Abstract
We consider the numerical solutions of a class of nonlinear (nonstandard) Volterra integral equation. We prove the existence and uniqueness of the one point collocation solutions and the solution by the repeated trapezoidal rule for the nonlinear Volterra integral equation. We analyze the convergence of the collocation methods and the repeated trapezoidal rule. Numerical experiments are used to illustrate theoretical results.
1. Introduction
In this paper we consider the nonlinear (nonstandard) Volterra integral equation: where and are continuous functions. Some examples of the nonlinear VIE (1) arise from nonlinear ordinary differential equations used to represent conservative systems. In [1] we provided sufficient conditions for the existence and uniqueness of the solution to (1). We also approximated the solutions for (1) using collocation methods, the repeated trapezoidal rule, and repeated Simpson’s rule.
Collocation solutions for Volterra integral equations have been the subject of several work over the years. Brunner [2] studied the existence and uniqueness of the collocation solution for linear Volterra integral equations of the second kind and showed that the methods yield global convergence of order . In [3] nonlinear Volterra integral equations with multiple proportional delays were studied; they provided sufficient conditions for the existence and uniqueness of the analytic and collocation solution. Moreover, [3] proved that the collocation methods yield global convergence of order . Other authors such as [4–6] also analysed the convergence of collocation methods for Volterra integral equations with different types of kernels. In this work we study the conditions for the existence and uniqueness of the numerical solutions of (1) and perform convergence analysis for the collocation methods and repeated trapezoidal rule.
2. Existence and Uniqueness of the Numerical Solution
Consider the nonlinear VIE where , , and . We prove a theorem analogous to the one in [3], establishing the existence and uniqueness of the collocation solution for (2). The collocation solution to (2) is given as where
Let , and ; then
To solve (7) we use an iterative procedure; thus we rewrite (7) in the form where . Let Then (8) can be rewritten as
Theorem 1. Assume that the given functions and in the nonlinear VIE (2) are continuous in their respective domains and . Then there exists a constant , for any uniform mesh with , such that (3) defines a unique collocation solution that belongs to the piecewise constant polynomial , for (2).
Proof. Since the kernel in (2) is continuous on , then in (10) is bounded. Whenever , . Hence, for in (10), exists and is bounded.
For some assume that is bounded. Then, arguing as above, exists and is bounded if .
The above holds if there is an such that for a uniform mesh with , the condition holds for all . Hence there exists a unique solution for (3).
Corollary 2. Under the same conditions as in Theorem 1, defines a unique solution to (2) by the repeated trapezoidal rule.
Proof. Expanding (11) we get where , , and for . Let then we have The result follows from Theorem 1 by taking
Corollary 3. Applying collocation methods or the repeated trapezoidal rule results in a unique solution for (1).
Proof. Existence and uniqueness for the case , follows from Theorem 1 and Corollary 2. For the case , existence and uniqueness for the collocation solution is established in [2]. For the case , the existence and uniqueness of the solution by the repeated trapezoidal rule follows from continuity of and . Combining these two cases establishes the existence and uniqueness of the collocation solution and the solution by the repeated trapezoidal rule for (1).
3. Convergence for the Numerical Methods
3.1. Global Convergence for the Collocation Methods
The global convergence for the case where in (1) has been analysed (see [2]); thus we will study the case where and where both and are nonzero. We use a procedure analogous to the one used in [3] to analyse the global convergence of the collocation solution , with .
Theorem 4. Assume that , , and , defined by (3), are the collocation solution for (2).
Then for all sufficiently small , we have
where is the Lagrange interpolation operator corresponding to the collocation parameter and the constant does not depend on .
Proof. Define as follows:
then the operator formulations for the VIE (2) and its collocation equation are given by
Based on the solvability of the VIE and its collocation equation we implement an iterative procedure and obtain
where and .
Then the error between and can be written as
which implies that
From the error estimates of the interpolation , we have
and with appropriate assumptions on and ,
which leads to
On the other hand, recall that the collocation solution to (1) is given by
Corollary 5. If the solution defined by (25) is the collocation solution to (1), then, for a sufficiently small , holds for any set of collocation point with . The constant depends on the but not on .
Proof. From [2] we know that the collocation solution for the VIE, satisfies Using the triangle inequality we have
3.2. Repeated Trapezoidal Rule
Consider the solution to (2) by repeated trapezoidal rule
Theorem 6. The approximate method given by (30) is convergent and its order of convergence is at least one.
Proof. Putting in (2), we have
Using the Lipschitz condition, (31) can be written as
where and are the errors of the integration rule.
Then, let ; hence
Then we have
For the functions and with at least first-order derivatives, we have . Hence we have
Corollary 7. The repeated trapezoidal solution for (1), defined by is convergent and its order of convergence is at least one.
Proof. Since for the repeated trapezoidal rule when used to solve (2) and (27), it follows that for (1) we have
4. Numerical Results
In this section we present numerical results obtained from the collocation methods and the repeated trapezoidal rule. To obtain the estimates for the convergence rates we used the following quantity: where , , and denote approximations to using the step sizes , , and . The results are shown in Figures 1 and 2, which indicates first-order convergence and this is in agreement with the results of the theorems in Section 3. The approximations for the convergence rates are done using results from the following examples.
(a) The convergence rates of the implicit Euler
(b) The convergence rates of the implicit midpoint
(c) The convergence rates of the repeated trapezoidal rule
(a) The convergence rates of the implicit Euler
(b) The convergence rates of the implicit midpoint
(c) The convergence rates of the repeated trapezoidal rule
Example 1. Consider
Example 2. Consider
5. Discussion
In this work we provided sufficient conditions for the existence and uniqueness of the collocation solution and the solution by the repeated trapezoidal rule for (2). We performed a numerical analysis of the nonlinear (nonstandard) Volterra integral equation (1). In Theorem 4 and Corollary 5 we proved that the collocation methods yield global convergence order of one. We also proved in Section 3.2 that the repeated trapezoidal rule has convergence order of one. Numerical approximations of the convergence orders of the implicit Euler, implicit midpoint, and repeated trapezoidal rule were made and are shown in Section 4. The computed orders of convergence are in agreement with the theoretical results in Section 3.
Conflict of Interests
The authors declare that there is no conflict of interests in respect of this paper.