Numerical solution for special non-linear Fredholm integral equation by HPM

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Applied Mathematics and Computation 195 (2008) 681–687 www.elsevier.com/locate/amc

Numerical solution for special non-linear Fredholm integral equation by HPM J. Biazar a

a,*

, H. Ghazvini

a,b

Department of Mathematics, Faculty of Sciences, University of Guilan, P.O. Box 1914, P.C. 41938, Rasht, Iran b Department of Mathematics, School of Mathematical Sciences, Shahrood University of Technology, P.O. Box 316, P.C. 3619995161, Shahrood, Iran

Abstract The aim of this paper is to apply homotopy perturbation method (HPM) to solve a kind non-linear integral equation of Fredholm type. Two examples are presented to show the ability of the method. The results reveal that the method is very effective and simple.  2007 Elsevier Inc. All rights reserved. Keywords: Homotopy perturbation method; Non-linear Fredholm integral equation

1. Introduction Homotopy perturbation method introduced by He [1–4] has been used by many mathematicians and engineers to solve various functional equations. In this method the solution is considered as the sum of an infinite series which converges rapidly to the accurate solutions. Using homotopy technique in topology, a homotopy is constructed with an embedding parameter p 2 [0, 1] which is considered as a ‘‘small parameter’’. A considerable research works have been conducted recently in applying this method to a class of linear and non-linear equations. This method was further developed and improved by He and applied to non-linear oscillators with discontinuities [5], non-linear wave equations [6], boundary value problem [7], limit cycle and bifurcation of non-linear problems [8], and many other subjects [1–4]. It can by say that He’s homotopy perturbation method is a universal one, is able to solve various kinds of non-linear functional equations. For examples it was applied to non-linear Schro¨dinger equations [9], to non-linear equations arising in heat transfer [10], to the quadratic Ricatti differential equation [11], and to other equations [12–15]. In [7], a comparison of (HPM) and homotopy analysis method was made, revealing that the former is more powerful than latter. We extend the method to solve non-linear Fredholm integral equation of second kind. Furthermore we will show that considerably better approximations related to the accuracy, level would be obtained. To demonstrate the above idea, numerical examples are given. *

Corresponding author. E-mail addresses: [email protected], [email protected] (J. Biazar), [email protected] (H. Ghazvini).

0096-3003/$ - see front matter  2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.05.015

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2. Basic idea of He’s homotopy perturbation method [1] To illustrate the basic ideas of this method, we consider the following non-linear differential equation: AðuÞ  f ðrÞ ¼ 0;

r 2 X;

ð1Þ

with the boundary conditions   ou B u; ¼ 0; r 2 C; ð2Þ on where A is a general differential operator, B a boundary operator, f(r) a known analytical function and C is the boundary of the domain X. The operator A can be divided into two parts, which are L and N, where L is a linear, but N is non-linear. Eq. (1) can be, therefore, rewritten as follows: LðuÞ þ N ðuÞ  f ðrÞ ¼ 0: ð3Þ By the homotopy technique, we construct a homotopy U ðr; pÞ : X  ½0; 1 ! R, which satisfies: H ðU ; pÞ ¼ ð1  pÞ½LðU Þ  Lðu0 Þ þ p½AðU Þ  f ðrÞ ¼ 0;

p 2 ½0; 1; r 2 X;

ð4Þ

or H ðU ; pÞ ¼ LðU Þ  Lðu0 Þ þ PLðu0 Þ þ p½N ðU Þ  f ðrÞ ¼ 0; ð5Þ where p 2 [0, 1] is an embedding parameter, u0 is an initial approximation of Eq. (1), which satisfies the boundary conditions. Obviously, from Eqs. (4) and (5) we will have ð6Þ H ðU ; 0Þ ¼ LðU Þ  Lðu0 Þ ¼ 0; H ðU ; 1Þ ¼ AðU Þ  f ðrÞ ¼ 0: ð7Þ The changing process of p form zero to unity is just that of U(r, p) from u0(r) to u(r). In topology, this is called homotopy. According to the (HPM), we can first use the embedding parameter p as a small parameter, and assume that the solution of Eqs. (4) and (5) can be written as a power series in p: U ¼ U 0 þ pU 1 þ p2 U 2 þ   

ð8Þ

Setting p = 1, results in the approximate solution of Eq. (1) u ¼ lim U ¼ U 0 þ U 1 þ U 2 þ    p!1

ð9Þ

The combination of the perturbation method and the homotopy method is called the homotopy perturbation method (HPM), which has eliminated the limitations of the traditional perturbation methods. On the other hand, this technique can have full advantage of the traditional perturbation techniques. The series (9) is convergent for most cases. However the convergent rate date on non-linear operator A(U), the following opinions are suggested by He [1]: (1) The second derivative of N(U)with respect to U must be small because the parameter may be relatively large, i.e., p ! 1. (2) The norm of L1 oN must be smaller than one so that the series converges. oV 3. Method of solution In this section we present homotopy perturbation method for solving a non-linear integral equation of Fredholm type: Z b P /ðxÞ ¼ f ðxÞ þ l kðx; tÞð/ðtÞÞ dt; x 2 ½a; b; P 2 N ; P P 2; ð10Þ a

where l is a real number, the kernel k(x, t) is a continuous function in [a, b] · [a, b] and f(x) is a given continuous function defined in [a, b].

J. Biazar, H. Ghazvini / Applied Mathematics and Computation 195 (2008) 681–687

For solving Eq. (10), by homotopy perturbation method we construct a homotopy as follows: Z b kðx; tÞðUðtÞÞP dtÞ ¼ 0; ð1  pÞðUðxÞ  /0 ðxÞÞ þ pðUðxÞ  f ðxÞ  l

683

ð11Þ

a

Suppose the solution of Eq. (11) has the form UðxÞ ¼ U0 ðxÞ þ pU1 ðxÞ þ p2 U2 ðxÞ þ   

ð12Þ

where Ui ðxÞi ¼ 1; 2; . . . are functions yet to be determined. The initial approximation U0(x) or /0(x) can be freely chosen, here we set U0 ðxÞ ¼ /0 ðxÞ ¼ f ðxÞ:

ð13Þ

Substituting (13) into (12) and arranging the coefficients powers p0 : U0 ðxÞ  /0 ðxÞ ¼ 0; Z b P kðx; tÞðU0 ðtÞÞ dt ¼ 0; p1 : U1 ðxÞ þ /0 ðxÞ  f ðxÞ  l a 8 Z b > > kðx; tÞð2U0 U1 Þ dt ¼ 0 if P ¼ 2; > U2 ðxÞ  l > > a > > Z b > > > > < U2 ðxÞ  l kðx; tÞð3ðU0 Þ2 U1 Þ dt ¼ 0 if P ¼ 3; 2 a p : Z b > > > 3 > > U2 ðxÞ  l kðx; tÞð4ðU0 Þ U1 Þ dt ¼ 0 if P ¼ 4; > > > a > > > : .. . .. . 8 Z b j1 X > > > U ðxÞ  l kðx; tÞ ðUk Ujk1 Þ dt ¼ 0 if P ¼ 2; > j > > > a k¼0 > > > Z b > j1 ji1 > X X > > U ðxÞ  l < kðx; tÞ ðUi Uk Ujki1 Þ dt ¼ 0 if P ¼ 3; j a pj : i¼0 k¼0 > Z b > j1 ji1 > X jik1 X X > > > ðxÞ  l kðx; tÞ ðUi Uk Ul Ujlki1 Þ dt ¼ 0 if P ¼ 4; U > j > > a > i¼0 k¼0 l¼0 > > > > : .. . .. . The approximate solutions of (10), therefore, can be obtained by setting p = 1 1 X /ðxÞ ¼ lim UðxÞ ¼ Uj ðxÞ: p!1

j¼0

4. Numerical results In this part, two examples are provided to illustrate this method Example 1. Let us solve the following non-linear Fredholm integral equation: Z 1 1 3 /ðxÞ ¼ sinðpxÞ þ cosðpxÞ sinðptÞð/ðtÞÞ dt; x 2 ½0; 1: 5 0 pffiffiffiffiffi For which the exact solution is /ðxÞ ¼ sinðpxÞ þ 203 391 cosðpxÞ:

ð14Þ

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We construct a homotopy X  ½0; 1 ! R, which satisfies Z 1 1 3 ð1  pÞðUðxÞ  /0 ðxÞÞ þ pðsinðpxÞ  cosðpxÞ sinðptÞðUðtÞÞ dtÞ ¼ 0; 5 0

ð15Þ

where UðxÞ ¼ U0 ðxÞ þ pU1 ðxÞ þ p2 U2 ðxÞ þ   

ð16Þ

Starting with U0 ðxÞ ¼ /0 ðxÞ ¼ sinðpxÞ: Comparison of identical powers of p yields to the following scheme: Z j1 ji1 X X 1 1 Uj ðxÞ ¼ cosðpxÞ sinðptÞ ðUi ðtÞUk ðtÞUjki1 ðtÞÞ dt; j ¼ 1; 2; 3; . . . 5 0 i¼0 k¼0 Pi Suppose Ui ¼ j¼0 Ui , by using (17) we obtain the following results:

ð17Þ

U1 ¼ 0:075000000000000000000 cosðpxÞ þ sinðpxÞ; U2 ¼ 0:074998492035526276899 cosðpxÞ þ sinðpxÞ; U3 ¼ 0:075420693016447334813 cosðpxÞ þ sinðpxÞ; U4 ¼ 0:075420749589026786486 cosðpxÞ þ sinðpxÞ; U5 ¼ 0:075425499142616942979 cosðpxÞ þ sinðpxÞ; .. . Suppose /ðxÞ  U5 , some numerical results of these solutions are presented in Table 1 and Fig. 1. Example 2. Consider the following integral equation with the exact solution x ln(x + 1):   Z 53 1 8 241 1 1 x þ ln 2 x þ 2  x ln 2  þ /ðxÞ ¼ x ln x  ðx  tÞð/ðtÞÞ2 dt; x 2 ½0; 1: 108 3 3 576 2 0 Homotopy perturbation method consist of the following scheme:   53 1 8 x  ln 2 x þ 2  x ln 2 ð1  pÞðU0 ðxÞ  /0 ðxÞÞ þ pðUðxÞ  x ln x þ 108 3 3 Z 1 241 1 þ  ðx  tÞðUðtÞÞ2 dtÞ ¼ 0: 576 2 0

ð18Þ

Table 1 Numerical results of Example 1 xi

/exact

/homotopy

j/exact  /homotopyj

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.754266889049371628E1 0.38075203836055527390 0.648806725544599958698 0.85335168974252179804 0.97436464499621144860 1 0.92774838759409569563 0.76468229900737305013 0.52676377913894667133 0.23728195038933957433 0.754266889049371628E1

0.75425499142616942979E1 0.38075090682934778652 0.64880576290806326229 0.85335099041777623963 0.97436427733943523375 1 0.92774875525087191048 0.76468299833211860853 0.52676474167688299602 0.23728308192054706171 0.75425499142616943016E1

1.1897623202198E6 1.1315312074873E6 9.6253793632469E7 6.99324474555841E7 3.6765677621485E7 0 3.6765677621485E7 6.9932474555840E7 9.6253793632469E7 1.1315312074873E6 1.1897623202197E6

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Fig. 1. The plots of approximation and exact solutions of Example 1.

Suppose the solution of Eq. (18) has the form UðxÞ ¼ U0 ðxÞ þ pU1 ðxÞ þ p2 U2 ðxÞ þ   

ð19Þ

We take   53 1 8 241 x þ ln 2 x þ 2  x ln 2  ; U0 ðxÞ ¼ x ln x  108 3 3 576 substituting (19) into (18) and equating the terms with the identical powers of p, we will have Uj ðxÞ ¼ Let Ui ¼

Pi

1 2

Z

1

ðx  tÞ 0

j¼0 Ui ,

j1 X

ðUk ðtÞUjk1 ðtÞÞ dt;

j ¼ 1; 2; 3; . . .

ð20Þ

k¼0

by using (20), the approximation solution as follows:

Table 2 Numerical results of Example 2 xi

/exact

/homotopy

j/exact  /homotopyj

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.95310179804324860044E2 0.36464311358790925242E1 0.78709279340247315612E1 0.13458889464848517220 0.20273255405408219099 0.28200217754744133219 0.37143977574351927736 0.47022933192169520655 0.57766849755515529839 0.69314718055994530942

0.2667965855283617446E6 0.95307597846324244667E2 0.36464061763776330528E1 0.78709038346018187722E1 0.13458866225504151113 0.20273233026142399675 0.28200196235556860477 0.37143956915243201677 o.47022913393139341278 0.57766830816563897144 0.69314699977121444930

2.6679658552836E7 2.5819580006154E7 2.4959501459471E7 2.4099422912789E7 2.3239344366107E7 2.2379265819424E7 2.1519187272742E7 2.0659108726059E7 1.9799030179377E7 1.8938951632695E7 1.8078873086012E7

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Fig. 2. The plots of approximation and exact solutions of Example 2.

U1 ¼ 1:3670778747472905  103 x  3:4015961796285688  104 þ x lnðx þ 1Þ; U2 ¼ 9:69991132410618  105 x  8:08420640778075055  105 þ x lnðx þ 1Þ; U3 ¼ 3:95147178266119  105 x þ 2:9702104910894865  105 þ x lnðx þ 1Þ; U4 ¼ 2:0702585390001  106 x  8:2836727360052977600  108 þ x lnðx þ 1Þ; U5 ¼ 8:600785466824  108 x  2:667965855283617446  108 þ x lnðx þ 1Þ; .. . Suppose /ðxÞ  U5 the results and the exact solution are presented in Table 2 and Fig. 2. 5. Conclusion In this article, He’s homotopy perturbation method has been successfully applied to find the solution of non-linear Fredholm integral equations of the second kind are presented in Tables 1 and 2, for differential results of x to show the stability of the method. The approximate solutions obtained by the homotopy perturbation method are compared with exact solutions. It can be concluded that the He’s homotopy perturbation method is very powerful and efficient technique in finding exact solutions for wide classes of problems. In our work, we use the MAPLE 10 package to carry the computations. References [1] J.H. He, The homotopy perturbation method for nonlinear oscillators with discontinuities, Applied Mathematics and Computation 151 (2004) 287–292. [2] J.H. He, Application of homotopy perturbation method to nonlinear wave equations, Chaos, Solitons and Fractals 26 (2005) 695– 700. [3] J.H. He, Homotopy perturbation method for solving boundary value problems, Physics Letters A 350 (2006) 87–88. [4] J.H. He, Limit cycle and bifurcation of nonlinear problems, Chaos, Solitons and Fractals 26 (3) (2005) 827–833. [5] J.H. He, Homotopy perturbation technique, Computer Methods in Applied Mechanics and Engineering 178 (1999) 257–262. [6] J.H. He, A coupling method of homotopy technique and perturbation technique for nonlinear problems, International Journal of Non-Linear Mechanics 35 (1) (2000) 37–43.

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[7] J.H. He, Comparison of homotopy perturbation method and homotopy analysis method, Applied Mathematics and Computation 156 (2004) 527–539. [8] J.H. He, Homotopy perturbation method: a new nonlinear analytical technique, Applied Mathematics and Computation 135 (2003) 73–79. [9] J. Biazar, H. Ghazvini, Exact solutions for nonlinear Schro¨dinger equations by He’s homotopy perturbation method, Physics Letters A 366 (2007) 79–84. [10] D.D. Ganji, The application of He’s homotopy perturbation method to nonlinear equations arising in heat transfer, Physics Letters A 355 (2006) 337–341. [11] S. Abbasbandy, Numerical solutions of the integral equations: Homotopy perturbation method and Adomian’s decomposition method, Applied Mathematics and Computation 173 (2006) 493–500. [12] Z. Odibat, S. Momani, Modified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order, Chaos, Solitons and Fractals, in press. [13] A.M. Siddiqui, R. Mahmood, Q.K. Ghori, Homotopy perturbation method for thin film flow of a third grade fluid down an inclined plane, Chaos, Solitons and Fractals, in press. [14] L. Cveticanin, Homotopy–perturbation method for pure nonlinear differential equation, Chaos, Solitons and Fractals 30 (2006) 1221– 1230. [15] J. Biazar, H. ghazvini, He’s homotopy perturbation method for solving system of Volterra integral equations of the second kind, Chaos, Solitons and Fractals, in press.