An improvement to homotopy perturbation method for solving system of linear equations

Computers and Mathematics with Applications 58 (2009) 2231–2235

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An improvement to homotopy perturbation method for solving system of linear equations Elçin Yusufoğlu ∗ Dumlupinar University, Art-Science Faculty, Department of Mathematics, 43100, Kutahya, Turkey

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abstract

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Keywords: Homotopy perturbation method Embedding parameter Accelerating vector System of linear equations Exact solution

In this paper, we present an efficient numerical algorithm to find exact solutions for the system of linear equations based on homotopy perturbation method (HPM). A reliable modification is proposed, and the modified method is employed to solve the system of linear equations; the results are compared with those obtained by the original homotopy perturbation method. Two examples are given to illustrate the ability and reliability of the modified method. The results reveal that the modified method is very simple and effective. © 2009 Elsevier Ltd. All rights reserved.

1. Introduction Many asymptotic techniques including homotopy perturbation method [1–3], energy method [4] (D’Acunto called this method as He’s variational method [5]), modifications of Lindstedt–Poincare method [6,7], bookkeeping parameter method [8], parametrized perturbation method [9] (Cai et al. called this method as He’s perturbation method [10]), iteration perturbation method [11], and Exp-method [12] were suggested by Prof. He during 1999–2008. For a relatively comprehensive survey on the method and its applications, the reader is referred to Prof. He’s review article [13]. The HPM, proposed first by He [1–3], was further developed and improved by scientists and engineers [13–15]. The method, which is a coupling of the traditional perturbation method and homotopy in topology, deforms continuously to a simple problem which is easily solved. This method, which does not require a small parameter in an equation, has a significant advantage in that it provides an analytical approximate solution to a wide range of nonlinear problems in applied sciences. The HPM is applied to nonlinear oscillators [16], to bifurcation of nonlinear problems [17], to the system of linear equations [18], to nonlinear partial differential equations of fractional order [19], to boundary value problems [20], to solve integral [21] and functional integral [22] equations, to solve the system of integral equations [23,24], and to other fields [25–41]. In this work, we present an improvement of the HPM to find exact solutions for the system of linear equations. An accelerating vector based on the HPM is introduced which leads to a rapid convergence and gives exact solutions. Two examples are given to illustrate the efficiency of the modified method. 2. HPM Consider the system of linear equations [18] Ax = b,

(1)

where A = aij ,

 

∗

x = xj ,

 

b = [bi ] ,

i = 1, 2, . . . , n, j = 1, 2, . . . , n.

Tel.: +90 274 2652031; fax: +90 274 2652056. E-mail address: [email protected].

0898-1221/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.camwa.2009.03.010

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To explain the HPM, we reconstitute (1) as L(u) = Au − b = 0,

(2)

with solution u = x, and we define the homotopy H (u, p) by H (u, 0) = F (u),

H (u, 1) = L(u),

(3)

where F (u) is a functional operator with solution, say, u0 , which can be obtained easily. We may choose a convex homotopy H (u, p) = (1 − p)F (u) + pL(u) = 0,

(4)

and continuously trace an implicitly defined curve from a starting point H (u0 , 0) to a solution H (x, 1). The embedding parameter p monotonically increases from zero to one as the trivial problem F (u) = 0 is continuously deformed to the original problem L(u) = 0. The embedding parameter p ∈ [0, 1] can be considered as an expanding parameter [1–3] u = u0 + pu1 + p2 u2 + · · ·

(5)

when p → 1, Eq. (4) corresponds to Eq. (2) and Eq. (5) becomes the approximate solution of Eq. (2), i.e., x = lim u0 + pu1 + p2 u2 + · · · =




p→1

∞ X

uk .

(6)

k=0

Taking F (u) = u − w0 , we substitute (5) into (4) and equate the terms with identical powers of p, obtaining u0 − w0 = 0,

p0 :

u0 = w0 ,

(7)

1

(A − I) u0 + u1 − w0 − b = 0,

2

(A − I) u1 + u2 = 0,

p : p :

u1 = b − (A − I) u0 + w0 ,

u2 = − (A − I) u1 ,

(8) (9)

and in general un+1 = − (A − I) un ,

n = 1, 2, . . . ,

(10)

if we take u0 = w0 = 0, then we have u1 = b,

(11)

u2 = − (A − I) u1 = − (A − I) b,

(12)

u3 = (A − I) b,

(13)

2

.. . un+1 = (−1)n (A − I)n b,

(14)

hence, the solution can be of the form u = I − (A − I) + (A − I)2 − · · · b.





(15)

The convergence of the series (15) has been proved by Keramati in [18]. In practice, all terms of series (6) cannot be determined and so we use an approximation of the solution by the following truncated series: m−1

um =

X

uk .

(16)

k=0

3. An improvement to HPM In this section we propose a scheme to accelerate the rate of HPM applied to system of linear equations. We define new homotopy H (u, p, α) by H (u, 0, α) = F (u),

H (u, 1, α) = L(u),

(17)

and typically, a convex homotopy as follows H (u, p, α) = (1 − p) F (u) + pL(u) + p(1 − p)α = 0,

(18)

where α = [αi ] is called the accelerating vector, and for α = 0 we define H (u, p, 0) = H (u, p), which is the standard HPM. t

E. Yusufoğlu / Computers and Mathematics with Applications 58 (2009) 2231–2235

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By substituting Eq. (5) in Eq. (18) and equating the terms with identical powers of p, we obtain u0 − w0 = 0,

p0 :

u0 = w0 ,

(19)

1

(A − I) u0 + u1 − w0 − b + α = 0,

2

(A − I) u1 + u2 − α = 0,

3

(A − I) u2 + u3 = 0,

p : p : p :

u1 = b − (A − I) u0 + w0 − α,

u2 = − (A − I) u1 + α,

u3 = − (A − I) u2 ,

(20) (21) (22)

.. . (A − I) un + un+1 = 0,

pn+1 :

un+1 = − (A − I) un ,

(23)

if we take u0 = w0 = 0, then we have u1 = b − α,

(24)

u2 = − (A − I) u1 + α = − (A − I) (b − α) + α,

(25)

u3 = − (A − I) u2 ,

(26)

.. . un+1 = − (A − I) un .

(27)

We try to find the parameters α as such that u2 = 0.

(28)

Hence from (25) we should have

− (A − I) (b − α) + α = 0,

(29)

Aα = (A − I) b.

(30)

or

From Eq. (30) we conclude that


α = I − A−1 b.

(31)

Thus from (27), we have u3 = u4 = · · · = 0, and the exact solution will be obtained as u = u0 + u1 = 0 + u1 = u1 .

(32)

4. Numerical results Example 4.1. Solve the system Au = b, where 4 −1 0

" A=

1 6 1

# −1

2 , −3

7 9 , 5

" # b=

x y . z

" # u=

(33)

The true solution is u = [1, 2, −1]t . By HPM and using six terms, Keramati [18] obtained the approximate solution as ut ≈ 0.99565,



2.00984,

 −1.00087 .

According to H (u, p, m) method, by using (30) we obtain

"

4 −1 0

1 6 1

#" # " α1 3 2 α2 = −1 −3 α3 0

4 −1 0

1 6 1

#" # " # α1 25 2 α2 = 48 . −3 α3 −11

−1

1 5 1

−1

7 2 · 9 . −4 5

# " #

(34)

or

"

−1

(35)

The solution of this system gives as α = [6, 7, 6]t . Thus the solution of Eq. (33) becomes u = u1 = b − α = [7, 9, 5]t − [6, 7, 6]t = [1, 2, −1]t , which is the exact solution of Example 4.1.

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Example 4.2. Solve the system Au = b, where

" A=

0.5 0.1 0.1

0.5 0.3 0.1

0.2 0.1 , 0.3

#

1.2 0.5 , 0.5

" b=

#

x y . z

" # u=

(36)

The exact solution is u = [1, 1, 1]t . By HPM and using ten iterations, Keramati [18] obtains an approximation to the solution as ut ≈ [1.12159, 0.94454, 0.94454]. Now we will apply the H (u, p, α) method to the given system. From Eq. (30) we have

#" # " −0.5 α1 α2 = 0.1 α3 0.1

0.5 0.1 0.1

0.5 0.3 0.1

0.2 0.1 0.3

0.5 0.1 0.1

0.5 0.3 0.1

0.2 0.1 0.3

"

0.5 −0.7 0.1

0.2 1.2 0.1 · 0.5 . −0.7 0.5

# "

#

(37)

or

"

#" # " # α1 −0.25 α2 = −0.18 . α3 −0.18

(38)

The solution of this system can therefore be identified as α = [0.2, −0.5, −0.5]t , consequently u = u1 = b − α = [1.2, 0.5, 0.5]t − [0.2, −0.5, −0.5]t = [1, 1, 1]t , which is the exact solution of Example 4.2. 5. Conclusions In this work, we first proposed a modification to the HPM by introducing accelerating parameters to solve the system linear equations. Instead of constructing a homotopy, H (u, p), in this new modification, a new homotopy, H (u, p, α), was constructed where α = [αi ]t is called the accelerating vector. The accelerating vector leads to a fast convergent rate, since only one iteration leads to exact solutions. The examples analyzed illustrate the ability and reliability of the modified method presented in this paper and reveal that the presented improvement of HPM is a simple and very effective tool for calculating the exact solutions. One can also apply this new modification to other linear and nonlinear equations. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]

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