MPE Mathematical Problems in Engineering 1563-5147 1024-123X Hindawi Publishing Corporation 760515 10.1155/2013/760515 760515 Research Article Solving Hammerstein Type Integral Equation by New Discrete Adomian Decomposition Methods Bakodah Huda O. 1 http://orcid.org/0000-0002-4245-4364 Darwish Mohamed Abdalla 1, 2 Kaya Metin O. 1 Department of Mathematics Sciences Faculty for Girls King Abdulaziz University Jeddah Saudi Arabia kau.edu.sa 2 Department of Mathematics Faculty of Science Damanhour University Damanhour Egypt damanhour.edu.eg 2013 17 11 2013 2013 02 05 2013 04 10 2013 2013 Copyright © 2013 Huda O. Bakodah and Mohamed Abdalla Darwish. 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.

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: (1)λx(t)=y(t)+abk(t,s)u(x(s))ds,hhhhhhhhhhλ0;atb. 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. ).

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  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 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  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 .

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

From formula (2), the zeros for x in [-1,1] of Tn(x) must correspond to the zeros for θ in [0,π] of cosnθ, so that (3)nθ=(2j-1)π2,j=1,2,3,,n. Hence, the zeros of Tn(x) are (4)xj=cos(2j-1)π2n,j=1,2,3,,n. The internal extrema of Tn(x) correspond to the extreme values of cosnθ, namely, the zeros of sinnθ, since (d/dx)Tn(x)=sinnθ/sinθ. Hence, including those at x=±1, the extrema of Tn(x) on [-1,1] are (5)xj=cosjπn,j=1,2,3,,n.

Definition 2.

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

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

Clenshaw-Curtis quadrature method proposed by Clenshaw and Curtis  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 (8)-11f(x)dx=0πf(cosθ)sin(θ)dθ.

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 Tj(x) is that, by (2), Tj(cosθ)=cos(jθ), and so the cosine series is really an approximation of f(x) by Chebyshev polynomials: (9)f(x)=a02T0(x)+j=1najTj(x),x[-1,1], and thus we are integrating f(x) by integrating its approximate expansion in terms of Chebyshev polynomials. The evaluation points xj=cos(jθ/n) correspond to the extrema of the Chebyshev polynomial Tn(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 Tj(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].

Let n2 be a given fixed integer, and define the (n+1) quadrature nodes on the interval [-1,1] as the extrema of the Chebyshev polynomial Tn(x) arguments by the boundary points: (10)xj=cosϑj,ϑj=jπn,j=0,1,2,,n. Fejér’s first rule  is obtained by using the well-known Chebyshev points as nodes, that is, xj from (10) with j=1/2,3/2,,n-1/2, namely, (11)R1Fn(f)=j=0nwjf(xj)=j=0nwjf(cosjπn), with the corresponding weights (12)wj=2n[1-2m=1[n/2]cos(mϑj+1)4m2-1],j=1,2,,n-1. Fejér’s second rule  is obtained by omitting the nodes x0=1 and xn=-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=wn=0. Then Fejér’s second rule is given by (13)R2Fn(f)=j=0nwj*f(xj*)=j=0nwj*  f(cosjπn+1) with the corresponding weights (14)wj*=4nsinϑjm=1[n/2]sin[(2m-1)ϑj]2m-1,j=0,2,,n.

By applying ADM, the solution x of (1) is given by the following series form: (15)x(t)=k=0xk(t), where the components xk(t), k0, can be computed later on. We represent the nonlinear term u(x(t)) by the Adomian polynomials, Ak(t), as follows: (16)u(x(t))=k=0Ak[x0(t),x1(t),,xk(t)], where Ak(t) can be evaluated by the following formula : (17)Ak[x0(t),x1(t),,xk(t)]=1k!dkdβk[u(k=0βkxk)]β=0. By substituting (15) and (16) into (1), we obtain (18)k=0xk(t)=1λy(t)+1λk=0abk(t,s)Ak(s)ds. Now, we can compute the components xk(t), k0, by using the following recursive relations : (19)xk+1(t)=1λabk(t,s)Ak(s)ds,k0,x0(t)=y(t)λ. It is noticed that the computation of each component xk(t), k0, 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 (20)τ=12[(a+b)+(b-a)t].

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: (21)I(f)=-11f(x)dxR2F3(f)=23  [f(12)+f(0)+f(-12)] and the Simpson 1/3 rule of precision three: (22)I(f)=-11f(x)dxRS(f)=13  [f(-1)+4f(0)+f(1)]. Then, we obtain the mixed quadrature rule of precision five for the approximate evaluation of I(f); namely, (23)R2F3S(f)=115[f(-1)+8f(-12)+12f(0)+8f(12)+f(1)] with a truncation error E2F3S given by (24)E2F3S(f)=137800f(vi)(0)+.

Rule 2.

In this rule the mixed quadrature rule will be of precision seven .

We consider the Fejér second rule of precision five: (25)R2F5(f)=245[7f(32)+9f(12)+13f(0)+9f(-12)+7f(-32)] and the Boole rule of precision five: (26)RB(f)=145[7f(-1)+32f(-12)+12f(0)+32f(12)+7f(1)]. Then, we obtain the mixed quadrature rule of precision seven for the approximate evaluation of I(f); namely, (27)R2F5B(f)=1315[9f(-1)+80f(-32)+144f(-12)+164f(0)+144f(12)+80f(32)+9f(1)] with a truncation error E2F5B given by (28)E2F5B(f)=1140  ×  9!f(viii)(0)+.

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: (29)|em(t)|=|xexact(t)-xapp.(t)|, 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: (30)10x(t)=10t-14(e-1)exp(t4)+01exp(s4+t4)(x(s))3ds. 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 xe(t)=t, .

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  are shown.

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

R 2 F 3
t | e 3 ( t ) | | e 4 ( t ) | | e 5 ( t ) |
0.14645 4.93549 × 10 - 3 4.43266 × 10 - 3 4.26484 × 10 - 3
0.50000 5.251397 × 10 - 3 4.71637 × 10 - 3 4.53780 × 10 - 3
0.85355 8.38783 × 10 - 3 7.53327 × 10 - 3 7.24806 × 10 - 6

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

R 2 F 5 S
t | e 3 ( t ) | | e 4 ( t ) | | e 5 ( t ) |
0.00000 1 . 68344 × 10 - 3 9 . 07620 × 10 - 4 6 . 12675 × 10 - 4
0.14645 1 . 68421 × 10 - 3 9 . 08038 × 10 - 4 6 . 12957 × 10 - 4
0.50000 1 . 79201 × 10 - 3 9 . 66157 × 10 - 4 6 . 52190 × 10 - 4
0.85355 2 . 86231 × 10 - 3 1 . 54320 × 10 - 3 1 . 04172 × 10 - 4
1.00000 4 . 57606 × 10 - 3 2 . 46717 × 10 - 3 1 . 66540 × 10 - 3

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

R 2 F 7 B
t | e 3 ( t ) | | e 4 ( t ) | | e 5 ( t ) |
0.00000 1 . 35339 × 10 - 3 2 . 35630 × 10 - 4 1 . 91978 × 10 - 6
0.06699 1 . 35388 × 10 - 3 2 . 35636 × 10 - 4 1 . 91988 × 10 - 6
0.25000 1 . 35916 × 10 - 3 2 . 36553 × 10 - 4 1 . 92729 × 10 - 6
0.50000 1 . 44117 × 10 - 3 2 . 50828 × 10 - 4 2 . 04359 × 10 - 6
0.75000 1 . 85770 × 10 - 3 3 . 23330 × 10 - 4 2 . 63429 × 10 - 6
0.93301 2 . 88530 × 10 - 3 5 . 02733 × 10 - 4 4 . 09596 × 10 - 6
1.00000 3.00680 × 10 - 3 6.40511 × 10 - 4 5 . 21850 × 10 - 6

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

t    | e 3 ( t ) | | e 4 ( t ) | | e 5 ( t ) |
0.00000 1 . 1673 × 10 - 3 3 . 2941 × 10 - 4 3 . 0618 × 10 - 6
0.25000 1 . 1718 × 10 - 3 3 . 3070 × 10 - 4 3 . 0738 × 10 - 6
0.50000 1 . 2426 × 10 - 3 3 . 5066 × 10 - 4 3 . 2593 × 10 - 6
0.75000 1 . 6017 × 10 - 3 4 . 5201 × 10 - 4 4 . 2014 × 10 - 6
1.00000 3 . 1730 × 10 - 3 8 . 9543 × 10 - 4 8 . 3229 × 10 - 6
Example 2.

Consider the nonlinear integral equation: (31)20x(t)=20t+cos(e+t)-cos(1+t)+01exp(x(s))sin(t+es)ds. Here, λ=20, y(t)=20t+cos(e+t)-cos(1+t), k(t,s)=sin(t+es), and u(x(t))=exp(x(s)). Equation (31) has an exact solution xe(t)=t, .

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 .

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

R 2 F 3
t | e 3 ( t ) | | e 4 ( t ) | | e 5 ( t ) |
0.14645 3 . 02305    ×    10 - 4    3 . 24491    ×    10 - 4    3 . 26736    ×    10 - 4
0.50000 1 . 64103    ×    10 - 4    1 . 82928    ×    10 - 4    1 . 84830    ×    10 - 4
0.85355 5 . 60069 × 10 - 6    1 . 87365 × 10 - 5    2 . 00742 × 10 - 5

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

R 2 F 5 S
t | e 3 ( t ) | | e 4 ( t ) | | e 5 ( t ) |
0.00000 3 . 48603 × 10 - 5 1 . 19222 × 10 - 5 9 . 59700 × 10 - 6
0.14645 3 . 69623 × 10 - 5 1 . 46203 × 10 - 5 1 . 23539 × 10 - 5
0.50000 3 . 87246 × 10 - 5 1 . 97467 × 10 - 5 1 . 78171 × 10 - 5
0.85355 3 . 56966 × 10 - 5 2 . 24303 × 10 - 5 2 . 16749 × 10 - 5
1.00000 3 . 31078 × 10 - 5 2 . 27365 × 10 - 5 2 . 16749 × 10 - 5

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

R 2 F 7 B
t | e 3 ( t ) | | e 4 ( t ) | | e 5 ( t ) |
0.00000 2 . 56882 × 10 - 5 4 . 05683 × 10 - 7 1 . 23148 × 10 - 7
0.06699 2 . 54489 × 10 - 5 3 . 99928 × 10 - 7 1 . 19841 × 10 - 7
0.25000 2 . 42181 × 10 - 5 3 . 75164 × 10 - 7 1 . 08116 × 10 - 7
0.50000 2 . 12422 × 10 - 5 3 . 21320 × 10 - 7 8 . 63619 × 10 - 8
0.75000 1 . 69456 × 10 - 5 2 . 47497 × 10 - 7 5 . 92382 × 10 - 8
0.93301 1 . 31145 × 10 - 5 1 . 83399 × 10 - 7 3 . 69417 × 10 - 8
1.00000 1 . 15954 × 10 - 5 1 . 58286 × 10 - 7 2 . 84312 × 10 - 8

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

t    | e 3 ( t ) | | e 4 ( t ) | | e 5 ( t ) |
0.00000 3.4816 × 10 - 3 2.7837 × 10 - 4 5.1932 × 10 - 5
0.25000 3.2659 × 10 - 3 2.6006 × 10 - 4 5.3149 × 10 - 5
0.50000 2.8471 × 10 - 3 2.2559 × 10 - 4 5.1061 × 10 - 5
0.75000 2.2513 × 10 - 3 1.7709 × 10 - 4 4.5798 × 10 - 5
1.00000 1.5155 × 10 - 3 1.1758 × 10 - 4 3.7688 × 10 - 5
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  (see Tables 4 and 8), and we found that our results are more accurate than results obtained in .

Appell J. Chen C.-J. How to solve Hammerstein equations Journal of Integral Equations and Applications 2006 18 3 287 296 10.1216/jiea/1181075392 MR2269724 ZBL1156.45004 Deimling K. Nonlinear Functional Analysis 1985 Berlin, Germany Springer MR787404 El-Bary A. A. Sobolev's method for Hammerstein integral equations Mathematical & Computational Applications 2006 11 2 91 94 MR2200054 ZBL1127.65097 Liu X.-L. On a nonlinear Hammerstein integral equation with a parameter Nonlinear Analysis. Theory, Methods & Applications 2009 70 11 3887 3893 10.1016/j.na.2008.07.038 MR2515307 ZBL1211.45005 O'Regan D. Meehan M. Existence Theory For Nonlinear Integral and Integrodierential Equations 1998 Dordrecht, The Netherlands Kluwer Academic Publishers Arora H. L. Abdelwahid F. I. Solution of non-integer order differential equations via the Adomian decomposition method Applied Mathematics Letters 1993 6 1 21 23 10.1016/0893-9659(93)90140-I MR1347748 ZBL0772.34009 Hosseini M. M. Nasabzadeh H. Modified Adomian decomposition method for specific second order ordinary differential equations Applied Mathematics and Computation 2007 186 1 117 123 10.1016/j.amc.2006.07.094 MR2316497 ZBL1114.65078 Kaya D. The use of Adomian decomposition method for solving a specific nonlinear partial differential equations Bulletin of the Belgian Mathematical Society 2002 9 3 343 349 MR2016573 Shah S. Shaikh A. Sandilo S. H. Modied decomposition method for nonlinear Volterra-Fredholm integrodierential equation Journal of Basic & Applied Sciences 2010 6 1 13 16 Behiry S. H. Abd-Elmonem R. A. Gomaa A. M. Discrete Adomian decomposition solution of nonlinear Fredholm integral equation Ain Shams Engineering Journal 2010 1 1 97 101 Dash R. B. Das D. Identification of some Clenshaw-Curtis quadrature rules as mixed quadrature of Fejer and Newton-Cote type of rules International Journal of Mathematical Sciences & Applications 2011 1 3 1493 1496 MR2862483 ZBL1266.65044 Dash R. B. Das D. A mixed quadrature rule by blending Clenshaw-Curtis and Gauss-Legendre quadrature rules for approximation of real definite integrals in adaptive environment IMECS 2011 2011 202 205 Mason J. C. Handscomb D. C. Chebyshev Polynomials 2003 New York, NY, USA A CRC Press MR1937591 Clenshaw C. W. Curtis A. R. A method for numerical integration on an automatic computer Numerische Mathematik 1960 2 197 205 MR0117885 10.1007/BF01386223 ZBL0093.14006 Waldvogel J. Fast construction of the Fejér and Clenshaw-Curtis quadrature rules Numerical Mathematics 2006 46 1 195 202 10.1007/s10543-006-0045-4 MR2214855 ZBL1091.65028 Adomian G. Solving Frontier Problems of Physics: The Decomposition Method 1994 60 Dordrecht, The Netherlands Kluwer Academic Publishers Group Fundamental Theories of Physics With a preface by Yves Cherruault MR1282283 Wazwaz A.-M. A First Course in Integral Equations 1997 Singapore World Scientific Publishing MR1612107 Jerri A. J. Introduction to Integral Equations with Applications 1999 2nd John Wiley & Sons MR1800272