in the solution of equation (9) still turns out to be one of the most effective of the regularization methods.

3.2. Tikhonov’s Regularization Method for Equation of Convolution type (Sequential Tikhonov Regularization Method)

We consider the convolution type Volterra integral equation of the first kind

(22)

A complete discussion on this question has been provided by Savelova ([5], [6], [7], [8], [9], [10], [11]).

For the convolution type Volterra integral equation of the first kind

(23)

We first make use of the so-called techniques of extension [12]. We introduce the functions

and

and hence obtain from equation (23), a new statement

It is only from this moment that the Tikhonov’s regularization method may be applied.

3.3. Denisov’s Regularization Method

Consider the equation

(24)

Let and the functions be continuous and the exact solution . What is more, let it be given that Suppose further, that in place of the right member the approximate right member such that is known. Then, in place of equation (24), we propose the solution of the -regularizing equation

(25)

Such a procedure is close to the Lavrent-ev’s classical method. The asymptote of the solution of equation (25) is examined in the study by Blondel [13] when with the exact function (but without the term ).

In the study by Denisov [14], it is proved that

(26)

where k = constant. When we select such that For instance, if is chosen so that , we observe that , which is the algorithm obtained from equation (24).

We introduce yet another value of (call it the quasi-optimal regularizer and denote it by ), the minimizer of the right-hand side of equation (25). Hence, we obtain

(27)

(28)

An important feature of the method under consideration is the simple nature of equation (24). Furthermore, the Volterra structure of the original equation (as seen) is maintained. However, the method insists on an initial knowledge of the value of which, in a concrete situation, is not obtainable.

3.4. Apartsin’s Regularization Method

Further modification of Denisov’s regularization method in conjunction with Apartsin–Bakushinskii’s regularization method is called regularization method. In the said regularization method, it is clear from the onset that the procedure will involve the replacement of the integral of equation (24) with the corresponding approximate finite sum ( h-regularization) on the grid and instead of battling with the identification of the preconceived appropriate values of and , a system of linear algebraic equations is put in place of equation (24) and is independent of

Now, we consider a system of linear algebraic equations with a triangular matrix of coefficients:

(29)

where

Notice that the system in equation (28) is obtained by replacing the integral in equation (24) with a finite sum defined by the mid-point rectangular formula having a step-size of h = constant.

Let and Then, for a regularized framework satisfying equation (29), the following estimate in respect of the error of the approximate solution is valid:

(30)

where

Let us analyze the estimate in equation (29). The first component on the right-hand side

(31)

characterizes the influence on the exact regularized framework by the right-hand side error. The second component

is connected with the approximation of the quadrature integral, while the third component

reflects the contribution of the total error of distortion of the desired equation (28) due to the additional -component.

It is seen that the error estimate in equation (29) unites the h–regularization estimate. This can be observed in equation (23) with the Denisov’s -regularization estimate in equation (25).

We call the quasi-optimal regularizer (QOR) of the parameters and h, the minimizer of the right-hand side of equation (29). This implies

(32)

(33)

and

(34)

Since, in practice, it is not clear which of the cases results in or , it is, therefore, good to select one of the relationships in the equation (32) for the asymptote but maintaining the condition in equation (32). Indeed, if we set , then

(35)

If we set , then

(36)

In all cases, however, irrespective of whether or , that is, the algorithm obtained from solving the system of linear algebraic equation (28) and also equation (22) taking advantage of asymptotes (32) and (33) are regularized. Besides, unlike Denisov’s, the method does not use the component, and the requirement of or is compensated by changing the dependence of the quasi-optimal asymptote and Finally, we note that as long as

then as , decreases faster than h (the carrier of the basic stability load) and so, parameters h and share homogeneous characterizations. As it now stands, the step size h should be regarded as the basic parameter while plays the role of the auxiliary. This is reflected by the fact that the presence or absence of does not influence the state of the asymptote This being so, as soon as , the -smoothing equation quickly requires the presence of the component ; whereas, when h ≠ 0, or , the asymptote is such that it replaces the function of and continues to support the method of regularization without the introduction of the component in the desired equation. However, the use of the parameter in place of h increases the effectiveness of the method since, for a small h (not to mention when h→0), the error of the solution is notably compressed.

3.5. Sergeev’s Regularization Method

Consider Volterra integral equation of the first kind

(37)

Let

(38)

If and , we can divide both sides of equation (37) by to obtain an equation with property (38).

Now, let the function be continuous with respect to x and s. Let Then after the nth differentiation of equation (37), we obtain a second kind equation

(39)

If

Then equation (39) has a unique solution (by virtue of the given exact data). But, previously, it was assumed that However, if f(x) together with the error is known, then the determination of the derivative f⁽ⁿ⁾(x) becomes an ill-posed problem and, hence, requires the use of any of the regularization methods. In the studies by Lavrent’ev et al and Sergeev ([15], [16]), a regularization method which also transforms the equation into a second kind equation but does not require the calculation of the said derivative of f(x) without losing the Volterra structure is proposed.

Suppose the values

, are known. We introduce the function

(40)

For this equation, let

Let it be assumed that in place of the exact right – hand member f(x), the approximate right – hand member is known such that .

Therefore, in place of the exact equation (37) or formally, its equivalent equation (39), we propose the solution of the following approximate Volterra equation of the second kind:

(41)

where

and

The estimate

(42)

where

,

is valid. The minimum value of the right-hand side of equation (42) is obtained if

(43)

For such value of ,

(44)

It is clear that as , and what is more,

(45)

that is, the algorithm obtained from equation (41) is regularizing.

In what appears to be the simplest case, where n = m = 1, it is expected to solve the equation

(46)

where

and

Consequently, we have that

(47)

The minimum value of the right-hand side of equation (47) is attained when

(48)

For such ,

(49)

Here,

3.6. Magnicki’s Regularization Method

This is closely related to Sergeev’s method but has a wider range of applications. We consider the equation

(50)

Let

(51)

Furthermore, let

Then, differentiating equation (50) n times, we obtain the equation

(52)

which has a unique solution . Again, similar to Sergeev’s method, we assume that and that are known.

Therefore, in place of equation (50) or (52), we propose the solution of a Volterra integral equation of the second kind

(53)

Subject to the conditions:

a class of functions defined by the following conditions:

and there exist such that

The following estimate is valid

(54)

where

The minimum value of the right-hand side is attained when

For such values of

As that is, the algorithm obtained from equation (53) is regularizing.

For ,

are continuous in x and s,

,

we propose a special equation (which is not necessarily a consequence of equation (53)):

where

The following estimate (for an optimal value of ) is valid:

Now, with conditions similar to those of equation (51), it becomes necessary to solve the following consequence of equation (51):

We obtain the following estimates

4. Conclusions

Although this study primarily examined regularization methods for first kind Volterra integral equations that maintain the Volterra structure, we never lost sight of the distinguishing features associated with this class of equations, which reflect in the uncertainty and difficulties surrounding their nature when attempting to find their solutions using one or other existing numerical methods. On the other hand, an attempt to solve Volterra integral equations of the first kind using analytic techniques may not be fruitful after all. This is so, because Volterra integral equations of the first kind, in a sense appears to be situated mid-way between Volterra integral equations of the second kind and Fredholm integral equation of the first kind. Precisely, if a Volterra integral equation of the second kind is well-posed and can be effectively solved by any classical means, then a Fredholm integral equation of the first kind is ill-posed, given any preconceived functional space and solvable by special approximating methods. Again, Volterra integral equation of the first kind may be well- posed or ill-posed depending on the choice of the solution space and can only be solved effectively by regularization methods.