Numerical Solutions of a Class of Nonlinear Volterra Integral Equations

and Applied Analysis 3 Using u (t 2l+1 ) ≃ u (t 2l ) + u (t 2l+2 )


Introduction
In this paper we study the nonlinear (nonstandard) Volterra integral equation of the second kind of the form (  () + ∫  0   (, )())  ,  ∈ [0, ] , (1) where ( ∈ N,  ≥ 2), with   ∈ R, and   ,   are continuous functions.Volterra integral equations play an important part in scientific and engineering problems such as population dynamics, spread of epidemics, semiconductor devices, wave propagation, superfluidity, and travelling wave analysis, Saveljeva [1].In cases where the kernel is of convolution type ((, ) = ( − )) the solutions to (1) include elliptic functions and natural generalizations of these functions which also have wide applications in the fields of science and engineering [2].This class of Volterra integral equations was considered by Sloss and Blyth [2] who proved the existence and uniqueness of the solution in the Banach space  2 and applied the Corrington's Walsh function method to (1).
Much work has been done in the study of numerical solutions to Volterra integral equations using collocation methods [1,[3][4][5][6][7].Benitez and Bolos [8] pointed out that collocation methods have proven to be a very suitable technique for approximating solutions to nonlinear integral equations because of their stability and accuracy.Other authors such as [9][10][11][12] used quadrature rules like repeated trapezoidal and repeated Simpson's rule to solve linear Volterra integral equations.However, collocation methods and quadrature rules have not been used to approximate solutions to (1).

The Collocation Method.
In our work we focus on onepoint collocation methods (see [13]).
Thus for  =  ,1 :=   +  1 ℎ and 0 <  1 ≤ 1 the collocation equation (3) assumes the form Expressing the collocation equation in terms of the stage values we get Let  ∈   and define Then The term   ( ,1 ) is called the lag term corresponding to the collocation solution, [13].

Repeated Trapezoidal Rule.
Using the trapezoidal rule we construct the solution to the integral equation (1) (see [12]).Let The approximation of the integral in (11) by repeated trapezoidal rule will give the following system: 2.3.Repeated Simpson's Rule.We use repeated Simpson's rule to construct the solution to the integral equation (1) (see [9]).

Existence and Uniqueness of the Solution
The following theorem shows that when  = 2 and  1 = 0 the integral equation ( 1) has a unique solution in the space [0, ].Theorem 2 gives sufficient conditions for the solution to (1) to exist.We prove the existence and uniqueness of the solution using a procedure analogous to the one used in Sloss and Blyth [2].
Theorem 1.The integral equation with  ∈ [0, 1],  ∈ R, and (, ) ∈ ([0, 1] × [0, 1]), has a unique solution  and the solution belongs to   = [0, ], 0 <  ≤ 1, with where Proof.The existence of the solution is shown in the corollary of Theorem 2 (in the next section).Here we prove the uniqueness of the solution.Let  and  + V be solutions of (18).Then, where   is a sequence of characteristic functions of intervals Then, Thus   is contractive if That is, Clearly, Suppose V() ̸ = 0 is a solution of (21), such that V may lie outside of [0, ].Then, which shows that   V is a fixed point of   for all .Since    → 0 in [0, 1] as   → 0, and for V ̸ = 0, we can select therefore Consequently

Numerical Computations
In our work we consider examples of (1) when  = 2.We use (6) to approximate the solutions considering two special cases:  1 = 1/2 (implicit midpoint method) and  1 = 1 (implicit Euler method).We also use the repeated trapezoidal and repeated Simpson's rule.Since the methods are implicit we perform an iterative procedure at each step implementing a tolerance of 10 −4 .For each method we used three different values of ℎ: ℎ = 0.01, ℎ = 0.005, and ℎ = 0.0025.

Example 1. Consider the nonlinear VIE
which arises from a nonlinear differential equation in [15] where  1 = 0 and  2 = 2.

Using the Iterated Collocation.
For  1 = 1/2 the iterated collocation solution of (48) is given as Integrate to obtain The iterated collocation solution of (48) with three different values of ℎ is shown in Figure 3.     (55) Figure 4 shows the solution to the VIE (48) for the three values of ℎ used.
The solution to (48) using repeated Simpson's rule is shown in Figure 5.  1 shows the errors in the solution of the integral equation (48) for the largest value of ℎ used.

Example 2. Consider
where  1 = 1 and  2 = 1/2.The integral equation (57) arises from nonlinear differential equations that represent conservative systems (see [16]).We used the four methods to approximate the solution to this example and Example 3, and we present tables for the absolute errors in the solution.
Table 2 shows the errors in the solution of (57) when ℎ = 0.01:    where  1 = 2 and  2 = 1.The nonlinear VIE arises from a nonlinear differential equation in [17].Shown in Table 3 are the errors in the solution of (58) when ℎ = 0.01.

Discussion
We approximated the solutions to Examples 1-3 using the implicit Euler method, implicit midpoint method, and repeated trapezoidal and repeated Simpson's rule using various values of the stepsize.At ℎ = 0.001 and below we obtained a similar solution from all the methods used; hence we take that as our "exact" solution.Therefore, for sufficiently small ℎ we get a good accuracy of the numerical solutions.When the stepsize is greater than 0.001 we obtained different numerical solutions from each of the four methods.We use the "exact" solution and absolute error to study the performance of each method when the stepsize is increased.
Tables 1-3 show the absolute errors in the solutions when ℎ = 0.01.From these tables we observe that the repeated Simpson's rule performs better followed by the implicit midpoint method then the repeated trapezoidal rule.Among the four methods used, the implicit Euler method gives a larger error as h is increased.We then found an iterated collocation solution for the implicit midpoint method and the accuracy of the method improved as shown in Figure 3.According to our numerical results, we conclude that the repeated Simpson's rule performs well since it gives better solutions when a reasonably large value of the stepsize is used.These observations are consistent for all three examples used.

Figure 4 :
Figure 4: The solution of (48) by the repeated trapezoidal rule.

Figure 5 :
Figure 5: The solution of (48) by the repeated Simpson's rule.