Solving Hammerstein Type Integral Equation by New Discrete Adomian Decomposition Methods
New discrete Adomian decomposition methods are presented by using some identified Clenshaw-Curtis quadrature rules. We investigate two mixed quadrature rules one of precision five and the other of precision seven. The first rule is formed by using the Fejér second rule of precision three and Simpson rule of precision three, while the second rule is formed by using the Fejér second rule of precision five and the Boole rule of precision five. Our methods were applied to a nonlinear integral equation of the Hammerstein type and some examples are given to illustrate the validity of our methods.
In this paper we study the problem of approximate solutions for the nonlinear integral equations of the Hammerstein type: Nonlinear integral equations arise naturally in many applications in describing numerous real world problems. For example, it occurs in solving several problems arising in economics, engineering, and physics. One of the most important frequently investigated nonlinear integral equations is the Hammerstein integral equation (cf. [1–5]).
On the other hand, there are significant interests in applying the Adomian decomposition method (ADM) for a wide class of nonlinear equations, for example, ordinary and partial differential equations, integral equations, and integrodifferential equations; see [6–9] and references therein.
In , Behiry et al. introduced a discrete version of the Adomian decomposition method and applied it to (1). This method is called a discrete Adomian decomposition method (DADM). DADM arises when the quadrature rules are used to approximate the definite integrals which cannot be computed analytically. The DADM gives the numerical solution at nodes used in the quadrature rules.
Dash and Das [11, 12] used mixed quadrature rules to approximate a definite integral, namely, mixed quadrature rules blending some Fejér and Newton-Cote type rule and a mixed quadrature rule blending Clenshaw-Curtis five-point rule and Gauss-Legendre three-point rule. Behiry and other  applied the Simpson rule with subinterval and step size . It is occasionally useful, both theoretically and practically, to have interpolatory formulas on sets of abscissas other than the equidistant set. A common choice is the set of zeros of an orthogonal polynomial.
Our main goal is to improve DADM, used by Behiry and other  to obtain approximate solutions of (1), by using mixed quadrature rules to approximate a definite integral. We use the advantage of the fact that the Fejér second rule of precision three and Simpson 1/3 rule of precision three to form a mixed quadrature rule of higher precision, that is, precision five. Also, we use the Fejér second rule of precision five and the Boole rule of precision five to form a mixed quadrature rule of precision seven. Our numerical examples show that our method gives approximate solutions to (1) more accurate than approximate solutions obtained in .
2. Quadrature Rules
In this section, we recall definitions of two quadrature rules which will be used throughout the paper, namely, Clenshaw-Curtis quadrature and Fejér quadrature. Clenshaw-Curtis quadrature and Fejér quadrature are based on an expansion of the integrand in terms of Chebyshev polynomials. So, let us first state some facts about Chebyshev polynomials. It is worth mentioning that Chebyshev polynomials are everywhere dense in numerical analysis .
Definition 1. The Chebyshev polynomial of the first kind is a polynomial in of degree , defined by the following relation:
From formula (2), the zeros for in of must correspond to the zeros for in of , so that Hence, the zeros of are The internal extrema of correspond to the extreme values of , namely, the zeros of , since . Hence, including those at , the extrema of on are
Definition 2. The Chebyshev polynomial of the second kind is a polynomial in of degree , defined by the following relation:
The zeros of are given by
2.1. Clenshaw-Curtis Quadrature
Clenshaw-Curtis quadrature method proposed by Clenshaw and Curtis  amounts to integrating via a change of variable . The algorithm is normally expressed for integration of a function over the interval ; any other interval can be obtained by appropriate rescaling. For this integral, we can write
That is, we have transformed the problem from integrating to one of integrating . This can be performed if we know the cosine series for . The reason that this is connected to the Chebyshev polynomials is that, by (2), , and so the cosine series is really an approximation of by Chebyshev polynomials: and thus we are integrating by integrating its approximate expansion in terms of Chebyshev polynomials. The evaluation points correspond to the extrema of the Chebyshev polynomial ; see (5). The fact that such Chebyshev approximation is just a cosine series under a change of variables is responsible for the rapid convergence of the approximation as more terms are included. A cosine series converges very rapidly for functions that are even, periodic, and sufficiently smooth. This is true here, since is even and periodic in by construction, and is -times differentiable everywhere if is -times differentiable on .
2.2. Fejér Quadrature
Let be a given fixed integer, and define the quadrature nodes on the interval as the extrema of the Chebyshev polynomial arguments by the boundary points: Fejér’s first rule  is obtained by using the well-known Chebyshev points as nodes, that is, from (10) with , namely, with the corresponding weights Fejér’s second rule  is obtained by omitting the nodes and and using the interpolating polynomial of degree . This may also be achieved by keeping the boundary points as nodes but preassigning the corresponding weights as . Then Fejér’s second rule is given by with the corresponding weights
3. DADM with New Nodes
By applying ADM, the solution of (1) is given by the following series form: where the components , , can be computed later on. We represent the nonlinear term by the Adomian polynomials, , as follows: where can be evaluated by the following formula : By substituting (15) and (16) into (1), we obtain Now, we can compute the components , , by using the following recursive relations : It is noticed that the computation of each component , , requires the computation of an integral in (19). If the evaluation of integral in (19) is analytically impossible, the ADM cannot be applied. In order to use numerical integration method for integral in (19), we transform the interval into the interval by using the transformation
Now, we will make use of the following two quadrature rules.
Rule 1. Here, the construction mixed quadrature rule will be of precision five .
We consider the Fejér second rule of precision three: and the Simpson rule of precision three: Then, we obtain the mixed quadrature rule of precision five for the approximate evaluation of ; namely, with a truncation error given by
Rule 2. In this rule the mixed quadrature rule will be of precision seven .
We consider the Fejér second rule of precision five: and the Boole rule of precision five: Then, we obtain the mixed quadrature rule of precision seven for the approximate evaluation of ; namely, with a truncation error given by
4. Numerical Examples
In this section we apply our methods to some integral equations of the Hammerstein type. These examples show the efficient and accuracy of our methods. The tables show computed absolute error: where is the number of the components . The computations associated with examples are performed using Mathematica .
Example 1. Consider the nonlinear integral equation:
Here, , , , and . Equation (30) has an exact solution , .
Tables 1, 2, and 3 the results of our example is stated while Table 4, the results obtained in  are shown.
Example 2. Consider the nonlinear integral equation:
Here, , , , and . Equation (31) has an exact solution , .
The results of this example represents in Tables 5, 6, and 7 while Table 8 represents the results given in .
In this paper, we use a new discrete Adomian decomposition method (NADM) to obtain numerical solutions of integral equations of the Hammerstein type. Our obtained results indicate that our method is a remarkably successful numerical technique for solving integral equations of the Hammerstein type. We make a comparison between our results in Tables 3 and 7 and results obtained in  (see Tables 4 and 8), and we found that our results are more accurate than results obtained in .
K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, Germany, 1985.View at: MathSciNet
D. O'Regan and M. Meehan, Existence Theory For Nonlinear Integral and Integrodierential Equations, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1998.
S. Shah, A. Shaikh, and S. H. Sandilo, “Modied decomposition method for nonlinear Volterra-Fredholm integrodierential equation,” Journal of Basic & Applied Sciences, vol. 6, no. 1, pp. 13–16, 2010.View at: Google Scholar
S. H. Behiry, R. A. Abd-Elmonem, and A. M. Gomaa, “Discrete Adomian decomposition solution of nonlinear Fredholm integral equation,” Ain Shams Engineering Journal, vol. 1, no. 1, pp. 97–101, 2010.View at: Google Scholar
R. B. Dash and D. Das, “A mixed quadrature rule by blending Clenshaw-Curtis and Gauss-Legendre quadrature rules for approximation of real definite integrals in adaptive environment,” IMECS, vol. 2011, pp. 202–205, 2011.View at: Google Scholar
J. C. Mason and D. C. Handscomb, Chebyshev Polynomials, A CRC Press, New York, NY, USA, 2003.View at: MathSciNet
G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, vol. 60 of Fundamental Theories of Physics, Kluwer Academic Publishers Group, Dordrecht, The Netherlands, 1994, With a preface by Yves Cherruault.View at: MathSciNet
A.-M. Wazwaz, A First Course in Integral Equations, World Scientific Publishing, Singapore, 1997.View at: MathSciNet
A. J. Jerri, Introduction to Integral Equations with Applications, John Wiley & Sons, 2nd edition, 1999.View at: MathSciNet