Solving Hammerstein Type Integral Equation by New Discrete Adomian Decomposition Methods

Abstract

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 1/3 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.

1. Introduction

In this paper we study the problem of approximate solutions for the nonlinear integral equations of the Hammerstein type: 𝜆𝑥(𝑡)=𝑦(𝑡)+^𝑏_𝑎𝑘(𝑡,𝑠)𝑢(𝑥(𝑠))𝑑𝑠,𝜆≠0;𝑎≤𝑡≤𝑏. (1) 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 [10], 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 [10] applied the Simpson rule with 𝑛 subinterval and step size ℎ=(𝑏−𝑎)/𝑛. 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 [10] 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 [10].

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 [13].

Definition 1. The Chebyshev polynomial 𝑇_𝑛(𝑥) of the first kind is a polynomial in 𝑥 of degree 𝑛, defined by the following relation: 𝑇_𝑛(𝑥)=cos𝑛𝜃,when𝑥=cos𝜃. (2)

From formula (2), the zeros for 𝑥 in [−1,1] of 𝑇_𝑛(𝑥) must correspond to the zeros for 𝜃 in [0,𝜋] of cos𝑛𝜃, so that 𝜋𝑛𝜃=2𝑗−12,𝑗=1,2,3,…,𝑛. (3) Hence, the zeros of 𝑇_𝑛(𝑥) are 𝑥_𝑗𝜋=cos2𝑗−12𝑛,𝑗=1,2,3,…,𝑛. (4) The internal extrema of 𝑇_𝑛(𝑥) correspond to the extreme values of cos𝑛𝜃, namely, the zeros of sin𝑛𝜃, since (𝑑/𝑑𝑥)𝑇_𝑛(𝑥)=sin𝑛𝜃/sin𝜃. Hence, including those at 𝑥=±1, the extrema of 𝑇_𝑛(𝑥) on [−1,1] are 𝑥_𝑗=cos𝑗𝜋𝑛,𝑗=1,2,3,…,𝑛. (5)

Definition 2. The Chebyshev polynomial 𝑈_𝑛(𝑥) of the second kind is a polynomial in 𝑥 of degree 𝑛, defined by the following relation: 𝑈_𝑛(𝑥)=sin(𝑛+1)𝜃sin𝜃,when𝑥=cos𝜃. (6)

The zeros of 𝑈_𝑛(𝑥) are given by 𝑥_𝑗=cos𝑗𝜋𝑛+1,𝑗=1,2,3,…,𝑛. (7)

2.1. Clenshaw-Curtis Quadrature

Clenshaw-Curtis quadrature method proposed by Clenshaw and Curtis [14] amounts to integrating via a change of variable 𝑥=cos(𝜃). The algorithm is normally expressed for integration of a function 𝑓(𝑥) over the interval [−1,1]; any other interval can be obtained by appropriate rescaling. For this integral, we can write ¹₋₁𝑓(𝑥)𝑑𝑥=^𝜋₀𝑓(cos𝜃)sin(𝜃)𝑑𝜃. (8)

That is, we have transformed the problem from integrating 𝑓(𝑥) to one of integrating 𝑓(cos𝜃)sin𝜃. This can be performed if we know the cosine series for 𝑓(cos𝜃). The reason that this is connected to the Chebyshev polynomials 𝑇_𝑗(𝑥) is that, by (2), 𝑇_𝑗(cos𝜃)=cos(𝑗𝜃), and so the cosine series is really an approximation of 𝑓(𝑥) by Chebyshev polynomials: 𝑓(𝑥)=𝑎₀2𝑇₀(𝑥)+^𝑛_𝑗=1𝑎_𝑗𝑇_𝑗(𝑥),𝑥∈[−1,1], (9) and thus we are integrating 𝑓(𝑥) by integrating its approximate expansion in terms of Chebyshev polynomials. The evaluation points 𝑥_𝑗=cos(𝑗𝜃/𝑛) correspond to the extrema of the Chebyshev polynomial 𝑇_𝑛(𝑥); 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 𝑇_𝑗(𝑥) are included. A cosine series converges very rapidly for functions that are even, periodic, and sufficiently smooth. This is true here, since 𝑓(cos𝜃) is even and periodic in 𝜃 by construction, and is 𝑗-times differentiable everywhere if 𝑓(𝑥) is 𝑗-times differentiable on [−1,1].

2.2. Fejér Quadrature

Let 𝑛≥2 be a given fixed integer, and define the (𝑛+1) quadrature nodes on the interval [−1,1] as the extrema of the Chebyshev polynomial 𝑇_𝑛(𝑥) arguments by the boundary points: 𝑥_𝑗=cos𝜗_𝑗,𝜗_𝑗=𝑗𝜋𝑛,𝑗=0,1,2,…,𝑛. (10) Fejér’s first rule [15] is obtained by using the well-known Chebyshev points as nodes, that is, 𝑥_𝑗 from (10) with 𝑗=1/2,3/2,…,𝑛−1/2, namely, 𝑅_1𝐹𝑛𝑓=^𝑛_𝑗=0𝑤_𝑗𝑓𝑥_𝑗=^𝑛_𝑗=0𝑤_𝑗𝑓cos𝑗𝜋𝑛, (11) with the corresponding weights 𝑤_𝑗=2𝑛1−2^[𝑛/2]_𝑚=1cos𝑚𝜗_𝑗+14𝑚²−1,𝑗=1,2,…,𝑛−1. (12) Fejér’s second rule [15] is obtained by omitting the nodes 𝑥₀=1 and 𝑥_𝑛=−1 and using the interpolating polynomial of degree 𝑛−2. This may also be achieved by keeping the boundary points as nodes but preassigning the corresponding weights as 𝑤₀=𝑤_𝑛=0. Then Fejér’s second rule is given by 𝑅_2𝐹𝑛𝑓=^𝑛_𝑗=0𝑤^∗_𝑗𝑓𝑥^∗_𝑗=^𝑛_𝑗=0𝑤^∗_𝑗𝑓cos𝑗𝜋𝑛+1 (13) with the corresponding weights 𝑤^∗_𝑗=4𝑛sin𝜗_𝑗^[𝑛/2]_𝑚=1sin(2𝑚−1)𝜗_𝑗2𝑚−1,𝑗=0,2,…,𝑛. (14)

3. DADM with New Nodes

By applying ADM, the solution 𝑥 of (1) is given by the following series form: 𝑥(𝑡)=^∞_𝑘=0𝑥_𝑘(𝑡), (15) where the components 𝑥_𝑘(𝑡), 𝑘≥0, can be computed later on. We represent the nonlinear term 𝑢(𝑥(𝑡)) by the Adomian polynomials, 𝐴_𝑘(𝑡), as follows: 𝑢(𝑥(𝑡))=^∞_𝑘=0𝐴_𝑘𝑥₀(𝑡),𝑥₁(𝑡),…,𝑥_𝑘(𝑡), (16) where 𝐴_𝑘(𝑡) can be evaluated by the following formula [16]: 𝐴_𝑘𝑥₀(𝑡),𝑥₁(𝑡),…,𝑥_𝑘(𝑡=1)𝑑𝑘!𝑑𝛽^𝑘𝑘𝑢^∞_𝑘=0𝛽^𝑘𝑥_𝑘𝛽=0. (17) By substituting (15) and (16) into (1), we obtain ^∞_𝑘=0𝑥_𝑘(𝑡)=1𝜆𝑦(𝑡)+1𝜆^∞_𝑘=0^𝑏_𝑎𝑘(𝑡,𝑠)𝐴_𝑘(𝑠)𝑑𝑠. (18) Now, we can compute the components 𝑥_𝑘(𝑡), 𝑘≥0, by using the following recursive relations [17]: 𝑥_𝑘+1(𝑡)=1𝜆^𝑏_𝑎𝑘(𝑡,𝑠)𝐴_𝑘(𝑠)𝑥𝑑𝑠,𝑘≥0,₀(𝑡)=𝑦(𝑡)𝜆. (19) It is noticed that the computation of each component 𝑥_𝑘(𝑡), 𝑘≥0, 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 [𝑎,𝑏] into the interval [−1,1] by using the transformation 1𝜏=2[(𝑎+𝑏)+(𝑏−𝑎)𝑡]. (20)

Now, we will make use of the following two quadrature rules.

Rule 1. Here, the construction mixed quadrature rule will be of precision five [11].

We consider the Fejér second rule of precision three: 𝐼𝑓=¹₋₁𝑓(𝑥)𝑑𝑥≃𝑅_2𝐹3𝑓=23𝑓1√2+𝑓(0)+𝑓−1√2 (21) and the Simpson 1/3 rule of precision three: 𝐼𝑓=¹₋₁𝑓(𝑥)𝑑𝑥≃𝑅_𝑆𝑓=13𝑓(−1)+4𝑓(0)+𝑓(1). (22) Then, we obtain the mixed quadrature rule of precision five for the approximate evaluation of 𝐼(𝑓); namely, 𝑅_2𝐹3𝑆𝑓=1𝑓()15−1+8𝑓−1√2(0)1+12𝑓+8𝑓√2+𝑓(1) (23) with a truncation error 𝐸_2𝐹3𝑆 given by 𝐸_2𝐹3𝑆𝑓=1𝑓37800^(𝑣𝑖)(0)+⋯. (24)

Rule 2. In this rule the mixed quadrature rule will be of precision seven [11].

We consider the Fejér second rule of precision five: 𝑅_2𝐹5𝑓=2√457𝑓321+9𝑓2+13𝑓(0)+9𝑓−12−√+7𝑓32 (25) and the Boole rule of precision five: 𝑅_𝐵𝑓=1()457𝑓−1+32𝑓−12(0)1+12𝑓+32𝑓2+7𝑓(1). (26) Then, we obtain the mixed quadrature rule of precision seven for the approximate evaluation of 𝐼(𝑓); namely, 𝑅_2𝐹5𝐵𝑓=1()−√3159𝑓−1+80𝑓32+144𝑓−12(0)1+164𝑓+144𝑓2√+80𝑓32+9𝑓(1) (27) with a truncation error 𝐸_2𝐹5𝐵 given by 𝐸_2𝐹5𝐵𝑓=1𝑓140×9!^{(𝑣𝑖𝑖𝑖)}(0)+⋯. (28)

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: 𝑒_𝑚(𝑡)=𝑥_exact(𝑡)−𝑥_app.(𝑡), (29) where 𝑚 is the number of the components 𝑥₁,𝑥₂,…,𝑚. The computations associated with examples are performed using Mathematica 6.

Example 1. Consider the nonlinear integral equation: (𝑡)110𝑥=10𝑡−4(𝑒−1)exp𝑡⁴+¹₀exp𝑠⁴+𝑡⁴(𝑥(𝑠))³𝑑𝑠. (30) Here, 𝜆=10, 𝑦(𝑡)=10𝑡−(1/4)(𝑒−1)exp(𝑡⁴), 𝑘(𝑡,𝑠)=exp(𝑠⁴+𝑡⁴), and 𝑢(𝑥(𝑡))=(𝑥(𝑠))³. Equation (30) has an exact solution 𝑥_𝑒(𝑡)=𝑡, [18].
Let 𝑥₀(𝑡)=𝑦(𝑡)/𝜆=𝑡−(1/40)(𝑒−1)exp(𝑡⁴).
Tables 1, 2, and 3 the results of our example is stated while Table 4, the results obtained in [10] are shown.

Table 1 

The effect of 𝑚 in the absolute error at 𝑛=3.

Table 2 

The effect of 𝑚 in the absolute error at 𝑛=5.

Table 3 

The effect of 𝑚 in the absolute error at 𝑛=7.

Table 4 

The effect of 𝑚 in the absolute error at 𝑛=8.

Example 2. Consider the nonlinear integral equation: (𝑡)()()+20𝑥=20𝑡+cos𝑒+𝑡−cos1+𝑡₁₀exp(𝑥(𝑠))sin𝑡+𝑒^𝑠𝑑𝑠. (31) Here, 𝜆=20, 𝑦(𝑡)=20𝑡+cos(𝑒+𝑡)−cos(1+𝑡), 𝑘(𝑡,𝑠)=sin(𝑡+𝑒^𝑠), and 𝑢(𝑥(𝑡))=exp(𝑥(𝑠)). Equation (31) has an exact solution 𝑥_𝑒(𝑡)=𝑡, [18].
Let 𝑥₀(𝑡)=𝑦(𝑡)/𝜆=𝑡+(1/20)cos(𝑒+𝑡)−(1/20)cos(1+𝑡).
The results of this example represents in Tables 5, 6, and 7 while Table 8 represents the results given in [10].

Table 5 

The effect of 𝑚 in the absolute error at 𝑛=3.

Table 6 

The effect of 𝑚 in the absolute error at 𝑛=5.

Table 7 

The effect of 𝑚 in the absolute error at 𝑛=7.

Table 8 

The effect of 𝑚 in the absolute error at 𝑛=8.

5. Conclusions

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 [10] (see Tables 4 and 8), and we found that our results are more accurate than results obtained in [10].