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Journal of Applied Mathematics
Volume 2012, Article ID 714973, 10 pages
Research Article

A New Direct Method for Solving Nonlinear Volterra-Fredholm-Hammerstein Integral Equations via Optimal Control Problem

1Faculty of Engineering, Umm Al-Qura University, P.O. Box 5555, Mecca, Saudi Arabia
2Faculty of Science, Helwan University, Helwan, Egypt

Received 28 December 2011; Accepted 28 January 2012

Academic Editor: Shuyu Sun

Copyright © 2012 M. A. El-Ameen and M. El-Kady. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


A new method for solving nonlinear Volterra-Fredholm-Hammerstein (VFH) integral equations is presented. This method is based on reformulation of VFH to the simple form of Fredholm integral equations and hence converts it to optimal control problem. The existence and uniqueness of proposed method are achieved. Numerical results are given at the end of this paper.

1. Introduction

The nonlinear integral equations arise in the theory of parabolic boundary value problems, engineering, various mathematical physics, and theory of elasticity [13]. In recent years, several analytical and numerical methods of this kind of problems have been presented [4, 5]. Analytically, the decomposition methods are used in [6, 7]. The classical method of successive approximations was introduced in [8], while some kind of appropriate projection such as Galerkin and collocation methods have been applied in [913]. These methods often transform integral or integrodifferential equations into a system of linear algebraic equations which can be solved by direct or iterative methods. In [14], the authors used Taylor series to solve the following nonlinear Volterra-Fredholm integral equation:𝑦(𝑥)=𝑓(𝑥)+𝜆1𝑥0𝑘1[](𝑥,𝑡)𝑦(𝑡)𝑝𝑑𝑡+𝜆210𝑘2[](𝑥,𝑡)𝑦(𝑡)𝑞𝑑𝑡,𝑝,𝑞𝑅,(1.1)

whereas the Legendre wavelets method for a special type was applied in [15] for solving the nonlinear Volterra-Fredholm integral equation of the form𝑦(𝑥)=𝑓(𝑥)+𝜆1𝑥0𝑘1[](𝑥,𝑡)𝐹(𝑦(𝑡))𝑑𝑡+𝜆210𝑘2(𝑥,𝑡)𝐺(𝑦(𝑡))𝑑𝑡,(1.2)

where 𝑓(𝑥) and the kernels 𝑘1(𝑥,𝑡) and 𝑘2(𝑥,𝑡) are assumed to be in 𝐿2(𝑅) on the interval 0𝑥,𝑡1. The nonlinear Volterra-Fredholm-Hammerstein integral equation is given in [16] as follows:𝑦(𝑡)=𝑓(𝑡)+𝜆1𝑡0𝑘1𝑔(𝑡,𝑠)1(𝑠,𝑦(𝑠))𝑑𝑠+𝜆210𝑘2(𝑡,𝑠)𝑔2(𝑠,𝑦(𝑠))𝑑𝑠,0𝑡,𝑠1.(1.3) In this paper, we introduce a method to find the numerical solution of a nonlinear Volterra-Fredholm-Hammerstein integral equation of the form:𝜙(𝑡)=𝑓(𝑡)+𝜆1𝑡0𝑉(𝑡,𝑠,𝜙(𝑠))𝑑𝑠+𝜆2𝑏𝑎𝐹(𝑡,𝑠,𝜙(𝑠))𝑑𝑠,0𝑡,𝑠1,(1.4) where 𝑓(𝑡), 𝑉(𝑡,𝑠,𝜙(𝑠)), and 𝐹(𝑡,𝑠,𝜙(𝑠)) are assumed to be in 𝐿2(𝑅) and satisfy the Lipschitz condition||𝐾𝑡,𝑠,𝜙1(𝑠)𝐾𝑡,𝑠,𝜙2||||𝜙(𝑠)𝑁(𝑡,𝑠)1(𝑠)𝜙2||(𝑠).(1.5) This paper is organized as follows. In Section 2, we present a form of (1.4) by Fredholm type integral equation, which can convert it into optimal control problem (OC). In Section 3, the existence and uniqueness are presented. The computational results are shown in Section 4.

2. Problem Reformulation

Let the VFH given in (1.4) be written in the form𝜙(𝑡)=𝑓(𝑡)+𝜆𝑏𝑎𝑘(𝑡,𝑠,𝜙(𝑠))𝑑𝑠,(2.1) such that𝐺𝑘(𝑡,𝑠,𝜙(𝑠))=𝐺(𝑡,𝑠,𝜙(𝑠))+𝐹(𝑡,𝑠,𝜙(𝑠)),(𝑡,𝑠,𝜙(𝑠))=𝑒(𝑡,𝑠,𝜙(𝑠))𝑉(𝑡,𝑠,𝜙(𝑠)),𝑒(𝑡,𝑠,𝜙(𝑠))=1𝑎<𝑠<𝑡<𝑏,0𝑠>𝑡,(2.2) and the kernel 𝑘(𝑡,𝑠,𝜙(𝑠))𝐶[𝑎,𝑏]×[𝑎,𝑏] satisfyies||||||||𝑘(𝑡,𝑠,𝜙(𝑠))𝑀,𝑓(𝑡)𝐾,(2.3) where M, K are arbitrary constants.

It is easy to see that (2.1) can be written as follows: 𝜙(𝑡)𝑓(𝑡)=𝑡𝑎̇̇[]=𝜙(𝑠)𝑓(𝑠)𝑑𝑠+𝜙(𝑎)𝑓(𝑎)𝑏𝑎𝛿̇𝜙̇𝑓[𝜙](𝑠)(𝑠)𝑑𝑠+(𝑎)𝑓(𝑎),where𝛿=1𝑎<𝑠<𝑡<𝑏,0𝑠>𝑡,(2.4) then𝑏𝑎𝛿̇̇[]𝜙(𝑠)𝑓(𝑠)𝑑𝑠+𝜙(𝑎)𝑓(𝑎)=𝜆𝑏𝑎𝑘(𝑡,𝑠,𝜙(𝑠))𝑑𝑠.(2.5) Since𝜙(𝑎)𝑓(𝑎)=𝜆𝑏𝑎𝑘(𝑎,𝑠,𝜙(𝑠))𝑑𝑡,(2.6) therefore,𝑏𝑎𝛿̇̇𝜙(𝑠)𝑓(𝑠)𝑑𝑠+𝜆𝑏𝑎[]𝑘(𝑎,𝑠,𝜙(𝑠))𝑘(𝑡,𝑠,𝜙(𝑠))𝑑𝑠=0.(2.7) Let𝐺(𝑡)=𝑏𝑎𝛿̇̇[]𝜙(𝑠)𝑓(𝑠)+𝜆𝑘(𝑎,𝑠,𝜙(𝑠))𝑘(𝑡,𝑠,𝜙(𝑠))𝑑𝑠=0,(2.8) that is, if||||𝐺(𝑡)=0.(2.9) By integrating (2.9), we have𝑏𝑎||||𝐺(𝑡)𝑑𝑡=0.(2.10) On the other hand, one can define the following equality:𝐹̇𝜙̇𝑓[𝑘](𝑡,𝑠,𝜙(𝑡),𝑢(𝑡))=𝛿(𝑠)(𝑠)+𝜆(𝑎,𝑠,𝜙(𝑠))𝑘(𝑡,𝑠,𝜙(𝑠)).(2.11) This will lead us to the following inequality:𝑏𝑎||||𝐺(𝑡)𝑑𝑡𝑏𝑎||||𝐹(𝑡,𝑠,𝜙(𝑡),𝑢(𝑡))𝑑𝑠𝑑𝑡,(2.12) wherė[]𝜙(𝑠)=𝑢(𝑠),𝑠𝑎,𝑏.(2.13) With the boundary conditions𝜙(𝑎)=𝑓(𝑎)+𝑏𝑎𝑘(𝑎,𝑠,𝜙(𝑠))𝑑𝑠,𝜙(𝑏)=𝑓(𝑏)+𝑏𝑎𝑘(𝑏,𝑠,𝜙(𝑠))𝑑𝑠.(2.14) At the end, we have the following OC problem:

minimize𝐼=Ω||||𝐹(𝑡,𝑠,𝜙(𝑡),𝑢(𝑡))𝑑𝑠𝑑𝑡(2.15) subject tȯ[]𝜙(𝑠)=𝑢(𝑠),𝑠𝑎,𝑏,(2.16)𝜙(𝑎) and 𝜙(𝑏) are defined in (2.14) where Ω=[𝑎,𝑏]×[𝑎,𝑏].

The existence and uniqueness of (2.1) will be considered in the next section by using the successive approximation method.

3. Existence and Uniqueness

The solution 𝜙(𝑡) of (2.1) can be approximated successively as follows:𝜙1(𝑡)𝜆𝑏𝑎𝑘𝑡,𝑠,𝜙0(𝑠)𝑑𝑠=𝑓(𝑡).(3.1) Thus, we obtain sequence of functions 𝜙0(𝑡),𝜙1(𝑡),,𝜙𝑛(𝑡), such that𝜙𝑛(𝑡)𝜆𝑏𝑎𝑘𝑡,𝑠,𝜙𝑛1(𝑠)𝑑𝑠=𝑓(𝑡),𝑛1,(3.2) with 𝜙0(𝑡)=𝑓(𝑡).

It is convenient to introduce𝜓𝑛(𝑡)=𝜙𝑛(𝑡)𝜙𝑛1(𝑡),𝑛1,(3.3) with 𝜓0(𝑡)=𝑓(𝑡).

Subtracting from (3.2), the same equation with replacing 𝑛 by 𝑛1, we get𝜙𝑛(𝑡)𝜙𝑛1(𝑡)=𝜆𝑏𝑎𝑘𝑡,𝑠,𝜙𝑛1(𝑠)𝑑𝑠𝜆𝑏𝑎𝑘𝑡,𝑠,𝜙𝑛2(𝑠)𝑑𝑠.(3.4)

Using (3.3), we have𝜓𝑛(𝑡)=𝜆𝑏𝑎𝑘𝑡,𝑠,𝜓𝑛1(𝑠)𝑑𝑠𝑛1.(3.5) Also, from (3.3), we deduce that𝜙𝑛(𝑡)=𝑛𝑖=0𝜓𝑖(𝑡).(3.6)

The existence and uniqueness of the solution can be followed.

Theorem 3.1 (2.1). If the kernel 𝑘(𝑡,𝑠,𝜙(𝑠)) and the function 𝑓(𝑡) are continuous and satisfy condition (2.3) in 𝑎<𝑠<𝑡<𝑏, then the integral equation (2.1) possesses a unique continuous solution.

Proof. From (3.5), we get ||𝜓𝑛||=||||𝜆(𝑡)𝑏𝑎𝑘𝑡,𝑠,𝜓𝑛1||||||𝑘(𝑠)𝑑𝑠𝜆𝑡,𝑠,𝜓𝑛1||(𝑠)𝑏𝑎𝑑𝑠𝜆(𝑏𝑎)𝑀.(3.7) We now show that this 𝜙(𝑡) satisfies (2.1).
The series (3.6) is uniformly convergent since the term 𝜓𝑖(𝑡) is dominated by𝜆(𝑏𝑎)𝑀. Then, 𝜆𝑏𝑎𝑘𝑡,𝑠,𝑖=0𝜓𝑖(𝑠)𝑑𝑠=𝑖=0𝜆𝑏𝑎𝑘𝑡,𝑠,𝜓𝑖=(𝑠)𝑑𝑠𝑖=0𝜓𝑖+1(𝑡)=𝑖=0𝜓𝑖+1(𝑡)+𝜓0(𝑡)𝜓0(𝑡).(3.8) Hence, we have 𝜆𝑏𝑎𝑘𝑡,𝑠,𝑖=0𝜓𝑖(𝑠)𝑑𝑠=𝑖=0𝜓𝑖(𝑡)𝑓(𝑡).(3.9) This proves that 𝜙(𝑡), defined in (3.6), satisfies (2.1). Since each of the 𝜓𝑖(𝑡) is clearly continuous, therefore 𝜙(𝑡) is continuous, where it is the limit of a uniformly convergent sequence of continuous functions.
To show that 𝜙(𝑡) is a unique continuous solution, suppose that there exists another continuous solution 𝜙(𝑡) of (2.1), Then, 𝜙(𝑡)𝜆𝑏𝑎𝑘𝑡,𝑠,𝜙(𝑠)𝑑𝑠=𝑓(𝑡).(3.10) Subtracting (3.10) from (2.1), we get 𝜙(𝑡)𝜙(𝑡)=𝜆𝑏𝑎𝑘𝑡,𝑠,𝜙(𝑠)𝜙(𝑠)𝑑𝑠.(3.11) Since 𝜙(𝑡) and 𝜙(𝑡) are both continuous, there exists a constant 𝐵 such that ||||𝜙(𝑡)𝜙(𝑡)𝐵.(3.12) By using the condition of (2.3), the inequality (3.12) becomes ||||𝜙(𝑡)𝜙(𝑡)𝜆(𝑏𝑎)𝑀𝐵.(3.13) For the large enough 𝑛, the right-hand side is arbitrary small, then 𝜙(𝑡)=𝜙(𝑡).(3.14) This completes the proof.

4. Computational Results

In this section, some numerical experiments will be carried out in order to compare the performances of the new method with respect to the classical collocation methods. The method has been applied to the following three test problems [16, 17].

Example 4.1. Consider the Volterra-Fredholm-Hammerstein integral equation 𝑥(𝑠)=2cos(𝑠)2+3𝑠0sin(𝑠𝑡)𝑥26(𝑡)𝑑𝑡+76cos(1)10(1𝑡)cos2(𝑠)(𝑡+𝑥(𝑡))𝑑𝑡.(4.1) The exact solution is given by 𝑥(𝑠)=cos(𝑠).
The computational maximum absolute errors for different values of 𝑁 are shown in Table 1. It is clear that the optimal control method is more accurate for small values of 𝑁. It seems that the errors for 𝑁=16, in case of OC method, are caused by machine error. The numerical solutions are computed by two methods and summarized in Figure 1, and it seems that our method compared very well with those obtained via the collocation method.

Table 1: Maximum errors for collocation and OC methods.
Figure 1: Observed results for Example 4.1 (𝑁=16).

Example 4.2. Consider the following VIE: 𝑥(𝑠)=1+sin2(𝑠)3𝑠0sin(𝑠𝑡)𝑥2[](𝑡)𝑑𝑡,𝑠0,10.(4.2) The exact solution is 𝑥(𝑠)=cos(𝑠). This example can be solved by using the proposed OC method. The numerical results together with computational effort of errors in boundaries and CPU time/iteration are given in Table 2. The computational efforts presented here proved that we could rearrange in a way to avoid the rounding errors in collocation and reducing the CPU time/iteration processes. Furthermore, this rearrangement of the computation leads to a much more accurate and robust method.
In Figure 2, the proposed OC method shows the observed results for Example 4.2 for 𝑁=16. It seems also that OC method is more accurate than collocation methods.

Table 2: Observed results for Example 4.2.
Figure 2: Observed results for Example 4.2 (𝑁=16).

Example 4.3. Consider the nonlinear Volterra-Fredholm integral equation 𝑥(𝑠)=𝑦(𝑠)+𝑠0(𝑠𝑡)𝑥2(𝑡)𝑑𝑡+10(𝑠+𝑡)𝑥(𝑡)𝑑𝑡,(4.3) with 𝑦(𝑠)=(1/30)𝑠6+(1/3)𝑠4𝑠2+(5/3)𝑠(5/4). We applied the OC method presented in this paper and solved (4.3). The computational results together with the exact solution 𝑥(𝑠)=𝑠22 are given in Figure 3.

Figure 3: Observed results for Example 4.3 (𝑁=16).

5. Conclusion

In this paper, the optimal control method is introduced to simplify the implementation of general nonlinear integral equations of the second kind. We have shown, in numerical examples, that this method is fast and gains better results compared with collocation method. The important thing to note is that the control-state constraint is satisfied everywhere. Furthermore, the structure of the optimal control agrees with the results obtained in [16].


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