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Application of homotopy perturbation method for systems of Volterra integral equations of the first kind
Chaos, Solitons and Fractals 42 (2009) 3020–3026
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Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos
Application of homotopy perturbation method for systems of Volterra integral equations of the first kind J. Biazar a,*, M. Eslami a, H. Aminikhah b a b
Department of Mathematics, Faculty of Sciences, The University of Guilan, P.O. Box 1914, P.C. 41938 Rasht, Iran Department of Mathematics, School of Mathematical Sciences, Shahrood University of Technology, P.O. Box 316, P.C. 3619995161 Shahrood, Iran
a r t i c l e
i n f o
a b s t r a c t
Article history: Accepted 3 April 2009
In this article, an application of He’s homotopy perturbation method is applied to solve systems of Volterra integral equations of the first kind. Some non-linear examples are prepared to illustrate the efficiency and simplicity of the method. Applying the method for linear systems is so easily that it does not worth to have any example. Ó 2009 Elsevier Ltd. All rights reserved.
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 have no such a small parameter. This method has been proposed to eliminate the small parameter [7,8]. In recent years the application of homotopy perturbation theory has appeared in many researches [9–17]. In this paper we propose homotopy perturbation method to solve systems of Volterra integral equations of the first kind. 2. The homotopy perturbation method applied to systems of Volterra integral equations of the first kind A system of integral equations of the first kind can be presented as the following:
Z
x
ki ðx; tÞg i ðu1 ðtÞ; u2 ðtÞ; . . . ; un ðtÞÞdt ¼ fi ðxÞ;
i ¼ 1; 2; . . . ; n:
ð1Þ
0
If gi(u1, . . . ,un) are linear, the system (1) can be represented as the following:
Z
x
0
n X
ki;j ðx; tÞuj ðtÞdt ¼ fi ðxÞ;
i ¼ 1; 2; . . . ; n:
j¼1
After differentiation we derive: n X j¼1
ki;j ðx; xÞuj ðxÞ þ
Z 0
x
n X @ki;j ðx; tÞ uj ðtÞdt ¼ fi0 ðxÞ: @x j¼1
ð2Þ
* Corresponding author. E-mail addresses: [email protected], [email protected] (J. Biazar), [email protected] (M. Eslami), [email protected], aminikhah@ shahroodut.ac.ir (H. Aminikhah). 0960-0779/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2009.04.016
J. Biazar et al. / Chaos, Solitons and Fractals 42 (2009) 3020–3026
3021
From ith equation, in (2), we write the ith unknown, ui, in terms of the other unknown, to derive the following linear system of Volterra integral equations of the second kind
ui ðxÞ ¼
n X f 0 ðxÞ ki;j ðx; tÞ uj ðxÞ ki;i ðx; xÞ j¼1 ki;i ðx; xÞ
Z
x
0
n X @ki;j ðx; tÞ=@x uj ðtÞdt: ki;i ðx; xÞ j¼1
ð3Þ
j–i
To illustrate the homotopy perturbation method, we consider (3) as
F i ðui Þ ¼ ui ðxÞ
n X ki;j ðx; tÞ f 0 ðxÞ þ uj ðxÞ þ ki;i ðx; xÞ j¼1 ki;i ðx; xÞ
Z 0
x
n X @ki;j ðx; tÞ=@x uj ðtÞdt ¼ 0: ki;i ðx; xÞ j¼1
ð4Þ
j–i
With the solution ui(x). We can define homotopy H(ui,p) by
Hðui ; pÞ ¼ ð1 pÞLi ðui Þ þ pF i ðui Þ ¼ 0;
ð5Þ
where Li(ui) is a functional operator with known solution ui 0, which can be obtained easily. Obviously, from Eq. (5) we have:
Hðui ; 0Þ ¼ Li ðui Þ;
Hðui ; 1Þ ¼ F i ðui Þ:
ð6Þ
The changing process of p from zero to unity is just that of H(ui,p) from a starting point H(ui,0) to a solution function H(ui,1). Applying the perturbation technique [6], due to the fact that, 0 6 p 6 1, being considered as a small parameter. We can assume that the solution of (5) can be expressed as a series in p, as follows:
ui ¼ ui;0 þ pui;1 þ p2 ui;2 þ
ð7Þ
Substituting (7) into (5) and equating the coefficients of p with the same power leads to f 0 ðxÞ
p0 : ui;0 ðtÞ ¼ k i ðx;xÞ ;
i ¼ 1; 2; . . . n;
i;i
p1 : ui;1 ðtÞ ¼
n P j¼1 j–i
p2 : ui;2 ðtÞ ¼
n P j¼1 j–i
ki;j ðx;tÞ u ðxÞ ki;i ðx;xÞ j;0
ki;j ðx;tÞ u ðxÞ ki;i ðx;xÞ j;1
n Rx P 0
j¼1
n Rx P 0
j¼1
@ki;j ðx;tÞ=@x uj;0 ðtÞdt; ki;i ðx;xÞ
i ¼ 1; 2; . . . n;
@ki;j ðx;tÞ=@x uj;1 ðtÞdt; ki;i ðx;xÞ
i ¼ 1; 2; . . . n;
.. . As p ? 1, (5) tends to Eqs. (4) and (7) to the solution of (4) in practice some terms of this series will serve as an approximaPm tion solution i.e., um j¼0 ui;j is a m + 1 terms approximated solution. This series is convergent for most cases, and the rate i ¼ of convergence depends on Fi(ui) [2].
3. Numerical example In this part three examples are provided. These examples are considered to illustrate ability and reliability of the method. Applying the method for linear systems is so easily that it does not worth to have any example. Example 1. Consider the following non-linear system of Volterra integral equations of the first kind
(Rx 0
1 4 f ðtÞ þ ðx tÞf ðtÞgðtÞdt ¼ 34 þ 12 x þ 12 x2 þ 12 x þ ex 14 e2x ;
0
1 4 gðtÞ þ ðx tÞf ðtÞgðtÞdt ¼ 54 þ 12 x þ 12 x2 þ 12 x ex 14 e2x :
Rx
With the exact solutions f(x) = x + ex, g(x) = x ex. We change the system of integral equations to the second kind by differentiation
(
f ðxÞ þ
Rx
gðxÞ þ
Rx
0
f ðtÞgðtÞdt ¼ 12 þ x þ 13 x3 þ ex 12 e2x ;
0
f ðtÞgðtÞdt ¼ 12 þ x þ 13 x3 ex 12 e2x :
Corresponding homotopies can be readily constructed as follows:
Hðf ; pÞ ¼ f ðxÞ þ p
Rx
Hðg; pÞ ¼ gðxÞ þ p
0
Rx 0
f ðtÞgðtÞdt 12 x 13 x3 ex þ 12 e2x ¼ 0; f ðtÞgðtÞdt 12 x 13 x3 þ ex þ 12 e2x ¼ 0:
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J. Biazar et al. / Chaos, Solitons and Fractals 42 (2009) 3020–3026
Equating the coefficients of the terms with identical powers of p, we have
( p0 : .. .
pnþ1 :
f0 ðxÞ ¼ 12 e2x þ ex þ 13 x3 þ x þ 12 ; g 0 ðxÞ ¼ 12 e2x ex þ 13 x3 þ x þ 12 ; 8 n Rx P > > fk ðtÞg nk ðtÞdt; > < fnþ1 ðxÞ ¼ 0 k¼0
n Rx P > > > fk ðtÞg nk ðtÞdt: : g nþ1 ðxÞ ¼ 0 k¼0
Therefore the approximations to the solutions of Example 1 can be readily obtained by
fm ¼
m X i¼0
fi ; g m ¼
m X
gi :
i¼0
The results corresponding absolute errors are presented in Table 1 and Fig. 1. Example 2. Consider the following non-linear system of Volterra integral equations of the first kind, with exact solutions, f ðxÞ ¼ 14 x2 þ 1, gðxÞ ¼ 13 x2 þ 32 ; hðxÞ ¼ 12 x þ 23.
8Rx 1 6 19 5 þ x tÞf ðtÞ þ ð12 x2 þ tÞgðtÞhðtÞÞdt ¼ 48 x þ 270 x þ 19 x4 þ 76 x3 þ x2 þ 5x; > 72 < R0 ðð5 x 1 2 2 1 2 1 5 35 4 x þ t f ðtÞ þ ð3 þ x tÞgðtÞ þ 4 ðx t ÞhðtÞ dt ¼ 24 x þ 288 x þ 17 x3 þ 54 x2 þ 92 x; 0 2 18 > :Rx 1 7 1 6 ðtf ðtÞgðtÞ xtg 2 ðtÞ 5hðtÞÞdt ¼ 54 x þ 72 x 14 x5 þ 17 x4 98 x3 12 x2 10 x: 0 96 3
By differentiation we derive,
Rx 8 ðxÞ þ 12 x2 þ x gðxÞhðxÞ þ 0 ðf ðtÞ þ xgðtÞhðtÞÞdt ¼ 18 x5 þ 19 x4 þ 19 x3 þ 72 x2 þ 2x þ 5; > 54 18 < 5f 1 2 Rx 5 4 x þ x f ðxÞ þ 3gðxÞ þ 0 ðxf ðtÞ þ gðtÞ þ 12 xhðtÞÞdt ¼ 24 x þ 35 x3 þ 17 x2 þ 52 x þ 92 ; 2 72 6 > Rx 2 : 2 2 7 6 1 5 5 4 17 3 xf ðxÞgðxÞ x g ðxÞ 5hðxÞ 0 ðtg ðtÞÞdt ¼ 54 x þ 12 x 4 x þ 24 x 27 x2 x 10 : 8 3 Corresponding homotopies can be readily constructed as follows:
Z x 1 2 1 1 1 5 19 4 19 3 7 2 2 Hðf ; pÞ ¼ f ðxÞ þ p ðf ðtÞ þ xgðtÞhðtÞÞdt x þ x gðxÞhðxÞ þ p x x x x x 1 ¼ 0; 10 5 5 0 40 270 90 10 5 Z x 1 2 1 1 1 5 4 35 3 17 2 5 3 xf ðtÞ þ gðtÞ þ xhðtÞ dt x x x x ¼ 0; Hðg; pÞ ¼ gðxÞ þ p x þ x f ðxÞ þ p 6 3 3 0 2 72 216 18 6 2 Z x 1 1 2 2 1 7 1 1 17 27 1 2 ðtg 2 ðtÞÞdt x6 þ x5 x4 þ x3 x2 x ¼ 0: Hðh; pÞ ¼ hðxÞ pxf ðxÞgðxÞ þ px g ðxÞ þ p 5 5 5 0 270 60 4 120 40 5 3 Equating the coefficients of the terms with identical powers of p, we have
8 1 5 19 4 19 3 7 2 2 > < f0 ðxÞ ¼ 40 x þ 270 x þ 90 x þ 10 x þ 5 x þ 1; 0 5 4 35 3 p : g 0 ðxÞ ¼ 72 x þ 216 x þ 17 x2 þ 56 x þ 32 ; 18 > : 7 6 1 5 1 4 17 3 ðxÞ ¼ x x þ x 120 x þ 27 x2 þ 15 x þ 23 ; h0 270 60 4 40 ... 8 n n 1 2 1 P Rx P > > fnþ1 ðxÞ ¼ 10 x þ 5x g k ðxÞhnk ðxÞ 15 0 fn ðxÞ þ x g k ðtÞhnk ðtÞ dt; > > > k¼0 k¼0 < R nþ1 p : g nþ1 ðxÞ ¼ 16 x2 þ 13 x fn ðxÞ 13 0x xfn ðtÞ þ g n ðtÞ þ 12 xhn ðtÞ dt; > > > n n n Rx P P P > > : hnþ1 ðxÞ ¼ 15 x fk ðxÞg nk ðxÞ 15 x2 g k ðxÞg nk ðxÞ þ 15 0 t fk ðtÞg nk ðtÞdt: k¼0
k¼0
k¼0
Table 1 The six terms approximations to the solution of Example 1 and their corresponding error. x
f 6(x)
g6(x)
e(f(xi))
e(g(xi))
0 0.1 0.2 0.3 0.4 0.5
1 1.205204395 1.421379286 1.649894364 1.891893287 2.148280450
1 1.005135605 1.021420714 1.049805636 1.091706713 1.149119550
0 2.6241 105 2.8069 105 4.2428 105 4.4465 105 2.29049 104
0 2.8077 105 2.2553 105 4.0044 105 5.3861 105 1.86507 104
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Fig. 1. The numerical results and exact solutions of Example 1.
Therefore the approximations to the solutions of Example 2 can be readily obtained by
fm ¼
m X
fi ;
gm ¼
i¼0
m X
gi;
m
h ¼
i¼0
m X
hi :
i¼0
The results corresponding absolute errors are presented in Table 2 and Fig. 2. Example 3. Consider the following non-linear system of Volterra integral equations of the first kind, with exact solutions, f1 ðxÞ ¼ x2 x þ 12 ; f 2 ðxÞ ¼ x; f 3 ðxÞ ¼ x þ 15 ; f 4 ðxÞ ¼ x3 14 :
8Rx 9 4 ðð5 þ x2 t2 Þf1 ðtÞ x2 f2 ðtÞf4 ðtÞ þ t2 f32 ðtÞÞdt ¼ 15 x7 þ 13 x5 40 x þ 151 x3 52 x2 þ 52 x; > 0 75 > > R > x < ðxtf ðtÞf ðtÞ þ x2 f ðtÞf ðtÞ þ 2f ðtÞ þ ðt xÞf ðtÞÞdt ¼ 1 x7 þ 1 x6 1 x5 þ 7 x4 þ 7 x3 þ 9 x2 ; 1 2 4 3 3 2 0 5 4 4 24 60 10 Rx 1 2 1 7 1 5 1 4 1 3 2 2 > ðf ðtÞÞð2 x Þ þ xtðf ðtÞ þ f ðtÞf ðtÞ þ f ðtÞÞdt ¼ x þ x þ x þ x x þ x; > 3 1 2 4 2 0 4 6 4 24 5 5 > > :Rx 7 5 7 4 7 3 xððx2 t2 Þðf22 ðtÞ þ f32 ðtÞÞ þ ðt 2 þ xÞf4 ðtÞ t 2 f1 ðtÞÞdt ¼ 60 x 20 x 50 x 18 x2 þ 12 : 0 By differentiation we get, Z
x
7 5 9 3 151 2 5 2xf1 ðtÞ 2xf2 ðtÞf4 ðtÞdt ¼ x6 þ x4 x þ x 5x þ ; 5 3 10 25 2 Z x 7 6 3 5 5 4 7 3 7 2 9 2 2 ðtf1 ðtÞf2 ðtÞ þ 2xf4 ðtÞf3 ðtÞ f3 ðtÞÞdt ¼ x þ x x þ x þ x f1 ðxÞf2 ðxÞ þ x f4 ðxÞf3 ðxÞ þ 2f 2 ðxÞ þ x þ x; 5 2 4 6 20 5 0 Z x 1 2 1 7 6 5 41 3 3 2 2 2 xf3 ðtÞ þ tðf1 ðtÞ þ f2 ðtÞf3 ðtÞ þ f2 ðtÞÞ dt ¼ x þ x x þ x 2x þ ; 2 x f3 ðxÞ þ x ðf1 ðxÞ þ f2 ðxÞf4 ðxÞ þ f2 ðxÞÞ þ 4 2 6 4 6 5 5 0 Z x 7 7 21 1 1 x4 x3 x2 x þ : ð2xðf22 ðtÞ þ f32 ðtÞÞ þ f4 ðtÞÞdt ¼ 2f 4 ðxÞ x2 f1 ðxÞ þ 12 5 50 4 2 0 5f 1 ðxÞ x2 f2 ðxÞf4 ðxÞ þ x2 f32 ðxÞ þ
0
Corresponding homotopies can be readily constructed as the following:
Table 2 The six terms approximations to the solution of Example 2 and their corresponding error. x
f 6(x)
g6(x)
h6(x)
e(f(xi))
e(g(xi))
e(h(xi))
0 0.1 0.2 0.3 0.4 0.5
1 1.002500008 1.010000779 1.022513450 1.040106812 1.063027966
1.5 1.503333358 1.513334992 1.530019156 1.553423083 1.583465931
0.6666666667 0.7166666173 0.7666635824 0.8166363465 0.8665574676 0.9166774932
0 9 109 7.77 108 1.3450 105 1.06813 104 5.27965 104
0 2.5 109 1.658 106 1.9156 105 8.9751 105 1.32599 104
0 4.94 108 3.0843 106 3.03201 105 1.091990 104 1.08264 105
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Fig. 2. The numerical results and exact solutions of Example 2.
Z x 1 1 1 7 6 1 4 9 3 151 2 1 x x þ x x þ x ¼ 0; Hðf1 ; pÞ ¼ f1 ðxÞ px2 f2 ðxÞf4 ðxÞ þ px2 f32 ðxÞ þ p ð2xf1 ðtÞ 2xf2 ðtÞf4 ðtÞÞdt þ 5 5 5 0 25 5 50 125 2 Z x 1 2 1 2 1 7 6 3 5 5 4 x x þ x Hðf2 ; pÞ ¼ f2 ðxÞ þ px f1 ðxÞf2 ðxÞ þ px f4 ðxÞf3 ðxÞ þ p ðtf1 ðtÞf2 ðtÞ þ 2xf4 ðtÞf3 ðtÞ f3 ðtÞÞdt þ 2 2 2 0 10 4 8 7 3 7 2 9 x x x ¼ 0; 12 40 10 Z x 1 1 1 1 Hðf3 ; pÞ ¼ f3 ðxÞ px2 f3 ðxÞ þ px2 ðf1 ðxÞ þ f2 ðxÞf4 ðxÞ þ f2 ðxÞÞ þ p xf3 ðtÞ þ tðf1 ðtÞÞ þ f2 ðtÞf4 ðtÞ þ f2 ðtÞ dt 8 2 2 0 2 7 6 5 4 1 3 3 2 1 x x x x þ x ¼ 0; 12 8 12 5 Z 10 x 1 2 1 7 4 7 3 21 2 1 1 2 Hðf4 ; pÞ ¼ f4 ðxÞ þ px f1 ðxÞ p x x x x þ ¼ 0: ð2xðf2 ðtÞ þ f32 ðtÞÞ þ f4 ðtÞÞdt þ 2 2 0 24 10 100 8 4 Equating the coefficients of the terms with identical powers of p, we derive
8 7 6 9 3 f1;0 ðxÞ ¼ 25 x þ 13 x4 50 x þ 151 x2 x þ 12 ; > 125 > > < 7 6 3 5 5 4 7 3 7 2 9 x þ 10 x; f2;0 ðxÞ ¼ 10 x þ 4 x 8 x þ 12 x þ 40 p0 : 7 5 1 3 1 6 4 3 2 > ðxÞ ¼ þ x þ x þ x þ x x þ ; f 3;0 > 12 8 12 10 5 > : 7 4 7 3 21 2 x þ 10 x þ 100 x þ 18 x 14 ; f4;0 ðxÞ ¼ 24 .. . n n 8 P P > 1 2 > f1;nþ1 ðxÞ ¼ 15 x2 f ðxÞf ðxÞ x f ðxÞf ðxÞ 2;k 4;nk 3;k 3;nk > 5 > > k¼0 k¼0 > > n > > R P > x 1 > > f2;k ðtÞf4;nk ðtÞ dt; > 5 0 2xf1;n ðtÞ 2x > > k¼0 > > n > n > P P > 1 2 1 2 > > f ðxÞ ¼ x f ðxÞf ðxÞ x f ðxÞf ðxÞ 2;nþ1 1;k 2;nk 4;k 3;nk > 2 2 > > k¼0 k¼0 > > n n < R P P 1 x nþ1 2 0 t f1;k ðtÞf2;nk ðtÞ þ 2x f3;k ðtÞf3;nk ðtÞ f4;n ðtÞ dt; p : > k¼0 k¼0 > > > > n P > > 1 2 1 2 > f3;nþ1 ðxÞ ¼ 8 x f3;n ðxÞ 2 x ðf1;n ðxÞ þ f2;k ðxÞf4;nk ðxÞ þ f2;n ðxÞÞ > > > k¼0 > > > > n R P > > 1 x 1 > xf ðtÞ þ tf ðtÞ þ tf ðtÞf ðtÞ þ f ðtÞ dt; 3;n 1;n 2;n 2;k 4;nk > 2 0 2 > > k¼0 > > > n n > Rx P P > > : f4;nþ1 ðxÞ ¼ 12 x2 f1;n ðxÞ þ 12 0 2x f2;k ðtÞf2;nk ðtÞ þ f3;k ðtÞf3;nk ðtÞ þ f4;n ðtÞ dt: k¼0
k¼0
The following approximations to the solution of Example 3 are considered as the following
f1m ¼
m X i¼0
f1;i ;
f2m ¼
m X i¼0
f2;i ;
f3m ¼
m X i¼0
f3;i ;
f4m ¼
m X
f4;i :
i¼0
Some values of the solutions and there corresponding errors are presented in Table 3 and Figs. 3 and 4.
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J. Biazar et al. / Chaos, Solitons and Fractals 42 (2009) 3020–3026 Table 3 The six terms approximations to the solution of Example 3 and their corresponding error. x
f16 ðxÞ
f26 ðxÞ
f36 ðxÞ
f46 ðxÞ
e(f1(xi))
e(f2(xi))
e(f3(xi))
e(f4(xi))
0 0.1 0.2 0.3 0.4 0.5
0.5 0.4100000000 0.3399999999 0.2900000001 0.2600001937 0.2500031735
0 0.1000000000 0.1999999996 0.2999999953 0.4000001658 0.5000033509
0.2 0.1000000001 0.0000000007 0.0999999603 0.1999995543 .02999991608
0.25 0.2489999999 0.2419999996 0.2229999823 0.1859997117 0.1249965911
0 0 0 2 1010 1.937 107 3.1736 106
0 1 1010 3 1010 4.7 1010 1.652 107 3.3511 106
0 0 6 1010 3.963 108 4.459 107 8.392 107
0 0 5 1010 1.75 108 2.881 107 3.4086 106
Fig. 3. The numerical results and exact solutions of Example 3.
Fig. 4. The numerical results and exact solutions of Example 3.
4. Conclusion In this work we had used homotopy perturbation method for solving linear and non-linear systems of Volterra integral equations of the first kind. For the sake of simplicity in explanation of the method the linear integral equations are considered, but examples are illustrated for non-linear systems. 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 strong ones.
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Contents lists available at ScienceDirect
Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos
Application of homotopy perturbation method for systems of Volterra integral equations of the first kind J. Biazar a,*, M. Eslami a, H. Aminikhah b a b
Department of Mathematics, Faculty of Sciences, The University of Guilan, P.O. Box 1914, P.C. 41938 Rasht, Iran Department of Mathematics, School of Mathematical Sciences, Shahrood University of Technology, P.O. Box 316, P.C. 3619995161 Shahrood, Iran
a r t i c l e
i n f o
a b s t r a c t
Article history: Accepted 3 April 2009
In this article, an application of He’s homotopy perturbation method is applied to solve systems of Volterra integral equations of the first kind. Some non-linear examples are prepared to illustrate the efficiency and simplicity of the method. Applying the method for linear systems is so easily that it does not worth to have any example. Ó 2009 Elsevier Ltd. All rights reserved.
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 have no such a small parameter. This method has been proposed to eliminate the small parameter [7,8]. In recent years the application of homotopy perturbation theory has appeared in many researches [9–17]. In this paper we propose homotopy perturbation method to solve systems of Volterra integral equations of the first kind. 2. The homotopy perturbation method applied to systems of Volterra integral equations of the first kind A system of integral equations of the first kind can be presented as the following:
Z
x
ki ðx; tÞg i ðu1 ðtÞ; u2 ðtÞ; . . . ; un ðtÞÞdt ¼ fi ðxÞ;
i ¼ 1; 2; . . . ; n:
ð1Þ
0
If gi(u1, . . . ,un) are linear, the system (1) can be represented as the following:
Z
x
0
n X
ki;j ðx; tÞuj ðtÞdt ¼ fi ðxÞ;
i ¼ 1; 2; . . . ; n:
j¼1
After differentiation we derive: n X j¼1
ki;j ðx; xÞuj ðxÞ þ
Z 0
x
n X @ki;j ðx; tÞ uj ðtÞdt ¼ fi0 ðxÞ: @x j¼1
ð2Þ
* Corresponding author. E-mail addresses: [email protected], [email protected] (J. Biazar), [email protected] (M. Eslami), [email protected], aminikhah@ shahroodut.ac.ir (H. Aminikhah). 0960-0779/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2009.04.016
J. Biazar et al. / Chaos, Solitons and Fractals 42 (2009) 3020–3026
3021
From ith equation, in (2), we write the ith unknown, ui, in terms of the other unknown, to derive the following linear system of Volterra integral equations of the second kind
ui ðxÞ ¼
n X f 0 ðxÞ ki;j ðx; tÞ uj ðxÞ ki;i ðx; xÞ j¼1 ki;i ðx; xÞ
Z
x
0
n X @ki;j ðx; tÞ=@x uj ðtÞdt: ki;i ðx; xÞ j¼1
ð3Þ
j–i
To illustrate the homotopy perturbation method, we consider (3) as
F i ðui Þ ¼ ui ðxÞ
n X ki;j ðx; tÞ f 0 ðxÞ þ uj ðxÞ þ ki;i ðx; xÞ j¼1 ki;i ðx; xÞ
Z 0
x
n X @ki;j ðx; tÞ=@x uj ðtÞdt ¼ 0: ki;i ðx; xÞ j¼1
ð4Þ
j–i
With the solution ui(x). We can define homotopy H(ui,p) by
Hðui ; pÞ ¼ ð1 pÞLi ðui Þ þ pF i ðui Þ ¼ 0;
ð5Þ
where Li(ui) is a functional operator with known solution ui 0, which can be obtained easily. Obviously, from Eq. (5) we have:
Hðui ; 0Þ ¼ Li ðui Þ;
Hðui ; 1Þ ¼ F i ðui Þ:
ð6Þ
The changing process of p from zero to unity is just that of H(ui,p) from a starting point H(ui,0) to a solution function H(ui,1). Applying the perturbation technique [6], due to the fact that, 0 6 p 6 1, being considered as a small parameter. We can assume that the solution of (5) can be expressed as a series in p, as follows:
ui ¼ ui;0 þ pui;1 þ p2 ui;2 þ
ð7Þ
Substituting (7) into (5) and equating the coefficients of p with the same power leads to f 0 ðxÞ
p0 : ui;0 ðtÞ ¼ k i ðx;xÞ ;
i ¼ 1; 2; . . . n;
i;i
p1 : ui;1 ðtÞ ¼
n P j¼1 j–i
p2 : ui;2 ðtÞ ¼
n P j¼1 j–i
ki;j ðx;tÞ u ðxÞ ki;i ðx;xÞ j;0
ki;j ðx;tÞ u ðxÞ ki;i ðx;xÞ j;1
n Rx P 0
j¼1
n Rx P 0
j¼1
@ki;j ðx;tÞ=@x uj;0 ðtÞdt; ki;i ðx;xÞ
i ¼ 1; 2; . . . n;
@ki;j ðx;tÞ=@x uj;1 ðtÞdt; ki;i ðx;xÞ
i ¼ 1; 2; . . . n;
.. . As p ? 1, (5) tends to Eqs. (4) and (7) to the solution of (4) in practice some terms of this series will serve as an approximaPm tion solution i.e., um j¼0 ui;j is a m + 1 terms approximated solution. This series is convergent for most cases, and the rate i ¼ of convergence depends on Fi(ui) [2].
3. Numerical example In this part three examples are provided. These examples are considered to illustrate ability and reliability of the method. Applying the method for linear systems is so easily that it does not worth to have any example. Example 1. Consider the following non-linear system of Volterra integral equations of the first kind
(Rx 0
1 4 f ðtÞ þ ðx tÞf ðtÞgðtÞdt ¼ 34 þ 12 x þ 12 x2 þ 12 x þ ex 14 e2x ;
0
1 4 gðtÞ þ ðx tÞf ðtÞgðtÞdt ¼ 54 þ 12 x þ 12 x2 þ 12 x ex 14 e2x :
Rx
With the exact solutions f(x) = x + ex, g(x) = x ex. We change the system of integral equations to the second kind by differentiation
(
f ðxÞ þ
Rx
gðxÞ þ
Rx
0
f ðtÞgðtÞdt ¼ 12 þ x þ 13 x3 þ ex 12 e2x ;
0
f ðtÞgðtÞdt ¼ 12 þ x þ 13 x3 ex 12 e2x :
Corresponding homotopies can be readily constructed as follows:
Hðf ; pÞ ¼ f ðxÞ þ p
Rx
Hðg; pÞ ¼ gðxÞ þ p
0
Rx 0
f ðtÞgðtÞdt 12 x 13 x3 ex þ 12 e2x ¼ 0; f ðtÞgðtÞdt 12 x 13 x3 þ ex þ 12 e2x ¼ 0:
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J. Biazar et al. / Chaos, Solitons and Fractals 42 (2009) 3020–3026
Equating the coefficients of the terms with identical powers of p, we have
( p0 : .. .
pnþ1 :
f0 ðxÞ ¼ 12 e2x þ ex þ 13 x3 þ x þ 12 ; g 0 ðxÞ ¼ 12 e2x ex þ 13 x3 þ x þ 12 ; 8 n Rx P > > fk ðtÞg nk ðtÞdt; > < fnþ1 ðxÞ ¼ 0 k¼0
n Rx P > > > fk ðtÞg nk ðtÞdt: : g nþ1 ðxÞ ¼ 0 k¼0
Therefore the approximations to the solutions of Example 1 can be readily obtained by
fm ¼
m X i¼0
fi ; g m ¼
m X
gi :
i¼0
The results corresponding absolute errors are presented in Table 1 and Fig. 1. Example 2. Consider the following non-linear system of Volterra integral equations of the first kind, with exact solutions, f ðxÞ ¼ 14 x2 þ 1, gðxÞ ¼ 13 x2 þ 32 ; hðxÞ ¼ 12 x þ 23.
8Rx 1 6 19 5 þ x tÞf ðtÞ þ ð12 x2 þ tÞgðtÞhðtÞÞdt ¼ 48 x þ 270 x þ 19 x4 þ 76 x3 þ x2 þ 5x; > 72 < R0 ðð5 x 1 2 2 1 2 1 5 35 4 x þ t f ðtÞ þ ð3 þ x tÞgðtÞ þ 4 ðx t ÞhðtÞ dt ¼ 24 x þ 288 x þ 17 x3 þ 54 x2 þ 92 x; 0 2 18 > :Rx 1 7 1 6 ðtf ðtÞgðtÞ xtg 2 ðtÞ 5hðtÞÞdt ¼ 54 x þ 72 x 14 x5 þ 17 x4 98 x3 12 x2 10 x: 0 96 3
By differentiation we derive,
Rx 8 ðxÞ þ 12 x2 þ x gðxÞhðxÞ þ 0 ðf ðtÞ þ xgðtÞhðtÞÞdt ¼ 18 x5 þ 19 x4 þ 19 x3 þ 72 x2 þ 2x þ 5; > 54 18 < 5f 1 2 Rx 5 4 x þ x f ðxÞ þ 3gðxÞ þ 0 ðxf ðtÞ þ gðtÞ þ 12 xhðtÞÞdt ¼ 24 x þ 35 x3 þ 17 x2 þ 52 x þ 92 ; 2 72 6 > Rx 2 : 2 2 7 6 1 5 5 4 17 3 xf ðxÞgðxÞ x g ðxÞ 5hðxÞ 0 ðtg ðtÞÞdt ¼ 54 x þ 12 x 4 x þ 24 x 27 x2 x 10 : 8 3 Corresponding homotopies can be readily constructed as follows:
Z x 1 2 1 1 1 5 19 4 19 3 7 2 2 Hðf ; pÞ ¼ f ðxÞ þ p ðf ðtÞ þ xgðtÞhðtÞÞdt x þ x gðxÞhðxÞ þ p x x x x x 1 ¼ 0; 10 5 5 0 40 270 90 10 5 Z x 1 2 1 1 1 5 4 35 3 17 2 5 3 xf ðtÞ þ gðtÞ þ xhðtÞ dt x x x x ¼ 0; Hðg; pÞ ¼ gðxÞ þ p x þ x f ðxÞ þ p 6 3 3 0 2 72 216 18 6 2 Z x 1 1 2 2 1 7 1 1 17 27 1 2 ðtg 2 ðtÞÞdt x6 þ x5 x4 þ x3 x2 x ¼ 0: Hðh; pÞ ¼ hðxÞ pxf ðxÞgðxÞ þ px g ðxÞ þ p 5 5 5 0 270 60 4 120 40 5 3 Equating the coefficients of the terms with identical powers of p, we have
8 1 5 19 4 19 3 7 2 2 > < f0 ðxÞ ¼ 40 x þ 270 x þ 90 x þ 10 x þ 5 x þ 1; 0 5 4 35 3 p : g 0 ðxÞ ¼ 72 x þ 216 x þ 17 x2 þ 56 x þ 32 ; 18 > : 7 6 1 5 1 4 17 3 ðxÞ ¼ x x þ x 120 x þ 27 x2 þ 15 x þ 23 ; h0 270 60 4 40 ... 8 n n 1 2 1 P Rx P > > fnþ1 ðxÞ ¼ 10 x þ 5x g k ðxÞhnk ðxÞ 15 0 fn ðxÞ þ x g k ðtÞhnk ðtÞ dt; > > > k¼0 k¼0 < R nþ1 p : g nþ1 ðxÞ ¼ 16 x2 þ 13 x fn ðxÞ 13 0x xfn ðtÞ þ g n ðtÞ þ 12 xhn ðtÞ dt; > > > n n n Rx P P P > > : hnþ1 ðxÞ ¼ 15 x fk ðxÞg nk ðxÞ 15 x2 g k ðxÞg nk ðxÞ þ 15 0 t fk ðtÞg nk ðtÞdt: k¼0
k¼0
k¼0
Table 1 The six terms approximations to the solution of Example 1 and their corresponding error. x
f 6(x)
g6(x)
e(f(xi))
e(g(xi))
0 0.1 0.2 0.3 0.4 0.5
1 1.205204395 1.421379286 1.649894364 1.891893287 2.148280450
1 1.005135605 1.021420714 1.049805636 1.091706713 1.149119550
0 2.6241 105 2.8069 105 4.2428 105 4.4465 105 2.29049 104
0 2.8077 105 2.2553 105 4.0044 105 5.3861 105 1.86507 104
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J. Biazar et al. / Chaos, Solitons and Fractals 42 (2009) 3020–3026
Fig. 1. The numerical results and exact solutions of Example 1.
Therefore the approximations to the solutions of Example 2 can be readily obtained by
fm ¼
m X
fi ;
gm ¼
i¼0
m X
gi;
m
h ¼
i¼0
m X
hi :
i¼0
The results corresponding absolute errors are presented in Table 2 and Fig. 2. Example 3. Consider the following non-linear system of Volterra integral equations of the first kind, with exact solutions, f1 ðxÞ ¼ x2 x þ 12 ; f 2 ðxÞ ¼ x; f 3 ðxÞ ¼ x þ 15 ; f 4 ðxÞ ¼ x3 14 :
8Rx 9 4 ðð5 þ x2 t2 Þf1 ðtÞ x2 f2 ðtÞf4 ðtÞ þ t2 f32 ðtÞÞdt ¼ 15 x7 þ 13 x5 40 x þ 151 x3 52 x2 þ 52 x; > 0 75 > > R > x < ðxtf ðtÞf ðtÞ þ x2 f ðtÞf ðtÞ þ 2f ðtÞ þ ðt xÞf ðtÞÞdt ¼ 1 x7 þ 1 x6 1 x5 þ 7 x4 þ 7 x3 þ 9 x2 ; 1 2 4 3 3 2 0 5 4 4 24 60 10 Rx 1 2 1 7 1 5 1 4 1 3 2 2 > ðf ðtÞÞð2 x Þ þ xtðf ðtÞ þ f ðtÞf ðtÞ þ f ðtÞÞdt ¼ x þ x þ x þ x x þ x; > 3 1 2 4 2 0 4 6 4 24 5 5 > > :Rx 7 5 7 4 7 3 xððx2 t2 Þðf22 ðtÞ þ f32 ðtÞÞ þ ðt 2 þ xÞf4 ðtÞ t 2 f1 ðtÞÞdt ¼ 60 x 20 x 50 x 18 x2 þ 12 : 0 By differentiation we get, Z
x
7 5 9 3 151 2 5 2xf1 ðtÞ 2xf2 ðtÞf4 ðtÞdt ¼ x6 þ x4 x þ x 5x þ ; 5 3 10 25 2 Z x 7 6 3 5 5 4 7 3 7 2 9 2 2 ðtf1 ðtÞf2 ðtÞ þ 2xf4 ðtÞf3 ðtÞ f3 ðtÞÞdt ¼ x þ x x þ x þ x f1 ðxÞf2 ðxÞ þ x f4 ðxÞf3 ðxÞ þ 2f 2 ðxÞ þ x þ x; 5 2 4 6 20 5 0 Z x 1 2 1 7 6 5 41 3 3 2 2 2 xf3 ðtÞ þ tðf1 ðtÞ þ f2 ðtÞf3 ðtÞ þ f2 ðtÞÞ dt ¼ x þ x x þ x 2x þ ; 2 x f3 ðxÞ þ x ðf1 ðxÞ þ f2 ðxÞf4 ðxÞ þ f2 ðxÞÞ þ 4 2 6 4 6 5 5 0 Z x 7 7 21 1 1 x4 x3 x2 x þ : ð2xðf22 ðtÞ þ f32 ðtÞÞ þ f4 ðtÞÞdt ¼ 2f 4 ðxÞ x2 f1 ðxÞ þ 12 5 50 4 2 0 5f 1 ðxÞ x2 f2 ðxÞf4 ðxÞ þ x2 f32 ðxÞ þ
0
Corresponding homotopies can be readily constructed as the following:
Table 2 The six terms approximations to the solution of Example 2 and their corresponding error. x
f 6(x)
g6(x)
h6(x)
e(f(xi))
e(g(xi))
e(h(xi))
0 0.1 0.2 0.3 0.4 0.5
1 1.002500008 1.010000779 1.022513450 1.040106812 1.063027966
1.5 1.503333358 1.513334992 1.530019156 1.553423083 1.583465931
0.6666666667 0.7166666173 0.7666635824 0.8166363465 0.8665574676 0.9166774932
0 9 109 7.77 108 1.3450 105 1.06813 104 5.27965 104
0 2.5 109 1.658 106 1.9156 105 8.9751 105 1.32599 104
0 4.94 108 3.0843 106 3.03201 105 1.091990 104 1.08264 105
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Fig. 2. The numerical results and exact solutions of Example 2.
Z x 1 1 1 7 6 1 4 9 3 151 2 1 x x þ x x þ x ¼ 0; Hðf1 ; pÞ ¼ f1 ðxÞ px2 f2 ðxÞf4 ðxÞ þ px2 f32 ðxÞ þ p ð2xf1 ðtÞ 2xf2 ðtÞf4 ðtÞÞdt þ 5 5 5 0 25 5 50 125 2 Z x 1 2 1 2 1 7 6 3 5 5 4 x x þ x Hðf2 ; pÞ ¼ f2 ðxÞ þ px f1 ðxÞf2 ðxÞ þ px f4 ðxÞf3 ðxÞ þ p ðtf1 ðtÞf2 ðtÞ þ 2xf4 ðtÞf3 ðtÞ f3 ðtÞÞdt þ 2 2 2 0 10 4 8 7 3 7 2 9 x x x ¼ 0; 12 40 10 Z x 1 1 1 1 Hðf3 ; pÞ ¼ f3 ðxÞ px2 f3 ðxÞ þ px2 ðf1 ðxÞ þ f2 ðxÞf4 ðxÞ þ f2 ðxÞÞ þ p xf3 ðtÞ þ tðf1 ðtÞÞ þ f2 ðtÞf4 ðtÞ þ f2 ðtÞ dt 8 2 2 0 2 7 6 5 4 1 3 3 2 1 x x x x þ x ¼ 0; 12 8 12 5 Z 10 x 1 2 1 7 4 7 3 21 2 1 1 2 Hðf4 ; pÞ ¼ f4 ðxÞ þ px f1 ðxÞ p x x x x þ ¼ 0: ð2xðf2 ðtÞ þ f32 ðtÞÞ þ f4 ðtÞÞdt þ 2 2 0 24 10 100 8 4 Equating the coefficients of the terms with identical powers of p, we derive
8 7 6 9 3 f1;0 ðxÞ ¼ 25 x þ 13 x4 50 x þ 151 x2 x þ 12 ; > 125 > > < 7 6 3 5 5 4 7 3 7 2 9 x þ 10 x; f2;0 ðxÞ ¼ 10 x þ 4 x 8 x þ 12 x þ 40 p0 : 7 5 1 3 1 6 4 3 2 > ðxÞ ¼ þ x þ x þ x þ x x þ ; f 3;0 > 12 8 12 10 5 > : 7 4 7 3 21 2 x þ 10 x þ 100 x þ 18 x 14 ; f4;0 ðxÞ ¼ 24 .. . n n 8 P P > 1 2 > f1;nþ1 ðxÞ ¼ 15 x2 f ðxÞf ðxÞ x f ðxÞf ðxÞ 2;k 4;nk 3;k 3;nk > 5 > > k¼0 k¼0 > > n > > R P > x 1 > > f2;k ðtÞf4;nk ðtÞ dt; > 5 0 2xf1;n ðtÞ 2x > > k¼0 > > n > n > P P > 1 2 1 2 > > f ðxÞ ¼ x f ðxÞf ðxÞ x f ðxÞf ðxÞ 2;nþ1 1;k 2;nk 4;k 3;nk > 2 2 > > k¼0 k¼0 > > n n < R P P 1 x nþ1 2 0 t f1;k ðtÞf2;nk ðtÞ þ 2x f3;k ðtÞf3;nk ðtÞ f4;n ðtÞ dt; p : > k¼0 k¼0 > > > > n P > > 1 2 1 2 > f3;nþ1 ðxÞ ¼ 8 x f3;n ðxÞ 2 x ðf1;n ðxÞ þ f2;k ðxÞf4;nk ðxÞ þ f2;n ðxÞÞ > > > k¼0 > > > > n R P > > 1 x 1 > xf ðtÞ þ tf ðtÞ þ tf ðtÞf ðtÞ þ f ðtÞ dt; 3;n 1;n 2;n 2;k 4;nk > 2 0 2 > > k¼0 > > > n n > Rx P P > > : f4;nþ1 ðxÞ ¼ 12 x2 f1;n ðxÞ þ 12 0 2x f2;k ðtÞf2;nk ðtÞ þ f3;k ðtÞf3;nk ðtÞ þ f4;n ðtÞ dt: k¼0
k¼0
The following approximations to the solution of Example 3 are considered as the following
f1m ¼
m X i¼0
f1;i ;
f2m ¼
m X i¼0
f2;i ;
f3m ¼
m X i¼0
f3;i ;
f4m ¼
m X
f4;i :
i¼0
Some values of the solutions and there corresponding errors are presented in Table 3 and Figs. 3 and 4.
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J. Biazar et al. / Chaos, Solitons and Fractals 42 (2009) 3020–3026 Table 3 The six terms approximations to the solution of Example 3 and their corresponding error. x
f16 ðxÞ
f26 ðxÞ
f36 ðxÞ
f46 ðxÞ
e(f1(xi))
e(f2(xi))
e(f3(xi))
e(f4(xi))
0 0.1 0.2 0.3 0.4 0.5
0.5 0.4100000000 0.3399999999 0.2900000001 0.2600001937 0.2500031735
0 0.1000000000 0.1999999996 0.2999999953 0.4000001658 0.5000033509
0.2 0.1000000001 0.0000000007 0.0999999603 0.1999995543 .02999991608
0.25 0.2489999999 0.2419999996 0.2229999823 0.1859997117 0.1249965911
0 0 0 2 1010 1.937 107 3.1736 106
0 1 1010 3 1010 4.7 1010 1.652 107 3.3511 106
0 0 6 1010 3.963 108 4.459 107 8.392 107
0 0 5 1010 1.75 108 2.881 107 3.4086 106
Fig. 3. The numerical results and exact solutions of Example 3.
Fig. 4. The numerical results and exact solutions of Example 3.
4. Conclusion In this work we had used homotopy perturbation method for solving linear and non-linear systems of Volterra integral equations of the first kind. For the sake of simplicity in explanation of the method the linear integral equations are considered, but examples are illustrated for non-linear systems. 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 strong ones.
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J. Biazar et al. / Chaos, Solitons and Fractals 42 (2009) 3020–3026
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