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Numerical solutions of a class of nonlinear Volterra integral equations.

1. Introduction

In this paper we study the nonlinear (nonstandard) Volterra integral equation of the second kind of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1)

where (r [member of] N, r [greater than or equal to] 2), with [b.sub.j] [member of] R, and [g.sub.j], [k.sub.j] 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 (K(t, s) = K(t - s)) 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 [L.sup.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, 3-7]. 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 [9-12] 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 [13]).

Let [t.sub.n] := nh (n = 0, 1, ..., N - 1) define a uniform partition for I = [0, T] and set [Z.sub.N] := [t.sub.0], ..., [t.sub.N], [I.sub.0] := [[t.sub.0], [t.sub.1]] [I.sub.n] := ([t.sub.n], [t.sub.n + 1]] (1 [less than or equal to] n [less than or equal to] N - 1). The solution to (1) will be approximated by using collocation in the piecewise constant polynomial space [S.sup.-1.sub.0]([Z.sub.N]).

For a given real number [c.sub.1], define the set [X.sub.N] := [t.sub.n, 1] of collocation points by

[t.sub.n, 1] = [t.sub.n] + [c.sub.1]h (0 [less than or equal to] [c.sub.1] [less than or equal to] 1, n=0, ..., N - 1). (2)

The collocation solution [u.sub.n] [member of] [S.sup.-1.sub.0] ([Z.sub.N]) is defined by the collocation equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3)

since

[u.sub.n] (t)= [u.sub.n] ([t.sub.n] + vh) = [L.sub.1] (v) [U.sub.n, 1], v [member of] (0, 1], (4)

where [L.sub.1](v) = 1 and is a Lagrange fundamental polynomial.

Thus for t = [t.sub.n1] := [t.sub.n] + [c.sub.1] h and 0 < [c.sub.1] [less than or equal to] 1 the collocation equation (3) assumes the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5)

Expressing the collocation equation in terms of the stage values we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6)

Let t [member of] [I.sub.n] and define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (7)

Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (8)

The term [F.sub.jn] ([t.sub.n, 1]) is called the lag term corresponding to the collocation solution, [13].

Iterated Collocation. The iterated approximation [u.sub.I] corresponding to u is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (9)

(see [4, 5, 14]).

Set t = [t.sub.n] [member of] [[bar.Z].sub.N] and use (4); we may write (9) in the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (10)

2.2. Repeated Trapezoidal Rule. Using the trapezoidal rule we construct the solution to the integral equation (1) (see [12]). Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (11)

The approximation of the integral in (11) by repeated trapezoidal rule will give the following system:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (12)

2.3. Repeated Simpson's Rule. We use repeated Simpsons rule to construct the solution to the integral equation (1) (see [9]).

If n is even, then Simpson's rule may be applied to each subinterval [[t.sub.2i,] [t.sub.2i + 1], [t.sub.2i + 2]]. For i = 0, 1, ..., (N/2) -1 we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (13)

Summing up,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (14)

We use (14) to solve the nonlinear (nonstandard) VIE. The approximation of (1) in the even nodes [t.sub.2m] is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (15)

Using

u ([t.sub.2l + 1]) [equivalent] u ([t.sub.2l]) + u ([t.sub.2l] + 2])/2. (16)

we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (17)

3. Existence and Uniqueness of the Solution

The following theorem shows that when r = 2 and [b.sub.1] = 0 the integral equation (1) has a unique solution in the space C[0,dj. 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

z (t) = b[(g(t) + [[integral].sup.2.sub.0] k(t, s) z(s)ds).sup.2], (18)

with g [member of] C[0, 1], b [member of] R, and k(t, s) [member of] C([0, 1] x [0, 1]), has a unique solution u and the solution belongs to [I.sub.d] = [0, d], 0 < d [less than or equal to] 1, with

0 < d < 1/K [1/2K [absolute value of b] - [[parallel]g[parallel].sub.[infinity]] - [[parallel]g[parallel].sub.[infinity]], (19)

where

K = sup [0, 1] x [0, 1] x [0, 1] [absolute value of k (t, s)]. (20)

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 u and u + v be solutions of (18).

Then,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (21)

Define [T.sub.n] : C[0, 1] x C[0, 1] by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (22)

where [X.sub.n] is a sequence of characteristic functions of intervals [0, [A.sub.n]] c [0, 1].

Let [v.sub.1] + [v.sub.2] [member of] C[0, 1]; consider

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (23)

then,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (24)

Let d = [max.sub.0[less than or equal to]t[less than or equal to]1] ([absolute value of [v.sub.1] (t)], [absolute value of [v.sub.2] (t)]) and take [T.sub.n] : [0, d] [right arrow] [0, 1].

Then,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (25)

therefore

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (26)

Thus [T.sub.n] is contractive if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (27)

That is,

0 < d < 1/K [1/2K [absolute value of b] - [[parallel]g[parallel]].sub.[infinity]] - [[parallel]u[parallel].sub.[infinity]]. (28)

Clearly, [T.sub.n] maps C[0, d] onto itself if (19) is satisfied. Also, [T.sub.n](0) = 0.

Suppose v(t) [not equal to] 0 is a solution of (21), such that v may lie outside of [0, d]. Then,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (29)

which shows that [X.sub.n] v is a fixed point of [T.sub.n] for all n. Since [X.sub.n]Y [right arrow] 0 in C[0, 1] as [A.sub.n] [right arrow] 0, and for v [not equal to] 0, we can select [X.sub.n] v [member of] [0, d]. But this is impossible since 0 is the only solution in [0, d]. Therefore the solution u of (18) is unique in C[0, d] if d > 0 exists that satisfies (19).

Theorem 2. There exists a solution u of (1), where u [member of] C[0, d] provided that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (30)

where [N.sub.b] is the number of nonzero [b.sub.j].

Proof. Let T'v(t) = [[summation].sup.r.sub.j=i] [b.sub.j] ([g.sub.j] (t) + [[integral].sup.t.sub.0] [k.sub.j] [(t, s) v (s) ds).sup.j] and [v.sub.1], [v.sub.2] [member of] [0, d] for a suitable d, and consider

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (31)

So

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (32)

therefore

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (33)

Consequently T' is a contraction mapping if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (34)

We need to show that T' : C[0, d] [right arrow] C[0, d]. Observe that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (35)

Therefore

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (36)

thus T' : C[0, d] [right arrow] C[0, d] if

[r.summation over (j=1)] [absolute value of [b.sub.j]] [([[parallel][g.sub.j][parallel].sub.[infinity]] + [K.sub.j]d).sub.j] < d. (37)

Hence the map T' is a contraction and maps [0, d] into itself provided (30) is satisfied. ?

Corollary 3. There exists a solution u to the integral equation

u(t) = b[(g(t) + [[integral].sup.t.sub.0] k(t, s)u(s)ds).sup.2], (38)

where u [member of] C [0, d] with

0 < d < 1/K [1/2[absolute value of b]K - [[parallel]g[parallel].sub.[infinity]], (39)

if

[[parallel]g[parallel].sub.[infinity]] < 1/4K [absolute value of b]. (40)

if

Proof. From Theorem 2 we get sufficient conditions for the existence of a solution

2K [absolute value of b] ([[parallel]g[parallel].sub.[infinity]] + Kd) < 1, (41)

[absolute value of b] [([[parallel]g[parallel].sub.[infinity]]+ Kd).sup.2] < d. (42)

Inequality (41) is solved by any d < [bar.d], where

[bar.d] = 1/K [1/2 [absolute value of b] K - [[parallel]g[parallel].sub.[infinity]]], (43)

and inequality (42) is equivalent to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (44)

This is satisfied by d [member of] ([d.sup.-], [d.sup.+]), where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (45)

If the regularity condition

1-4K [absolute value of b] [[parallel]g[parallel].sub.[infinity]] > 0 (46)

is satisfied, [d.sub.+] and [d.sub.-] are real and positive. Furthermore,

[bar.d] = [d.sup.-] + [d.sub.+]/2, (47)

so that (46) ensures d [member of] ([d.sub.-], [bar.d]) satisfies both inequalities (41) and (42) in Theorem 2.

4. Numerical Computations

In our work we consider examples of (1) when r = 2. We use (6) to approximate the solutions considering two special cases: [c.sub.1] = 1/2 (implicit midpoint method) and [c.sub.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.sup.-4]. For each method we used three different values of h: h = 0.01, h = 0.005, and h = 0.0025.

4.1. Example 1. Consider the nonlinear VIE

u (t) = 2[(1 + [[integral].sup.t.sub.0] (t - s)u(s)ds).sup.2] 0 [less than or equal to] t [less than or equal to] 1, (48)

which arises from a nonlinear differential equation in [15] where [b.sub.1] = 0 and [b.sub.2] = 2.

4.1.1. Using Implicit Euler Method. When [c.sub.1] = 1 and [t.sub.n, 1] = [t.sub.n] + h, the collocation solution of (48) is given by

[U.sub.n, i] = 2[(1 + [F.sub.n] ([t.sub.n, 1]) + [U.sub.n, 1] [h.sup.2]/2).sup.2], (49)

where

[F.sub.n] ([t.sub.n, 1]) = h [n-1.summation over (i=0)] ([t.sub.n] - [t.sub.i] + [h/2]) [U.sub.i, 1]. (50)

Figure 1 shows the solution to (48) at h = 0.01, h = 0.005, and h = 0.0025.

4.1.2. Using Implicit Midpoint Method. When [c.sub.1] = 1/2 and [t.sub.n, 1] = [t.sub.n] + (h/2), the collocation solution of (48) is given by

[U.sub.n, 1] = 2 [(1 + [F.sub.n] ([t.sub.n, 1]) + [U.sub.n, 1] [h.sup.2]/8).sup.2], (51)

where

[F.sub.n] ([t.sub.n, 1]) = h [n-1.summation over (i=0)] ([t.sub.n] - [t.sub.i]) [U.sub.i, 1]. (52)

Figure 2 shows the solution to (48) at h = 0.01, h = 0.005, and h = 0.0025.

4.1.3. Using the Iterated Collocation. For [c.sub.1] = 1/2 the iterated collocation solution of (48) is given as

[u.sub.I] ([t.sub.n]) = 2[(1 + h [n-1.summation over (i=0)] [[integral].sup.1.sub.0] ([t.sub.n] - [t.sub.i] - sh) ds [U.sub.i1]).sup.2]. (53)

Integrate to obtain

[u.sub.I] ([t.sub.n]) = 2[(1 + h [n-1.summation over (i=0)] ([t.sub.n] - [t.sub.i] - h/2) [U.sub.i1]).sup.2]. (54). (54)

The iterated collocation solution of (48) with three different values of h is shown in Figure 3.

4.1.4. Using Repeated Trapezoidal Rule. For the VIE (48) u(0) = 2 and

u ([t.sub.n]) = 2 [(1 + [h/2] [t.sub.n] u(0)+ h [n-1.summation over (i=1)] ([t.sub.n] - [t.sub.n-1]) [U.sub.n-1]).sup.2] (55)

Figure 4 shows the solution to the VIE (48) for the three values of h used.

4.1.5. Using Repeated Simpson's Rule. When t = 0, u(0) = 2 for (48) and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (56)

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 h used.

4.2. Example 2. Consider

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (57)

where [b.sub.1] = 1 and [b.sub.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:

4.3. Example 3. Consider the integral equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (58)

where [b.sub.1] =2 and [b.sub.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.

5. 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.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

http://dx.doi.org/10.1155/2014/652631

References

[1] D. Saveljeva, "Quadratic and cubic spline collocation for Volterra integral equations," 2006, http://dspace.utlib.ee/dspace/ handle/10062/1163.

[2] B. G. Sloss and W F. Blyth, "Corrington's Walsh function method applied to a nonlinear integral equation," Journal of Integral Equations and Applications, vol. 6, no. 2, pp. 239-256, 1994.

[3] H. Brunner, "Iterated collocation methods for Volterra integral equations with delay arguments," Mathematics of Computation, vol. 62, no. 206, pp. 581-599, 1994.

[4] J. G. Blom and H. Brunner, "The numerical solution of nonlinear Volterra integral equations of the second kind by collocation and iterated collocation methods," Society for Industrial and Applied Mathematics. Journal on Scientific and Statistical Computing, vol. 8, no. 5, pp. 806-830, 1987

[5] T. Diogo, "Collocation and iterated collocation methods for a class of weakly singular Volterra integral equations," Journal of Computational and Applied Mathematics, vol. 229, no. 2, pp. 363-372, 2009.

[6] T. Diogo and P. Lima, "Collocation solutions of weakly singular Volterra integral equation," Matematica Aplicada e Computacional, vol. 8, pp. 229-238, 2007

[7] V. Horvat, "On collocation methods for Volterra integral equations with delay arguments," Mathematical Communications, vol. 4, no. 1, pp. 93-109, 1999.

[8] R. Benitez and V. J. Bolos, "Blow-up collocation solutions of some Volterra integral equations," 2011, http://arxiv.org/abs/1112.4658.

[9] M. Aigo, "On the numerical approximation of Volterra integral equations of the second kind using quadrature rules," International Journal of Advanced Scientific and Technological Research, vol. 1, pp. 558-564, 2013.

[10] R. Katani and S. Shahmorad, "Block by block method for the systems of nonlinear Volterra integral equations," Applied Mathematical Modelling, vol. 34, no. 2, pp. 400-406, 2010.

[11] F. Mirzaee, "A computational method for solving linear Volterra integral equations," Applied Mathematical Sciences, vol. 6, no. 17-20, pp. 807-814, 2012.

[12] J. Saberi-Nadjafi and M. Heidari, "A quadrature method with variable step for solving linear Volterra integral equations of the second kind," Applied Mathematics and Computation, vol. 188, no. 1, pp. 549-554, 2007.

[13] H. Brunner, Collocation Methods For Volterra Integral And Related Functional Differential Equations, Cambridge University Press, Cambridge, Mass, 2004.

[14] H. Brunner, "Iterated collocation methods and their discretizations for Volterra integral equations," SIAM Journal on Numerical Analysis, vol. 21, no. 6, pp. 1132-1145, 1984.

[15] M. S. Corrington, "Solution of differential and integral equations with walsh functions," IEEE Trans Circuit Theory, vol. 20, no. 5, pp. 470-476, 1973.

[16] R. H. Rand, Lecture notes on nonlinear vibrations, 2003, http://www.math.cornell.edu/~rand/randdocs/nlvibe52.pdf.

[17] D. G. Zill, W S. Wright, and M. R. Cullen, Differential Equations with Boundary Value Problems, Richard Stratton, 2012.

H. S. Malindzisa and M. Khumalo

Department of Pure and Applied Mathematics, University of Johannesburg, PO. Box 524, Auckland Park 2006, South Africa

Correspondence should be addressed to H. S. Malindzisa; hlukaphi@gmail.com

Received 10 April 2014; Revised 17 June 2014; Accepted 26 June 2014; Published 9 July 2014

Academic Editor: Chun-Gang Zhu

TABLE 1: Absolute errors in the solution of (48) when h = 0.01.

t     Implicit   Implicit   Repeated      Repeated
      euler      midpoint   trapezoidal   Simposon

0.1   0.0077     0.0038     0.0074        --
0.2   0.0161     0.0078     0.0154        0.0001
0.3   0.0266     0.0127     0.0252        --
0.4   0.0406     0.0190     0.0377        0.0001
0.5   0.0607     0.0275     0.0548        0.0005
0.6   0.0907     0.0397     0.0180        0.0004
0.7   0.1371     0.0576     0.1146        0.0007
0.8   0.2115     0.0849     0.1689        0.0013
0.9   0.3360     0.1280     0.2549        0.0023
1     0.5530     0.1992     0.3966        0.0044

TABLE 2: Absolute errors in the solution of (57) when h = 0.01.

t     Implicit   Implicit   Repeated      Repeated
      euler      midpoint   trapezoidal   Simposon

0.1   0.0028     0.0014     0.0027        --
0.2   0.0057     0.0028     0.0055        --
0.3   0.0089     0.0043     0.0086        --
0.4   0.0127     0.0061     0.0121        0.0001
0.5   0.0171     0.0081     0.0162        0.0001
0.6   0.0226     0.0105     0.0209        0.0001
0.7   0.0295     0.0135     0.0239        --
0.8   0.0385     0.0172     0.0342        0.0001
0.9   0.0501     0.0219     0.0436        0.0001
1.0   0.0659     0.0279     0.0553        0.0002

TABLE 3: Absolute errors in the solution of (58) when h = 0.01.

t     Implicit   Implicit   Repeated      Repeated
      euler      midpoint   trapezoidal   Simposon

0.1   0.0116     0.0057     0.0110        --
0.2   0.0239     0.0116     0.0229        0.0001
0.3   0.0389     0.0187     0.0369        --
0.4   0.0586     0.0274     0.0543        0.0002
0.5   0.0857     0.0390     0.0774        0.0003
0.6   0.1247     0.0549     0.1089        0.0005
0.7   0.1829     0.0773     0.1537        0.0009
0.8   0.2727     0.1104     0.2196        0.0016
0.9   0.4163     0.1607     0.3196        0.0028
1.0   0.6541     0.2399     0.4771        0.0049
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Date:Jan 1, 2014
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