Biorthogonal Systems Approximating the Solution of the Nonlinear Volterra Integro-Differential Equation

The aim of this paper is to introduce a numerical method to approximate the solution of the nonlinear Volterra integro-differential equation, which generalizes that developed in [1]. Let us consider the nonlinear Volterra integro-differential equation

(1.1)

where and and are continuous functions satisfying a Lipschitz condition with respect to the last variables: there exist such that

(1.2)

for and for . In the sequel, these conditions will be assumed. It is a simple matter to check that a function is a solution of (1.1) if, and only if, it is a fixed point of the self-operator of the Banach space (usual supnorm) given by the formula

(1.3)

Section 2 shows that operator satisfies the hypothesis of the Banach fixed point theorem and thus the sequence converges to the solution of (1.1) for any However, such a sequence cannot be determined in an explicit way. The method we present consists of replacing the first element of the convergent sequence, by the new easy to calculate function and in such a way that the error is small enough. By repeating the same process for the function and so on, we obtain a sequence that approximates the solution of (1.1) in the uniform sense. To obtain such sequence, we will make use of some biorthogonal systems, the usual Schauder bases for the spaces and , as well as their properties. These questions are also reviewed in Section 2. In Section 3 we define the sequence described above and we study the error . Finally, in Section 4 we apply the method to two examples.

Volterra integro-differential equations are usually difficult to solve in an analytical way. Many authors have paid attention to their study and numerical treatment (see for instance [2–15] for the classical methods and recent results). Among the main advantages of our numerical method as opposed to the classical ones, such as collocation or quadrature, we can point out that it is not necessary to solve algebraic equation systems; furthermore, the integrals involved are immediate and therefore we do not have to require any quadrature method to calculate them. Let us point out that our method clearly applies to the case where the involved functions are defined in , although we have chosen the unit interval for the sake of simplicity. Schauder bases have been used in order to solve numerically some differential and integral problems (see [1, 16–20]).

We first show that operator also satisfies a suitable Lipschitz condition. This result is proven by using an inductive argument. The proof is similar to that of the linear case (see [1, Lemma ]).

Lemma 2.1.

For any and , we have

(2.1)

where

In view of the Banach fixed point theorem and Lemma 2.1, has a unique fixed point and

(2.2)

Now let us consider a special kind of biorthogonal system for a Banach space. Let us recall that a sequence in a Banach space is said to be a Schauder basis if for every there exists a unique sequence of scalars such that The associated sequence of (continuous and linear) projections is defined by the partial sums We now consider the usual Schauder basis for the space (supnorm), also known as the Faber-Schauder basis: for a dense sequence of distinct points with and we define and for all we use to stand for the piecewise linear function with nodes at the points with for all and It is straightforward to show (see [21]) that the sequence of projections satisfies the following interpolation property:

(2.3)

In order to define an analogous basis for the Banach space (supnorm), let us consider the mapping given by (for a real number , denotes its integer part)

(2.4)

If is a Schauder base for the space , then the sequence

(2.5)

with , is a Schauder basis for (see [21]). Therefore, from now on, if is a dense subset of distinct points in , with and , and is the associated usual Schauder basis, then we will write to denote the Schauder basis for obtained in this "natural" way. It is not difficult to check that this basis satisfies similar properties to the ones for the one-dimensional case: for instance, the sequence of projections satisfies, for all and for all with ,

(2.6)

Under certain weak conditions, we can estimate the rate of convergence of the sequence of projections. For this purpose, consider the dense subset of distinct points in and let be the set ordered in an increasing way for Clearly, is a partition of . Let denote the norm of the partition . The following remarks follow easily from the interpolating properties (2.3) and (2.6) and the mean-value theorems for one and two variables:

(2.7)

(2.8)

We now turn to the main purpose of this paper, that is, to approximate the unique fixed point of the nonlinear operator given by (1.3), with the adequate conditions. We then define the approximating sequence described in the Introduction.

Theorem 3.1.

Let and Let be a set of positive numbers and, with the notation above, define inductively, for and the functions

(3.1)

(3.2)

where

(1) is a natural number such that

(2) is a natural number such that with

(3.3)

Then, for all it is satisfied that

(3.4)

Proof.

In view of condition ( ) we have, by applying (2.7), that for all , the inequality

(3.5)

is valid. Analogously, it follows from condition ( ) and (2.8) that for all

(3.6)

As a consequence, we derive that for all we have

(3.7)

and therefore,

(3.8)

as announced.

The next result is used in order to establish the fact that the sequence defined in Theorem 3.1 approximates the solution of the nonlinear Volterra integro-differential equation, as well as giving an upper bond of the error committed.

Proposition 3.2.

Let and be any subset of Then

(3.9)

with being the fixed point of the operator and

Proof.

We know from Lemma 2.1 that

(3.10)

for , which implies

(3.11)

The proof is complete by applying (2.2) to and taking into account that

(3.12)

As a consequence of Theorem 3.1 and Proposition 3.2, if is the exact solution of the nonlinear Volterra integro-differential (1.1), then for the sequence of approximating functions the error is given by

(3.13)

where In particular, it follows from this inequality that given there exists such that

In order to choose and (projections and in Theorem 3.1), we can observe the fact, which is not difficult to check, that the sequences and are bounded (and hence conditions (1.1) and (1.3)) in Theorem 3.1 are easy to verify), provided that the scalar sequence is bounded, and are functions, and , , , and satisfy a Lipschitz condition at their last variables. Indeed in view of inequality (3.13),

(3.14)

and in particular is bounded. Therefore, taking into account that the Schauder bases considered are monotone (norm-one projections, see [21]), we arrive at

(3.15)

Take and to derive from the triangle inequality and the last inequality that

(3.16)

Finally, since the sequence is bounded, also is. Similarly, one proves that is bounded (sequences and are bounded and and are Lipschitz at their second variables) and is bounded (sequences and are bounded and , and are Lipschitz at the third variables).

We have chosen the Schauder bases above for simplicity in the exposition, although our numerical method also works by considering fundamental biorthogonal systems in and .

The behaviour of the numerical method introduced above will be illustrated with the following two examples.

Example 4.1.

([22, Problem ]). The equation

(4.1)

has exact solution

Example 4.2.

Consider the equation

(4.2)

whose exact solution is

The computations associated with the examples were performed using Mathematica 7. In both cases, we choose the dense subset of

(4.3)

to construct the Schauder bases in and . To define the sequence introduced in Theorem 3.1, we take and (for all ) in the expression (3.2), that is

(4.4)

In Tables 1 and 2 we exhibit, for and , the absolute errors committed in eight points ( ) of when we approximate the exact solution by the iteration . The results in Table 1 improve those in [22].