Numerical Solutions of a Class of Nonlinear Volterra Integral Equations
We consider numerical solutions of a class of nonlinear (nonstandard) Volterra integral equations. We first prove the existence and uniqueness of the solution of the Volterra integral equation in the context of the space of continuous functions over a closed interval. We then use one-point collocation methods with a uniform mesh to construct solutions of the nonlinear (nonstandard) VIE and quadrature rules. We conclude that the repeated Simpson's rule gives better solutions when a reasonably large value of the stepsize is used.
In this paper we study the nonlinear (nonstandard) Volterra integral equation of the second kind of the form where , , with , 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 . 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 . This class of Volterra integral equations was considered by Sloss and Blyth  who proved the existence and uniqueness of the solution in the Banach space 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, 3–7]. Benitez and Bolos  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 [9–12] 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).
2. The Numerical Methods
2.1. The Collocation Method
In our work we focus on one-point collocation methods (see ).
Let define a uniform partition for and set ,. The solution to (1) will be approximated by using collocation in the piecewise constant polynomial space .
For a given real number , define the set of collocation points by
The collocation solution is defined by the collocation equation since where and is a Lagrange fundamental polynomial.
Thus for and 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 is called the lag term corresponding to the collocation solution, .
Iterated Collocation. The iterated approximation corresponding to is defined by (see [4, 5, 14]).
Set and use (4); we may write (9) in the form
2.2. Repeated Trapezoidal Rule
Using the trapezoidal rule we construct the solution to the integral equation (1) (see ). 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 ).
If is even, then Simpson’s rule may be applied to each subinterval . For we have
Summing up, We use (14) to solve the nonlinear (nonstandard) VIE. The approximation of (1) in the even nodes is given by
3. Existence and Uniqueness of the Solution
The following theorem shows that when and the integral equation (1) has a unique solution in the space . 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 .
Theorem 1. The integral equation with , , and , has a unique solution and the solution belongs to , , 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 be solutions of (18).
Define by where is a sequence of characteristic functions of intervals .
Let ; consider then,
Let and take .
Then, therefore Thus is contractive if That is,
Clearly, maps onto itself if (19) is satisfied. Also, .
Suppose is a solution of (21), such that may lie outside of . Then, which shows that is a fixed point of for all . Since in as , and for , we can select . But this is impossible since is the only solution in . Therefore the solution of (18) is unique in if exists that satisfies (19).
Theorem 2. There exists a solution of (1), where provided that where is the number of nonzero .
Proof. Let and for a suitable , and consider
Consequently is a contraction mapping if
We need to show that . Observe that
thus if Hence the map is a contraction and maps into itself provided (30) is satisfied.
Corollary 3. There exists a solution to the integral equation where with if
Proof. From Theorem 2 we get sufficient conditions for the existence of a solution
Inequality (41) is solved by any , where and inequality (42) is equivalent to This is satisfied by , where
If the regularity condition is satisfied, and are real and positive. Furthermore, so that (46) ensures satisfies both inequalities (41) and (42) in Theorem 2.
4. Numerical Computations
In our work we consider examples of (1) when . We use (6) to approximate the solutions considering two special cases: (implicit midpoint method) and (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 . For each method we used three different values of : , , and .
4.1. Example 1
Consider the nonlinear VIE which arises from a nonlinear differential equation in  where and .
4.1.1. Using Implicit Euler Method
When and , the collocation solution of (48) is given by where
Figure 1 shows the solution to (48) at , , and .
4.1.2. Using Implicit Midpoint Method
When and , the collocation solution of (48) is given by where
Figure 2 shows the solution to (48) at , , and .
4.1.3. Using the Iterated Collocation
For the iterated collocation solution of (48) is given as
Integrate to obtain
The iterated collocation solution of (48) with three different values of is shown in Figure 3.
4.1.4. Using Repeated Trapezoidal Rule
For the VIE (48) and
Figure 4 shows the solution to the VIE (48) for the three values of used.
4.1.5. Using Repeated Simpson’s Rule
When , for (48) and
The solution to (48) using repeated Simpson’s rule is shown in Figure 5.
Table 1 shows the errors in the solution of the integral equation (48) for the largest value of used.
4.2. Example 2
Consider where and . The integral equation (57) arises from nonlinear differential equations that represent conservative systems (see ). 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 :
4.3. Example 3
Consider the integral equation where and . The nonlinear VIE arises from a nonlinear differential equation in . Shown in Table 3 are the errors in the solution of (58) when .
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 and below we obtained a similar solution from all the methods used; hence we take that as our “exact” solution. Therefore, for sufficiently small we get a good accuracy of the numerical solutions. When the stepsize is greater than 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 . 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.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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