Research Article | Open Access
Convergence Analysis of Legendre Pseudospectral Scheme for Solving Nonlinear Systems of Volterra Integral Equations
We are concerned with the extension of a Legendre spectral method to the numerical solution of nonlinear systems of Volterra integral equations of the second kind. It is proved theoretically that the proposed method converges exponentially provided that the solution is sufficiently smooth. Also, three biological systems which are known as the systems of Lotka-Volterra equations are approximately solved by the presented method. Numerical results confirm the theoretical prediction of the exponential rate of convergence.
Volterra-type integral equations (VIEs) are the mathematical model of many evolutionary problems with memory arising from biology, chemistry, physics, and engineering. For instance, in several heat transfer problems in physics, the equations are usually replaced by systems of Volterra integral equations (SVIEs). Since just few of these equations (i.e., VIEs and SVIEs) can be solved analytically, it is often necessary to apply appropriate numerical techniques.
Among numerical approaches, spectral methods are very powerful tools for approximating the solutions of many kinds of differential equations arising in various fields of science and engineering [1–5]. Spectral (exponential) accuracy and ease of applying these methods are two effective properties which have encouraged many authors to use them for integral equations (IEs) too. Spectral methods have been widely used by many authors in numerical analysis [6–13] for different kinds of IEs. In , Tang et al. proposed a Legendre spectral method (LSM) and its error analysis for the linear VIEs of the second kind. In this paper, we extend the LSM  to the numerical solution of the SVIEs of the second kind, including giving a convergence analysis for the nonlinear case. Thus, we consider the following nonlinear SVIEs: where and are given, whereas is the unknown function. We will consider the case that the solution of (1) is sufficiently smooth.
The remainder of this paper is organized as follows. The LSM is introduced in Section 2. Convergence analysis of the proposed method is discussed in Section 3. Section 4 states three applications of the desired equation in the biological systems. In Section 5, four types of biological models that are known as Lotka-Volterra system of equations are solved by the LSM to show the efficiency of the presented method and to verify the theoretical results obtained in Section 3. Also some comparisons are made with the existing results. Finally, Section 6 includes some concluding remarks.
2. Implementation of the Legendre Spectral Method
In this section, we apply the basic idea of Tang et al. , which was previously used by the authors in [10, 12], for discretizing the nonlinear SVIEs (1). In the procedure of approximation, Legendre Gauss quadrature rule together with Lagrange interpolation is used.
In order to use the spectral method, we consider the collocation points as the set of Legendre-Gauss points (i.e., the roots of , where is the th Legendre polynomial). Assume that the system (1) holds at : Gauss quadrature rules can be used to compute approximately the integral terms in (2). To this end, we make the change of variable and obtain By applying the -point Gauss quadrature formula associated with the Legendre weights , we get where and the points coincide with the collocation points .
Let and , for , and represent the Lagrange interpolation polynomials of and , in which is the th Lagrange basis function. The use of these interpolation polynomials for representing and in terms of and implies that Let and denote the approximation of and , respectively. Then, from (7), we obtain the discrete problem which is a system of nonlinear algebraic equations and unknown coefficients and . The nonlinear system (8) can be solved by an appropriate numerical method and the Lagrange interpolation of the solutions can be then obtained from (6). The numerical experiments show that the nonlinear system (8) can be solved easily by fsolve command in the Maple software.
3. Convergence Analysis
In this section, convergence analysis of the proposed method for the system of Volterra integral equations (1) will be provided. We show that the rate of convergence is exponential.
For convenience, we need some definitions and lemmas for providing the proof of the main Theorem. These lemmas include integral error of the Gauss quadrature rules, estimates of the interpolation error, Lebesgue constant of the Legendre series, and finally the Gronwall inequality.
Definition 1. Let be a bounded interval of , and let . One denotes by the space of the measurable functions such that . Endowed with the norm it is a Banach space.
Definition 2. Let be a bounded interval of , and let be an integer. One defines to be the vector space of the functions such that all the distributional derivatives of of order up to can be represented by functions in . In short, is endowed with the inner product for which is a Hilbert space. The associated norm is
Lemma 3 (integration error from Gauss quadrature [14, page 290]). Assume that a -point Gauss or Gauss-Radau or Gauss-Lobatto quadrature formula relative to the Legendre weight is used to integrate the product , where with for some and . Then, there exists a constant independent of such that where
Lemma 4 (estimates for the interpolation error [14, page 289]). Assume that and denote as its interpolation polynomial associated with the -point Gauss or Gauss-Radau or Gauss-Lobatto points , namely, Then, the following estimates hold:
Lemma 5 (Lebesgue constant for the Legendre series ). Assume that is the th Lagrange interpolation polynomials associated with the Gauss or Gauss-Radau or Gauss-Lobatto points. Then, where is a bounded constant.
Lemma 6 (Gronwall inequality). If a nonnegative integrable function satisfies where is an integrable function, then
Here, we assume that the kernel has the two following properties which are required for the proof of the convergence analysis:(i)the Lipschitz property; in other words where and ;(ii).
In the following, we will provide the main theorem of this section in . A similar technique could be designed in by using an extrapolation between and .
Theorem 7. Let be the exact solution of Volterra equation (1) and assume that where is given by (8) and is the th Lagrange basis function associated with the Gauss points . If , then, for , we have provided that is sufficiently large, where is a constant independent of .
Proof. According to notation (15), if then the numerical schemes (5) can be written as which gives where From Lemma 3 and the assumption (or ), we have On the other hand, (27) can be rewritten as follows: Multiplying on both sides of (30) and summing up from to yield which can be restated in the following form: where is defined by (23) and the interpolation operator is defined by (16). It follows from (32) and (1) that Let , , denote the error function. Then, we have Consequently, where According to the Lipschitz property of the kernel , we have The use of the Gronwall inequality with yields From (29) and Lemma 5, we have Using -error bounds for the interpolation polynomials (i.e., Lemma 4) gives By letting in (17), we have The above estimates, together with (38), yield which leads to (24) provided that is sufficiently large. This completes the proof.
The Lotka-Volterra equations model the dynamic behavior of an arbitrary number of competitors . Though originally formulated to describe the time history of a biological system, these equations find their application in a number of engineering fields such as nonlinear control. The accurate solutions of the Lotka-Volterra equations may become a difficult task either if the equations are stiff (even with a small number of species) or when the number of species is large . Therefore, it is necessary to apply the robust numerical techniques to achieve the best approximations. We refer the interested reader to [18–22] for more information on the biological models and the Lotka-Volterra equations.
First, we consider the prey-predator model: Lotka-Volterra system as an interacting species model to be governed by [23, 24] where , , , , , and are appropriate constants. Here, is the prey (e.g., rabbits) population and is the predator (e.g., foxes) population at time .
As the second system, we consider the simple 2-species Lotka-Volterra competition model with each species and having logistic growth in the absence of the other [23, 24]: where , , , , , , , and are all positive constants and the ’s are the linear birth rates and the ’s are the carrying capacities. The and measure the competitive effect of on and on , respectively, and they are generally not equal. If we nondimensionalize this system by writing the system (44) would be changed into the following system: where and are the new initial values.
As the final system, we consider the following version of the Lotka-Volterra equations : where , , , , and are constants.
These models can be easily transformed into their associated systems of Volterra integral equations. This idea (i.e., changing the IVPs into their associated Volterra integral forms) has been done by many authors like [25, 26]. This approach of transforming the IVPs into their Volterra form has many interesting advantages such as imposing the initial conditions to the new equations and ease of applying high order Gauss quadrature rules for getting highly accurate approximations.
For instance, the first model (43) can be transformed into a system of Volterra integral equations as follows: For the ease of applying the spectral method, as , we make the change of variable and observe the problem where Similar procedures can be applied to restate the other models (46)-(47) as Volterra integral equations system in .
5. Numerical Examples
In this section, some numerical examples are considered to illustrate the efficiency and accuracy of the proposed method. In all examples, we consider as the Legendre-Gauss points with the corresponding weights where is the th Legendre polynomial. Also, the nonlinear algebraic systems are solved directly by using fsolve command in Maple 13 software with the Digits environment variable assigned to be 30. All calculations are run on a Pentium 4 PC with 2.70 GHz CPU and 4 GB RAM. In order to show the efficiency of the LSM, we compare our results with those of the other methods that were proposed recently in the literature like Bessel collocation method  (BCM), HPM , and VIM . It should be noted that, in all tables, the absolute values of residuals are provided in the uniform nodes , , , , , and .
Example 1. Let us first consider the following problem for model (43): where , , and . We transfer this model to the system of Volterra integral equations as follows: where and .
For comparing the LSM and BCM , for , , , the absolute values of residual functions for the approximate solutions obtained by the LSM and BCM are provided in Table 5. These comparisons are also depicted in Figure 1. Moreover, Figure 2 displays the residual functions which are obtained by our method for . From Table 1 and Figure 1, we observe that the presented method is very effective and the obtained results are better than those of the BCM. In , one can see that the BCM is better than the ADM  and HPM ; thus, the current method is more effective than these methods too. From Figure 2, one can conclude that numerical solution with high accuracy is furnished by the presented method.
Example 2. We now consider the following problem for model (46): where and .
We apply the presented method to find the approximate solutions of the equivalent system of Volterra integral equations. Table 2 shows the numerical results of the residual functions obtained by LSM for , , . In order to compare the results of our method for with the five-term HPM solutions , the residual functions of these methods are depicted in Figures 3 and 4. From Table 2, one can conclude that the LSM provides the numerical solutions with high accuracy. Also, from Figures 3 and 4, we see that the results obtained by the presented method are better than those obtained by the HPM.