Exact Solutions for Non-linear Volterra-Fredholm Integro-Differential ...

ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.9(2010) No.3,pp.285-289

Exact Solutions for Non-linear Volterra-Fredholm Integro-Differential Equations by He’s Homotopy Perturbation Method J. Biazar ∗ , M. Eslami Department of Mathematics, Faculty of Sciences, University of Guilan P.O. Box 413351914 P.C. 4193833697, Rasht, Iran (Received 21 September 2009, accepted 10 December 2009)

Abstract: In this article, an application of He’s homotopy perturbation method is applied to solve non-linear Volterra-Fredholm integro-differential equations. Some non-linear examples are prepared to illustrate the efficiency and simplicity of the method. Keywords: homotopy perturbation method; non-linear Volterra-Fredholm integro-differential equations

1

Introduction

Homotopy perturbation method has been used by many mathematicians and engineers to solve various functional equations. This method continuously deforms difficult problem, mostly because of non-linearly, into a simple, linear, equation [1-5]. Almost all perturbation methods are based on the assumption of the existence of a small parameter in the equation. But most non-linear problems do not have such a small parameter. This method has been proposed to eliminate the small parameter [6, 7]. In recent years, the application of homotopy perturbation theory has appeared in many researches [1014]. In this paper, we propose homotopy perturbation method to solve non-linear Volterra-Fredholm integro-differential equations. Consider the following equation [8] 𝑚 ∑

𝑝𝑗 (𝑥)𝑦 𝑗 (𝑥) =𝑓 (𝑥) + 𝜆1

𝑗=0

∫

𝑥

𝑎

∫

𝑝

𝑘1 (𝑥, 𝑡) [𝑦(𝑡)] 𝑑𝑡 + 𝜆2

𝑏 𝑎

𝑞

𝑘2 (𝑥, 𝑡) [𝑦(𝑡)] 𝑑𝑡,

(1)

where 𝑝𝑗 (𝑥)(𝑗 = 0 . . . 𝑚), 𝑓 (𝑥), 𝑘1 (𝑥, 𝑡), 𝑘2 (𝑥, 𝑡) are function having 𝑛 th (𝑛 ⩾ 𝑚) derivatives on an integral 𝑎 ⩽ 𝑥, 𝑡 ⩽ 𝑏, and 𝑎, 𝑏, 𝜆1 and 𝜆2 are constants, 𝑝 and 𝑞 are positive integer.

2

Homotopy perturbation method

To illustrate the homotopy perturbation method, we consider (1) as 𝐿(𝑦) =

𝑚 ∑ 𝑗=0

𝑝𝑗 (𝑥)𝑦 𝑗 (𝑥)−𝑓 (𝑥) − 𝜆1

∫

𝑥 𝑎

𝑝

∫

𝑘1 (𝑥, 𝑡) [𝑦(𝑡)] 𝑑𝑡 − 𝜆2

𝑏 𝑎

𝑞

𝑘2 (𝑥, 𝑡) [𝑦(𝑡)] 𝑑𝑡,

(2)

with boundary conditions, 𝐵(𝑦, ∂𝑦/∂𝑛) = 0, and exact solution 𝑦(𝑥) = 𝑔(𝑥). By the homotopy technique, we can define homotopy 𝐻(𝑦, 𝑝) by 𝐻(𝑦, 𝑝) = (1 − 𝑝)𝐹 (𝑦) + 𝑝𝐿(𝑦),

(3)

where 𝐹 (𝑦) is a functional operator with known solution 𝑦0 which generally satisfies the boundary conditions. Obviously, from Eq. (3) we have 𝐻(𝑦, 0) = 𝐹 (𝑦), 𝐻(𝑦, 1) = 𝐿(𝑦), ∗ Corresponding

author.

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International Journal of NonlinearScience,Vol.9(2010),No.3,pp. 285-289

and the changing process of the parameter 𝑝 from 0 to 1, is just that of 𝐻(𝑦, 𝑝) from 𝐻(𝑦, 0) to 𝐻(𝑦, 1). In topology, it is called deformation, 𝐻(𝑦, 0) and 𝐻(𝑦, 1) are called homotopic. Applying the perturbation technique [10], due to the fact that 0 ⩽ 𝑝 ⩽ 1, can be considered as a small parameter. We can assume that the solution of (3) can be expressed as a series in as follows: 𝑦 = 𝑦0 + 𝑝𝑦1 + 𝑝2 𝑦2 + ⋅ ⋅ ⋅ . (4) As 𝑝 → 1, (3) tends to equation (2) and (4), in most cases it converges to an approximate solution of (2), i.e., 𝑔 = lim 𝑦 = 𝑦0 + 𝑦1 + 𝑦2 + ⋅ ⋅ ⋅ . 𝑝→1

3

Numerical Examples

In this part, three examples are provided. These examples are considered to illustrate the ability and reliability of the method. These examples are solved numerically in [15, 16]. Example1. Let’s solve the following non-linear Volterra-Fredholm integro-differential equation [15] ∫ 𝑥 ∫ 1 3 𝑦 ′ (𝑥) + 2𝑥𝑦(𝑥) = 𝑓 (𝑥) + (𝑥 + 𝑡)[𝑦(𝑡)] 𝑑𝑡 + (𝑥 − 𝑡)𝑦(𝑡)𝑑𝑡, 0

0

where 𝑓 (𝑥) = (− 23 𝑥 + 19 )𝑒3𝑥 + (2𝑥 + 1)𝑒𝑥 + ( 43 − 𝑒)𝑥 + 89 , with condition 𝑦(0) = 1 and exact solution 𝑦(𝑥) = 𝑒𝑥 . We construct a homotopy Ω × [0, 1] → ℝ which satisfies 𝑦 ′ (𝑥) + (1 − 𝑝)2𝑥𝑒𝑥 + 𝑝2𝑥𝑦(𝑥)− ∫ 1 ∫ 3 3 𝑓 (𝑥) − (1 − 𝑝)(𝑥 + 𝑡)[𝑒𝑡 ] + 𝑝(𝑥 + 𝑡)[𝑦(𝑡)] 𝑑𝑡 − 0

𝑥

0

(1 − 𝑝)(𝑥 − 𝑡)𝑒𝑡 + 𝑝(𝑥 − 𝑡)𝑦(𝑡)𝑑𝑡 = 0.

Substituting (4) in to (5), and equating the coefficients of the terms with the identical powers of 𝑝, we have: ∫ 1 3 (𝑥 + 𝑡)[𝑒𝑡 ] 𝑑𝑡 + (𝑥 − 𝑡)𝑒𝑡 𝑑𝑡 = 0 ⇒ 𝑦0 (𝑥) = 𝑒𝑥 . 0 0 ∫ 𝑥 3 3 1 ′ 𝑥 𝑝 : 𝑦 1 (𝑥) − 2𝑥𝑒 + 2𝑥𝑦0 (𝑥) = −(𝑥 + 𝑡)[𝑒𝑡 ] + (𝑥 + 𝑡)[𝑦0 (𝑡)] 𝑑𝑡

𝑝0 : 𝑦 ′ 0 (𝑥) + 2𝑥𝑒𝑥 = 𝑓 (𝑥) +

∫ +

0

1

∫

𝑥

0

−(𝑥 − 𝑡)𝑒𝑡 + (𝑥 − 𝑡)𝑦0 (𝑡)𝑑𝑡 = 0 ⇒ 𝑦1 (𝑥) = 0.

𝑝2 : 𝑦 ′ 2 (𝑥) + 2𝑥𝑦1 (𝑥) = 𝑝3 : 𝑦 ′ 3 (𝑥) + 2𝑥𝑦2 (𝑥) =

∫ ∫

𝑥 0

𝑥

0

3

∫

(𝑥 + 𝑡)[𝑦1 (𝑡)] 𝑑𝑡 + 3

(𝑥 + 𝑡)[𝑦2 (𝑡)] 𝑑𝑡 +

∫

1 0

0

1

(𝑥 − 𝑡)𝑦1 (𝑡)𝑑𝑡 = 0 ⇒ 𝑦2 (𝑥) = 0. (𝑥 − 𝑡)𝑦2 (𝑡)𝑑𝑡 = 0 ⇒ 𝑦3 (𝑥) = 0.

And by repeating this approach, we obtain 𝑦4 (𝑥) = 𝑦5 (𝑥) = ... = 0. Therefore, the approximation to the solution of Example 1 can be readily obtained by 𝑦=

∞ ∑

𝑦𝑖 = 𝑒𝑥 + 0 + 0 + ⋅ ⋅ ⋅ .

𝑖=0

And hence,

𝑦(𝑥) = 𝑒𝑥 ,

which is an exact solution of Example 1. Example 2. Consider the following Volterra-Fredholm integro-differential equation [15] ∫ 𝑥 ∫ 1 2 𝑦 ′′ (𝑥) − 𝑥𝑦 ′ (𝑥) + 𝑥𝑦(𝑥) = 𝑓 (𝑥) + (𝑥 − 2𝑡)[𝑦(𝑡)] 𝑑𝑡 + 𝑥𝑡𝑦(𝑡)𝑑𝑡, −1

−1

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(5)

J. Biazar and M. Eslami: Exact Solutions for Non-linear Volterra-Fredholm Integro-Differential Equations ⋅ ⋅ ⋅

287

2 6 23 where 𝑓 (𝑥) = 25 𝑥 − 13 𝑥4 + 𝑥3 − 2𝑥2 − 15 𝑥 + 53 , with condition 𝑦(0) = −1, 𝑦 ′ (0) = 0 and exact solution 𝑦(𝑥) = 𝑥2 − 1. By using the He’s homotopy perturbation method, we have

𝑦 ′′ (𝑥) − (1 − 𝑝)𝑥(2𝑥) − 𝑝𝑥𝑦 ′ (𝑥) + (1 − 𝑝)𝑥(𝑥2 − 1) + 𝑝𝑥𝑦(𝑥) − 𝑓 (𝑥) ∫ 𝑥 ∫ 1 2 2 − (1 − 𝑝)(𝑥 − 2𝑡)[𝑥2 − 1] + 𝑝(𝑥 − 2𝑡)[𝑦(𝑡)] 𝑑𝑡 + (1 − 𝑝)𝑥𝑡(𝑥2 − 1) + 𝑝𝑥𝑡𝑦(𝑡)𝑑𝑡 = 0. −1

(6)

−1

Substituting (4) in to (6), and equating the∫coefficients of the terms with the identical powers of 𝑝, we have:

𝑝0 : 𝑦 ′′ 0 (𝑥) − 𝑥(2𝑥) + 𝑥(𝑥2 − 1) − 𝑓 (𝑥) − ∫ +

1

−1

𝑥

−1

2

(𝑥 − 2𝑡)[𝑥2 − 1] 𝑑𝑡

𝑥𝑡(𝑥2 − 1)𝑑𝑡 = 0, ⇒ 𝑦0 (𝑥) = 𝑥2 − 1,

𝑝1 : 𝑦 ′′ 1 (𝑥) + 𝑥(2𝑥) − 𝑥𝑦 ′ 0 (𝑥) − 𝑥(𝑥2 − 1) + 𝑥𝑦0 (𝑥) ∫ 𝑥 ∫ 2 2 − −(𝑥 − 2𝑡)[𝑥2 − 1] + (𝑥 − 2𝑡)[𝑦0 (𝑡)] 𝑑𝑡 + −1

2

′′

∫

′

𝑝 : 𝑦 2 (𝑥) − 𝑥𝑦 1 (𝑥) + 𝑥𝑦1 (𝑥) − 𝑝3 : 𝑦 ′′ 3 (𝑥) − 𝑥𝑦 ′ 2 (𝑥) + 𝑥𝑦2 (𝑥) −

𝑥

−1 ∫ 𝑥 −1

1 −1

−𝑥𝑡(𝑥2 − 1) + 𝑥𝑡𝑦0 (𝑡)𝑑𝑡 = 0, ⇒ 𝑦1 (𝑥) = 0, ∫

2

(𝑥 − 2𝑡)[𝑦1 (𝑡)] 𝑑𝑡 + 2

(𝑥 − 2𝑡)[𝑦2 (𝑡)] 𝑑𝑡 +

1

−1 ∫ 1 −1

𝑥𝑡𝑦1 (𝑡)𝑑𝑡 = 0 ⇒ 𝑦2 (𝑥) = 0, 𝑥𝑡𝑦2 (𝑡)𝑑𝑡 = 0 ⇒ 𝑦3 (𝑥) = 0.

And by repeating this approach, we obtain 𝑦4 (𝑥) = 𝑦5 (𝑥) = ... = 0. Therefore the approximation to the solution of Example 2 can be readily obtained by 𝑦=

∞ ∑

𝑦𝑖 = 𝑥2 − 1 + 0 + 0 + ⋅ ⋅ ⋅ .

𝑖=0

And hence, 𝑦(𝑥) = 𝑥2 − 1, which is an exact solution of Example 2. Example 3. Consider the following non-linear Volterra-Fredholm integro-differential equation [16] ∫

′′′

𝑦 (𝑥) + 𝑦(𝑥) = 𝑓 (𝑥) +

𝑥 0

∫

2

𝑦 (𝑡)𝑑𝑡 +

0

1

(𝑥2 𝑡 + 𝑥𝑡2 )𝑑𝑡,

′ ′′ where 𝑓 (𝑥) = − 15 𝑥5 + 23 𝑥3 + 56 𝑥2 − 113 105 𝑥 − 1, with condition 𝑦(0) = 1, 𝑦 (0) = 0, 𝑦 (0) = −2, and exact solution 2 𝑦(𝑥) = 1 − 𝑥 He’s homotopy perturbation method consists of the following scheme

′′′

∫

2

𝑦 (𝑥) + (1 − 𝑝)(1 − 𝑥 ) + 𝑝𝑦(𝑥) − 𝑓 (𝑥) − ∫ −

0

1

2

2

2 2

2

0

𝑥

2

2

(1 − 𝑝)[1 − 𝑥2 ] + 𝑝[𝑦(𝑡)] 𝑑𝑡 (7)

2

2

(1 − 𝑝)(𝑥 𝑡 + 𝑥𝑡 )[1 − 𝑥 ] + 𝑝(𝑥 𝑡 + 𝑥𝑡 )[𝑦(𝑡)] 𝑑𝑡 = 0.

Substituting (4) in to (7), and equating the coefficients of the terms with the identical powers of 𝑝, we have:

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288

International Journal of NonlinearScience,Vol.9(2010),No.3,pp. 285-289

𝑝0 : 𝑦 ′′′ 0 (𝑥) + (1 − 𝑥2 ) − 𝑓 (𝑥) − ∫

1

∫

1

− 2

𝑝 :𝑦

0 ′′′

2

𝑥 0

2

[1 − 𝑥2 ] 𝑑𝑡

(𝑥 𝑡 + 𝑥𝑡 )[1 − 𝑥 ] 𝑑𝑡 = 0 ⇒ 𝑦0 (𝑥) = 1 − 𝑥2 , 0 ∫ 𝑥 2 2 1 ′′′ 𝑝 : 𝑦 1 (𝑥) − (1 − 𝑥2 ) + 𝑦0 (𝑥) − −[1 − 𝑥2 ] + [𝑦(𝑡)] 𝑑𝑡 −

2

∫

2 2

0

−(𝑥 𝑡 + 𝑥𝑡 )[1 − 𝑥 ] + (𝑥2 𝑡 + 𝑥𝑡2 )[𝑦0 (𝑡)]2 𝑑𝑡 = 0 ⇒ 𝑦1 (𝑥) = 0,

2 (𝑥)

2

2

2 2

∫ + 𝑦1 (𝑥) −

𝑝3 : 𝑦 ′′′ 3 (𝑥) + 𝑦2 (𝑥) −

∫

𝑥 0

0

𝑥

∫

2

𝑝[𝑦1 (𝑡)] 𝑑𝑡 −

0

∫

2

𝑝[𝑦2 (𝑡)] 𝑑𝑡 −

1

1

0

2

(𝑥2 𝑡 + 𝑥𝑡2 )[𝑦1 (𝑡)] 𝑑𝑡 = 0 ⇒ 𝑦2 (𝑥) = 0, 2

(𝑥2 𝑡 + 𝑥𝑡2 )[𝑦2 (𝑡)] 𝑑𝑡 = 0 ⇒ 𝑦3 (𝑥) = 0.

And by repeating this approach, we obtain 𝑦4 (𝑥) = 𝑦5 (𝑥) = ... = 0. Therefore the approximation to the solution of Example 3 can be readily obtained by 𝑦=

∞ ∑

𝑦𝑖 = 1 − 𝑥2 + 0 + 0 + ⋅ ⋅ ⋅ .

𝑖=0 2

And hence, 𝑦(𝑥) = 1 − 𝑥 , which is an exact solution of Example 3.

4

Conclusion

In this work, we used homotopy perturbation method for solving non-linear Volterra-Fredholm integro-differential equations. The results have been approved the efficiency of this method for solving these problems. The solution obtained by homotopy perturbation method is valid for not only weakly non-linear equations but also for strong ones. Furthermore, accurate solutions were derived from first-order approximations in the examples presented in this paper.

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