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The solution of Burgers’ and good Boussinesq equations using ADM–Padé technique
Chaos, Solitons and Fractals 32 (2007) 1008–1026 www.elsevier.com/locate/chaos
The solution of Burgers’ and good Boussinesq equations using ADM–Pade´ technique Tamer A. Abassy a, Magdy A. El-Tawil b
b,*
, Hassan K. Saleh
b
a Department of Basic Science, Benha Higher Institute of Technology, Benha, 13512, Egypt Department of Engineering Mathematics and Physics, Faculty of Engineering, Cairo University, Giza, Egypt
Accepted 10 November 2005
Communicated by Prof. Ji-Huan He
Abstract ADM–Pade´ technique is a combination of Adomian decomposition method (ADM) and Pade´ approximants. It is an approximate method, which can be adapted to solve nonlinear partial differential equations. In this paper, we solve Burgers’ and Boussinesq equation using ADM–Pade´ technique which gives the approximate solution with faster convergence rate and higher accuracy than using ADM alone. Ó 2005 Elsevier Ltd. All rights reserved.
1. Introduction The Burgers’ equation [24] ut þ uux uxx ¼ 0;
x 2 R;
uðx; 0Þ ¼ f ðxÞ;
ð1Þ
serves as a useful model for many interesting problems in applied mathematics. It models effectively certain problems of a fluid flow nature, in which either shocks or viscous dissipation is a significant factor. The first steady-state solutions of Burgers’ equation were given by Bateman [7] in 1915. However, the equation gets its name from the extensive research of Burgers [9] beginning in 1939. It can be used as a model for any nonlinear wave propagation problem subject to dissipation [14]. Depending on the problem being modeled, this dissipation may result from viscosity, heat conduction, mass diffusion, thermal radiation, chemical reaction, or other source. Burgers’ equation has not been solved in closed form, but for many combinations of initial and boundary conditions, an exact solution can easily be found. Hopf [24] and Cole [10] discovered a transformation that reduces the Burgers’ equation to the linear heat equation. However, these solutions are restricted, in most cases, to the infinite interval 1 < x < 1 (see [14]). Lyle used finite element in solving Burger’ equation [27].
*
Corresponding author. Tel.: +20 2 5625129; fax: +20 2 5723486. E-mail addresses: [email protected] (T.A. Abassy), [email protected] (M.A. El-Tawil).
0960-0779/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.11.029
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An example of a solitary wave producing equation is the Boussinesq equation utt ¼ uxx þ uxxxx þ ðu2 Þxx ;
x 2 R.
ð2Þ
This equation is known as the ‘‘bad’’ Boussinesq equation, has been studied by Hirota [22]. The related equation utt ¼ uxx uxxxx þ ðu2 Þxx ;
x2R
ð3Þ
is the ‘‘good’’ Boussinesq equation [28]. The constrain associated with the Boussinesq equations (2) and (3) are assumed to have the form uðx; 0Þ ¼ f ðxÞ;
ut ðx; 0Þ ¼ gðxÞ.
ð4Þ
The Boussinesq equation describes motions of long waves in shallow water under gravity and in a one-dimensional nonlinear lattice [21–23]. Eq. (2) is of special interest [11,12] because it admits inverse scattering formulation. A great deal of research work has been invested in recent years for the study of the Boussinesq equation. Ablowitz and Segur [3] implemented the inverse scattering transform method to handle the nonlinear equations of physical significance where soliton solutions were developed. Hirota [21–23] constructed the N-soliton solutions of the evolution equations by reducing it to the bilinear form. On the other hand, Nimmo and Freeman [31,32] introduced an alternative formulation of the N-soliton solutions in terms of some function of the Wronskian determinant of N functions. Recently, Kaptsov [26] implemented Hirota’s bilinear representation to construct a new set of exact solutions of Boussinesq equation. In particular, solutions that describe the propagation of wave packets and their interaction are obtained in [26,5]. The approach of Kaptsov [26] and Andreev et al. [5] introduced an efficient algorithm to handle the Boussinesq equation and to develop multisoliton solutions. Bratsos [8] approached the good Boussinesq equation and bad Boussinesq equation where the method of lines has been used to transform the equation into a first-order, nonlinear initial value problem. Manoranjan et al. [29] obtained a closed form solution for the two solitons interactions of Eq. (3) and carried out numerical experiments based on the Petrov–Galerkin method with linear ‘‘hat’’ functions to demonstrate the possibility of the break-up of an initial pulse into two solitons. In [30] it has been shown that the ‘‘good’’ Boussinesq equation possesses a highly complicated mechanism for the solitary waves interaction. The nonlinear stability and the convergence of some simple finite-difference schemes for the numerical solution of problems involving the ‘‘good’’ Boussinesq equation are presented in [33]. Most recently, Wazwaz [35] constructed the solution of Eq. (2) in the form of truncated Taylor series by using Adomian decomposition method ADM. Ji-Huan He suggests some other modified Adomian method in his review article [15], and the comparison of Adomian method and variational iteration method [16,17], for more details see [16–20]. ADM has received much attention in recent years. In most cases, we can not rely on the ADM in obtaining solutions in large intervals because the series solution we obtained using ADM converges in a limited interval [2] and outside it, high errors is obtained (see [1]). In this paper, we used the Pade´ approximants with ADM in overcoming this drawback. The ADM method together with Pade´ approximants (ADM–Pade´ technique) extends the domain of solution and gives better accuracy and better convergence than using ADM alone. This link is used before with ordinary differential equations (see [25,34]) and partial differential equations (see [1]). Mathematica is used powerfully in obtaining the different approximations. Different case studies and figures are illustrated to examine the method of analysis.
2. Analysis ADM method is used first to obtain the approximate truncated series solution then the Pade´ approximants is used to obtain an equivalent rational function approximation or the closed form solution (see [1]). 2.1. Applying ADM on Burgers’ equation Following the analysis of Adomian [4], Eq. (1) can be re-written in an operator form as the following: Lu Ru þ Nu ¼ 0;
x 2 R;
uðx; 0Þ ¼ f ðxÞ;
ð5Þ
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where the differential operator L and R are o ; ot 2 o R¼ 2; ox
L¼
ð6Þ ð7Þ
and the nonlinear term is ð8Þ
Nu ¼ uux .
The Adomian decomposition method assumes that the unknown function u(x, t) can be expressed by an infinite series of the form 1 X uðx; tÞ ¼ un ðx; tÞ; ð9Þ n¼0
and the nonlinear operator Nu can be decomposed by an infinite series of polynomials given by 1 X An ; Nu ¼
ð10Þ
n¼0
where the components un(x, t) will be determined recurrently and An are the so-called Adomian polynomials of u0, u1, . . ., un defined by " !# 1 X 1 dn i An ¼ N k ui ; n ¼ 0; 1; 2; 3; . . . ; ð11Þ n! dkn i¼0 k¼0
where N(u) = uux, these polynomials can be constructed for all nonlinearity according to algorithms set by Adomian [4]. The inverse operator L1 is an integral operator given by Z t ðÞ dt. ð12Þ L1 ðÞ ¼ 0
Applying L1 on Eq. (5) and using the constrain we find uðx; tÞ ¼ f ðxÞ þ L1 ðNu þ RuÞ.
ð13Þ
Substituting Eqs. (9) and (10) into Eq. (13) gives 1 X
un ðx; tÞ ¼ f ðxÞ þ L1
n¼0
1 X
An þ R
n¼0
1 X
!! un
ð14Þ
;
n¼0
where An are Adomian polynomials that represent the nonlinear term uux and given by A0 ¼ u0x u0 ; A1 ¼ u0x u1 þ u1x u0 ; A2 ¼ u0x u2 þ u1x u1 þ u2x u0 ; ð15Þ
A3 ¼ u0x u3 þ u1x u2 þ u1 u2x þ u3x u0 ; A4 ¼ u0x u4 þ u1x u3 þ u2x u2 þ u3x u1 þ u4x u0 ; .. . other polynomials can be generated in a like manner. The component of un(x, t) follows immediately upon setting: u0 ðx; tÞ ¼ f ðxÞ;
unþ1 ðx; tÞ ¼ L1 ðAn þ Run Þ;
n P 0.
ð16Þ
From these results we obtain the truncated power series, which we use to obtain [L/M] Pade´ approximants. 2.2. Applying ADM on Boussinesq equation Following the analysis of Adomian [4], Eqs. (2) and (3) can be re-written in an operator form as the following: Lu Ru Nu ¼ 0;
x 2 R;
uðx; 0Þ ¼ f ðxÞ;
ut ðx; 0Þ ¼ gðxÞ;
ð17Þ
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where the differential operator L and R are o2 ; ot2 o2 o4 R¼ 2 4; ox ox and the nonlinear term is Nu ¼ ðu2 Þxx . L¼
ð18Þ ð19Þ ð20Þ
1
The inverse operator L is an integral operator given by Z tZ t ðÞ dt dt. L1 ðÞ ¼ 0
ð21Þ
0
Applying L1 on Eq. (17) and using the constrain we find uðx; tÞ ¼ f ðxÞ þ gðxÞt þ L1 ðNu þ RuÞ.
ð22Þ
Substituting Eqs. (9) and (10) into Eq. (22) gives 1 X
1
un ðx; tÞ ¼ f ðxÞ þ gðxÞt þ L
n¼0
1 X n¼0
An þ R
1 X
!! un
ð23Þ
;
n¼0
where An are Adomian polynomials that represent the nonlinear term (u2)xx defined by " !# 1 X 1 dn i An ¼ N k u ; n ¼ 0; 1; 2; 3; . . . ; i n! dkn i¼0
ð24Þ
k¼0
where N(u) = (u2)xx, these polynomials can be constructed for all nonlinearity according to algorithms set by Adomian [4], and given by A0 ¼ 2ðu0x Þ2 þ 2u0 u0xx ; A1 ¼ 4u0x u1x þ 2u1 u0xx þ 2u0 u1xx ; A2 ¼ 2ðu21x þ 2u0x u2x þ u0xx u2 þ u1 u1xx þ u2xx u0 Þ;
ð25Þ
A3 ¼ 2ð2u1x u2x þ 2u0x u3x þ u0xx u3 þ u1xx u2 þ u2xx u1 þ u3xx u0 Þ; .. . other polynomials can be generated in a like manner. The component of un(x, t) follows immediately upon setting: u0 ðx; tÞ ¼ f ðxÞ þ gðxÞt;
unþ1 ðx; tÞ ¼ L1 ðAn þ Run Þ;
n P 0.
ð26Þ
From these results we obtain the truncated power series solution, which we use to obtain [L/M] Pade´ approximants. 2.3. Applying Pade´ approximants on the truncated series solution As we see in the previous two section we will obtain the truncated series solution of order at least (L + M) in t that we will use it to obtain Pade´ [L/M](x, t) approximate solution for u(x, t). We denote the L, M Pade´ approximants to A(x) by L P L ðxÞ ¼ ; ð27Þ M QM ðxÞ where PL(x) is a polynomial of degree at most L and QM(x) is a polynomial of degree at most M. The formal power series 1 X AðxÞ ¼ ai xi ; ð28Þ i¼1
P L ðxÞ ¼ OðxLþMþ1 Þ; AðxÞ QM ðxÞ determines the coefficients of PL(x) and QM(x) by the equation.
ð29Þ
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Since we can obviously multiply the numerator and denominator by constant and leave [L/M] unchanged, we impose the normalization condition QM ð0Þ ¼ 1:0. Finally we require that PL(x) and QM(x) have no common factors. If we write the coefficients of PL(x) and QM(x) as P L ðxÞ ¼ p0 þ p1 x þ p2 x2 þ þ pL xL ; QM ðxÞ ¼ q0 þ q1 x þ q2 x2 þ þ qM xM .
ð30Þ
ð31Þ
Then by (30) and (31) we may multiply (29) by QM(x), which linearizes the coefficient equations. We can write out (29) in more detail as 9 aLþ1 þ aL q1 þ þ aLMþ1 qM ¼ 0; > > > > aLþ2 þ aLþ1 q1 þ þ aLMþ2 qM ¼ 0; > = ð32Þ .. > > . > > > ; aLþM þ aLþm1 q1 þ þ aL qM ¼ 0; 9 a0 ¼ p0 ; > > > > > a1 þ a0 q1 ¼ p1 ; > > > = a2 þ a1 q1 þ a0 q2 ¼ p2 ; ð33Þ > > > .. > > . > > > ; aL þ aL1 q1 þ þ a0 qL ¼ pL . To solve these equations, starts with Eqs. (32), which is a set of linear equations for all the unknown q’s. Once the q’s are known, then Eqs. (33) gives an explicit formula for the unknown p’s, which complete the solution. If Eqs. (32) and (33) are nonsingular, then we can solve them directly and obtain Eq. (34) [6], where Eq. (34) holds and, if the lower index on a sum exceeds the upper, the sum is replaced by zero: aLMþ1 aLMþ2 aLþ1 .. .. .. .. . . . . det a a a L Lþ1 LþM PL j j PL ajM xj PL a x a x L j¼M j¼M 1 jMþ1 j¼0 j ð34Þ ; ¼ aLMþ1 aLMþ2 aLþ1 M .. .. .. .. . . . . det aL a a Lþ1 LþM xM xM1 1 To obtain diagonal Pade´ approximants of different order like, [2/2], [4/4] or [6/6] we can use Mathematica.
3. Case studies In the following, two test case studies are solved to illustrate the efficiency of the ADM–Pade´ technique. 3.1. Case study 1 Consider Eq. (1) with the following constraint: uðx; 0Þ ¼
k kx 1 tanh . 2 4
ð35Þ
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Substituting Eq. (35) in Eq. (16), we obtain the following resulting components: k kx u0 ðx; tÞ ¼ 1 tanh ; 2 4 k3 kx u1 ðx; tÞ ¼ sech2 t; 16 4 k5 kx kx 2 sech2 u2 ðx; tÞ ¼ tanh t ; 128 4 4 k7 kx kx 2 þ cosh t3 ; u3 ðx; tÞ ¼ sech4 4 2 3072 k9 kx kx 2kx 11 sinh þ sinh t4 ; u4 ðx; tÞ ¼ sech5 4 4 4 98304 k 11 kx kx 33 26 cosh þ cosh½kx t5 ; u5 ðx; tÞ ¼ sech6 4 2 3932160 k 13 kx kx 3kx 5kx sech7 302 sinh 75 sinh þ sinh t6 ; u6 ðx; tÞ ¼ 188743680 4 4 4 4 k 15 kx kx 3kx u7 ðx; tÞ ¼ sech8 1208 þ 1191 cosh 120 cosh ½kx þ cosh t7 ; 1569646080 4 2 2 k 17 kx kx 3kx 5kx 7kx u8 ðx; tÞ ¼ sech9 15619 sinh þ 4293 sinh 247 sinh þ sinh t8 . 676457349120 4 4 4 4 4 Considering these components, the solution can be approximated as n X um ðx; tÞ. /n ðx; tÞ ¼
1013
ð36Þ
ð37Þ
m¼0
It is known that the exact solution of this problem is [13] k k k 1 tanh x t . uðx; tÞ ¼ 2 4 2
ð38Þ
For k = 1, ADM truncated series solution (Eq. (37) with n = 8) gives good approximation in small interval of convergence. This is illustrated in Fig. 1 which shows ADM truncated series solution /8(x, t) compared with the exact solution at x = 0. The error between the exact solution and the ADM truncated series solution /8(x, t) at x = 0 is shown in Fig. 2, where we can notice high errors beyond the small interval. Using ADM–Pade´ approximation at x = 0, the rational approximations [2/2] and [4/4] take the form of Eqs. (39) and (40), respectively:
Fig. 1. The truncated ADM series solution /8(x, t) and the exact solution u(x, t) for Burgers’ equation at k = 1 and x = 0.
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Fig. 2. The error between the exact solution u(x, t) and the truncated series solution /8(x, t) for Burgers’ equation at k = 1 and x = 0.
2 0:5 þ 0:625t þ 0:00260417t2 ; ð0; tÞ ¼ 1 þ 0:00520833t2 2 4 0:5 þ 0:0625t þ 0:00334821t2 þ 0:000093t3 þ 1:16257 106 t4 ð0; tÞ ¼ . 4 1 þ 0:00669643t2 þ 2:32515 106 t4
ð39Þ ð40Þ
Fig. 3. The ADM–Pade´ rational approximation [2/2](x, t) and [4/4](x, t) for Burgers’ equation at k = 1 and x = 0.
Fig. 4. The error in ADM–Pade´ between the exact solution u(x, t) and the rational approximation [2/2](x, t) and [4/4](x, t) for Burgers’ equation at k = 1 and x = 0.
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The results are shown in Fig. 3 which illustrate how much the results are improved at x = 0. The error between the exact solution u(x, t) and [2/2](x, t) and [4/4](x, t) at x = 0 is also illustrated in Fig. 4 where we can notice that [4/4](x, t) is better than [2/2](x, t) and both are more reliable than ADM alone. Repeating the previous calculations at x = 1, we obtain the following rational approximations [2/2](x, t) and [4/4](x, t):
Fig. 5. The truncated ADM series solution /8(x, t) and the exact solution u(x, t) for Burgers’ equation at k = 1 and x = 1.
Fig. 6. The error between the exact solution u(x, t) and the truncated series solution /8(x, t) for Burgers’ equation at k = 1 and x = 1.
Fig. 7. The ADM–Pade´ rational approximation [2/2](x, t) and [4/4](x, t) for Burgers’ equation at k = 1 and x = 1.
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2 0:37754 þ 0:0471926t þ 0:00196636t2 ; ð1; tÞ ¼ 1 0:0306148t þ 0:00520833t2 2 4 0:37754 þ 0:04719t 0:00253t2 0:00007023t3 þ 8:778 107 t4 . ð1; tÞ ¼ 4 1 0:0306148t þ 0:0066964t2 0:00004556t3 þ 2:3252 106 t4
ð41Þ ð42Þ
Figs. 5–8 show how much the results improved after using ADM–Pade´ approximation. Also, Figs. 9–12 show the three-dimensional plot of /8(x, t), o/ot(/8(x, t)), the Pade´ [4/4](x, t) and o/ot([4/4](x, t)) in the interval indicated in the figures, respectively. Figs. 9 and 10 show the deterioration in the ADM solution beyond the interval of convergence. Figs. 11 and 12 show the improvement in results after using ADM–Pade´ approximation. It is clear that the interval of convergence has increased. Fig. 12 shows solitary wave of Burgers’ equation. Figs. 13 and 14 shows The approximate solution [4/4](x, t) and o/ot([4/4](x, t)), respectively, for Burgers’ equation at different time level.
Fig. 8. The error in ADM–Pade´ between the exact solution u(x, t) and the rational approximations [2/2](x, t) and [4/4](x, t) for Burgers’ equation at k = 1 and x = 1.
Fig. 9. The surface generated from the truncated series solution /8(x, t) for Burgers’ equation in the interval 15 < t < 15 and 15 < x < 15 at k = 1.
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3.2. Case study 2 Consider ‘‘good’’ Boussinesq equation (3) with the following constraint pffiffiffiffiffiffiffiffiffiffiffiffiffi h i hcxi hcxi 3c2 3c3 1 þ c2 2 cx ¼ f ðxÞ; ut ðx; 0Þ ¼ tanh ¼ gðxÞ. sech sech2 uðx; 0Þ ¼ 2 2 2 2 2
ð43Þ
Fig. 10. The surface generated from o/ot(/8(x, t)) for Burgers’ equation in the interval 5 < t < 5 and 5 < x < 5 at k = 1.
Fig. 11. The surface generated from the approximate solution [4/4](x, t) for Burgers’ equation in the interval 20 < t < 20 and 20 < x < 20 at k = 1.
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Substituting Eq. (43) in Eq. (26), we obtain the components un(x, t) then the solution can be approximated as n X um ðx; tÞ. /n ðx; tÞ ¼
ð44Þ
m¼0
Eq. (44) with n = 4 gives ADM truncated series solution /4(x, t) hcxi 3 pffiffiffiffiffiffiffiffiffiffiffiffi hcxi hcxi 3 hcxi 3 /4 ðx;tÞ ¼ c2 sech2 þ c3 1 c2 sech2 tanh t þ c4 ð1 þ c2 Þð2 þ cosh ½cxÞsech4 t2 2 2 2 2 2 8 2 hcxi hcxi 3 1 3cx 1 6 c5 ð1 c2 Þ2 sech5 11sinh þ sinh t3 c ð1 þ c2 Þ2 ð33 26 cosh ½cx þ 16 2 2 2 128 hcxi hcxi hcxi 5 1 3cx 5cx t4 þ c7 ð1 c2 Þ2 sech7 302sinh 57 sinh þ sinh t5 þ cosh ½2cxÞsech6 2 1280 2 2 2 2 hcxi 1 c8 ð1 þ c2 Þ3 ð1208 þ 1191 cosh ½cx 120cosh ½2cx þ cosh ½3cxÞsech8 t6 þ 15360 2 hcxi hcxi 7 1 3cx 5cx 7cx c9 ð1 c2 Þ2 sech9 15619sinh þ 4293sinh 247sinh þ sinh t7 215040 2 2 2 2 2 1 þ c10 ð1 þ c2 Þ4 ð78095 88234cosh ½cx þ 14608cosh ½2cx 502cosh ½3cx 3440640 hcxi hcxi hcxi 9 1 3cx t8 þ c11 ð1 c2 Þ2 sech11 1310354 sinh 455192sinh þ cosh ½4cxÞsech10 2 61931520 2 2 2 5cx 7cx 9cx 1013sinh þ sinh t9 þ O½t10 . þ47840sinh ð45Þ 2 2 2 It is known that the exact solution of this problem is [24] hc pffiffiffiffiffiffiffiffiffiffiffiffiffi i 3c2 sech2 ðx þ 1 c2 tÞ . uðx; tÞ ¼ 2 2
ð46Þ
Following the same procedure as we do in the Burgers’ equation on the Boussinesq equation, with c = 0.5 and ADM truncated series solution (Eq. (44) with n = 4), we obtained the following results and figures. The results and Figs. 15–25 enhance the conclusion that the ADM–Pade´ technique gives very good results than using ADM alone. Using ADM–Pade´ approximation at x = 0, the rational approximations [2/2] and [4/4] take the form of Eqs. (47) and (48), respectively:
Fig. 12. The surface generated from o/ot([4/4](x, t)) for Burgers’ equation in the interval 18 < t < 18 and 18 < x < 18 at k = 1.
T.A. Abassy et al. / Chaos, Solitons and Fractals 32 (2007) 1008–1026
Fig. 13. The approximate solution [4/4](x, t) for Burgers’ equation at k = 1 and different time level.
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Fig. 14. Plot of o/ot([4/4](x, t)) for Burgers’ equation at k = 1 and different time level.
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Fig. 15. The truncated ADM series solution /4(x, t) and the exact solution u(x, t) for good Boussinesq equation at c = 0.5 and x = 0.
Fig. 16. The error between the exact solution u(x, t) and the truncated series solution /4(x, t) for good Boussinesq equation at c = 0.5 and x = 0.
2 0:375 þ 0:00585938t2 ð0; tÞ ¼ ; 2 1 þ 0:03125t2 4 0:375 þ 0:003069t2 þ 0:000011335t4 ð0; tÞ ¼ . 4 1 þ 0:0386905t2 þ 0:000378999t4
ð47Þ ð48Þ
The results are shown in Fig. 17 which illustrate how much the results are improved at x = 0. The error between the exact solution u(x, t) and [2/2](x, t) and [4/4](x, t) at x = 0 is also illustrated in Fig. 18 where we can notice that [4/4](x, t) is better than [2/2](x, t) and both are more reliable than ADM alone. Repeating the previous calculations at x = 1, we obtain the following rational approximations [2/2](x, t) and [4/4](x, t): 2 0:35251 þ 0:013708t þ 0:00459358t2 ð1; tÞ ¼ ; ð49Þ 2 1 þ 0:0671662t þ 0:0325315t2 4 0:35251 þ 0:0071633t þ 0:002694t2 0:000053t3 8:6 106 t4 . ð50Þ ð1; tÞ ¼ 1 þ 0:0857319t þ 0:039889t2 þ 0:0016437t3 þ 0:00036964t4 4 Figs. 19–22 show how much the results improved after using ADM–Pade´ approximation. Also, Figs. 23 and 24 show the three-dimensional plot of /4(x, t) and the Pade´ [4/4](x, t) in the interval indicated in figures. Fig. 23 shows the deterioration in the ADM solution beyond the interval of convergence. Fig. 24 shows the improvement in results after using ADM–Pade´ approximation. It is clear that the interval of convergence has increased. Fig. 25 shows the solution at deferent time level.
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Fig. 17. The ADM–Pade´ rational approximation [2/2](x, t) and [4/4](x, t) for good Boussinesq equation at c = 0.5 and x = 0.
Fig. 18. The error in ADM–Pade´ between the exact solution u(x, t) and the rational approximation [2/2](x, t) and [4/4](x, t) for good Boussinesq equation at c = 0.5 and x = 0.
Fig. 19. The truncated ADM series solution /4(x, t) and the exact solution u(x, t) for good Boussinesq equation at c = 0.5 and x = 1.
Note. If we use the ‘‘bad’’ Boussinesq equation with the constrain pffiffiffiffiffiffiffiffiffiffiffiffiffi h i hcxi hcxi 3c2 3c2 1 þ c2 2 cx sech uðx; 0Þ ¼ sech2 ¼ f ðxÞ; ut ðx; 0Þ ¼ tanh ¼ gðxÞ. 2 2 2 2 2
ð51Þ
We will obtain the same results as in ‘‘good’’ Boussinesq equation but with no limit on the range of the constant ‘‘c’’.
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Fig. 20. The error between the exact solution u(x, t) and the truncated series solution /4(x, t) for good Boussinesq equation at c = 0.5 and x = 1.
Fig. 21. The ADM–Pade´ rational approximation [2/2](x, t) and [4/4](x, t) for good Boussinesq equation at c = 0.5 and x = 1.
Fig. 22. The error in ADM–Pade´ between the exact solution u(x, t) and the rational approximation [2/2](x, t) and [4/4](x, t) for good Boussinesq equation at c = 0.5 and x = 1.
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Fig. 23. The surface generated from the truncated series solution /4(x, t) for good Boussinesq equation in the interval 3 < t < 3 and 3 < x < 3 at c = 0.5.
Fig. 24. Shows the surface generated from the approximate solution [4/4](x, t) for good Boussinesq equation in the interval 10 < t < 10 and 20 < x < 20 at c = 0.5.
4. Conclusion From the previous results and the results of Abassy [1] we conclude that using ADM–Pade´ technique in solving solitary wave equations is very efficient than using ADM alone. For obtaining more accurate results and larger intervals of convergence we increase the order of Pade´ approximants we use like [6/6] and [8/8]. The ADM–Pade´ technique can be
T.A. Abassy et al. / Chaos, Solitons and Fractals 32 (2007) 1008–1026
1025
Fig. 25. The approximate solution [4/4](x, t) for good Boussinesq equation at c = 0.5 and different time level.
used in solving large scale of nonlinear partial differential equations. The use of Mathematica facilitates the calculations of the ADM–Pade´ technique.
References [1] Abassy TA, El-Tawil M, Kamel H. The solution of KdV and mKdV equations using Adomian Pade´ approximation. Int J Nonlinear Sci Numer Simulat 2004;5(4):327–39. [2] Abboui K, Cherruault Y. Convergence of Adomian method applied to differential equations. Comput Math Appl 1994;28(5):103–9.
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[3] Ablowitz MJ, Segur H. Solitons and the inverse scattering transform. Philadelphia (PA): SIAM; 1981. [4] Adomian G. Solving frontier problem of physics: the decomposition method. Boston (MA): Kluwer Academic Publishers; 1994. [5] Andreev VK, Kabstov OV, Pukhnachov VV, Rodionov AA. Applications of group-theoretical methods in hydrodynamics. Boston: Kluwer Academic Publishers; 1998. [6] Baker Jr GA. Essentials of Pade` approximants. London: Academic Press; 1975. [7] Bateman H. Some recent researches on the motion of fluids. Mon Weather Rev 1915;43. [8] Bratsos AG. The solution of the Boussinesq equation using the method of lines. Comput Meth Appl Mech Eng 1998;157:33–44. [9] Burgers JM. Mathematical examples illustrating relations occurring in the theory of turbulent fluid motion. Trans Roy Neth Acad Sci, Amsterdam 1939;17:1–53. [10] Cole JD. On a quasi-linear parabolic equation occurring in aerodynamics. Quart Appl Math 1951;9:225–36. [11] Debnath L. Nonlinear partial differential equations for scientists and engineers. Berlin: Brikhauser; 1998. [12] Debnath L. Nonlinear water waves. Boston: Academic Press; 1994. [13] Drazin PG, Jonson RS. Soliton: an introduction. New York: Cambridge University Press; 1993. [14] Fletcher CAJ. Burgers’ equation: a model for all reasons. In: Noye J, editor. Numerical solutions of partial differential equations. Amsterdam: North-Holland Pub. Co.; 1982. [15] He JH. A review on some new recently developed nonlinear analytical techniques. Int J Nonlinear Sci Numer Simulat 2000;1(1):51–70. [16] He JH. Variational iteration method: a kind of nonlinear analytical technique: some examples. Int J Nonlinear Mech 1999;34(4):699–708. [17] He JH. Variational iteration method for autonomous ordinary differential system. Appl Math Comput 2000;114(2/3):115–23. [18] He JH, Wan YQ, Guo Q. An iteration formulation for normalized diode characteristics. Int J Circ Theory Appl 2004;32(6):629–32. [19] He JH. Approximate analytical solution for seepage flow with fractional derivatives in porous media. Comput Meth Appl Mech Eng 1998;167(1–2):57–68. [20] He JH. Approximate solution of nonlinear differential equations with convolution product nonlinearities. Comput Meth Appl Mech Eng 1998;167(1–2):69–73. [21] Hirota R. Direct methods in soliton theory. In: Bullogh RK, Caudrey PJ, editors. Solitons. Berlin: Springer; 1980. [22] Hirota R. Exact envelope-soliton solutions of a nonlinear wave. J Math Phys 1973;14(7):805–9. [23] Hirota R. Exact N-soliton solutions of the wave equation of long waves in shallow water and in nonlinear lattices. J Math Phys 1973;14(7):810–4. [24] Hopf E. The partial differential equation ut + uux = luxx. Commun Pure Appl Math 1950;3:201–30. [25] Jiao YC, Yamamoto Y, Dang C, Hao Y. An after treatment technique for improving the accuracy of Adomian’s decomposition method. Comput Math Appl 2002;43:783–98. [26] Kabstov OV. Construction of exact solutions of the Boussinesq equation. J Appl Mech Tech Phys 1998;39(3):389–92. [27] Lyle CS. Finite element approximations of Burgers’ equation with Robin’s boundary conditions. MSc thesis, The Faculty of the Virginia Polytechnic Institute and State University, 1997. [28] Mckean HP. Boussinesq equation on the circle. Commun Pure Appl Math 1981;34:599–691. [29] Manoranjan VS, Mitchell AR, Morris JL. Numerical solutions of the ‘‘good’’ Boussinesq equation. SIAM J Sci Stat Comput 1984;5:946–57. [30] Manoranjan VS, Ortega T, Sang-Seran JM. Soliton and antisoliton interactions in the ‘‘good’’ Boussinesq equation. J Math Phys 1988;29:1964–8. [31] Nimmo JJC, Freeman NC. A method of obtaining the n-soliton solutions of the Boussinesq equation in terms of Wronskian. Phys Lett A 1983;95:4–6. [32] Nimmo JJC, Freeman NC. The use of Ba¨cklund transformations in obtaining the N-soliton solutions in Wronskian form. J Phys A 1984;17:1415–24. [33] Ortega T, Sang-Seran JM. Nonlinear stability and convergence of finite-difference methods for the ‘‘good’’ Boussinesq equation. Numer Math 1990;58:215–29. [34] Wazwaz AM. Analytical approximations and Pade´ approximants for Volterra’s population model. Appl Math Comput 1999;100:13–25. [35] Wazwaz AM. Construction of soliton solutions and periodic solutions of the Boussinesq equation by modified decomposition method. Chaos, Solitons & Fractals 2001;12:1549–56.
The solution of Burgers’ and good Boussinesq equations using ADM–Pade´ technique Tamer A. Abassy a, Magdy A. El-Tawil b
b,*
, Hassan K. Saleh
b
a Department of Basic Science, Benha Higher Institute of Technology, Benha, 13512, Egypt Department of Engineering Mathematics and Physics, Faculty of Engineering, Cairo University, Giza, Egypt
Accepted 10 November 2005
Communicated by Prof. Ji-Huan He
Abstract ADM–Pade´ technique is a combination of Adomian decomposition method (ADM) and Pade´ approximants. It is an approximate method, which can be adapted to solve nonlinear partial differential equations. In this paper, we solve Burgers’ and Boussinesq equation using ADM–Pade´ technique which gives the approximate solution with faster convergence rate and higher accuracy than using ADM alone. Ó 2005 Elsevier Ltd. All rights reserved.
1. Introduction The Burgers’ equation [24] ut þ uux uxx ¼ 0;
x 2 R;
uðx; 0Þ ¼ f ðxÞ;
ð1Þ
serves as a useful model for many interesting problems in applied mathematics. It models effectively certain problems of a fluid flow nature, in which either shocks or viscous dissipation is a significant factor. The first steady-state solutions of Burgers’ equation were given by Bateman [7] in 1915. However, the equation gets its name from the extensive research of Burgers [9] beginning in 1939. It can be used as a model for any nonlinear wave propagation problem subject to dissipation [14]. Depending on the problem being modeled, this dissipation may result from viscosity, heat conduction, mass diffusion, thermal radiation, chemical reaction, or other source. Burgers’ equation has not been solved in closed form, but for many combinations of initial and boundary conditions, an exact solution can easily be found. Hopf [24] and Cole [10] discovered a transformation that reduces the Burgers’ equation to the linear heat equation. However, these solutions are restricted, in most cases, to the infinite interval 1 < x < 1 (see [14]). Lyle used finite element in solving Burger’ equation [27].
*
Corresponding author. Tel.: +20 2 5625129; fax: +20 2 5723486. E-mail addresses: [email protected] (T.A. Abassy), [email protected] (M.A. El-Tawil).
0960-0779/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.11.029
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An example of a solitary wave producing equation is the Boussinesq equation utt ¼ uxx þ uxxxx þ ðu2 Þxx ;
x 2 R.
ð2Þ
This equation is known as the ‘‘bad’’ Boussinesq equation, has been studied by Hirota [22]. The related equation utt ¼ uxx uxxxx þ ðu2 Þxx ;
x2R
ð3Þ
is the ‘‘good’’ Boussinesq equation [28]. The constrain associated with the Boussinesq equations (2) and (3) are assumed to have the form uðx; 0Þ ¼ f ðxÞ;
ut ðx; 0Þ ¼ gðxÞ.
ð4Þ
The Boussinesq equation describes motions of long waves in shallow water under gravity and in a one-dimensional nonlinear lattice [21–23]. Eq. (2) is of special interest [11,12] because it admits inverse scattering formulation. A great deal of research work has been invested in recent years for the study of the Boussinesq equation. Ablowitz and Segur [3] implemented the inverse scattering transform method to handle the nonlinear equations of physical significance where soliton solutions were developed. Hirota [21–23] constructed the N-soliton solutions of the evolution equations by reducing it to the bilinear form. On the other hand, Nimmo and Freeman [31,32] introduced an alternative formulation of the N-soliton solutions in terms of some function of the Wronskian determinant of N functions. Recently, Kaptsov [26] implemented Hirota’s bilinear representation to construct a new set of exact solutions of Boussinesq equation. In particular, solutions that describe the propagation of wave packets and their interaction are obtained in [26,5]. The approach of Kaptsov [26] and Andreev et al. [5] introduced an efficient algorithm to handle the Boussinesq equation and to develop multisoliton solutions. Bratsos [8] approached the good Boussinesq equation and bad Boussinesq equation where the method of lines has been used to transform the equation into a first-order, nonlinear initial value problem. Manoranjan et al. [29] obtained a closed form solution for the two solitons interactions of Eq. (3) and carried out numerical experiments based on the Petrov–Galerkin method with linear ‘‘hat’’ functions to demonstrate the possibility of the break-up of an initial pulse into two solitons. In [30] it has been shown that the ‘‘good’’ Boussinesq equation possesses a highly complicated mechanism for the solitary waves interaction. The nonlinear stability and the convergence of some simple finite-difference schemes for the numerical solution of problems involving the ‘‘good’’ Boussinesq equation are presented in [33]. Most recently, Wazwaz [35] constructed the solution of Eq. (2) in the form of truncated Taylor series by using Adomian decomposition method ADM. Ji-Huan He suggests some other modified Adomian method in his review article [15], and the comparison of Adomian method and variational iteration method [16,17], for more details see [16–20]. ADM has received much attention in recent years. In most cases, we can not rely on the ADM in obtaining solutions in large intervals because the series solution we obtained using ADM converges in a limited interval [2] and outside it, high errors is obtained (see [1]). In this paper, we used the Pade´ approximants with ADM in overcoming this drawback. The ADM method together with Pade´ approximants (ADM–Pade´ technique) extends the domain of solution and gives better accuracy and better convergence than using ADM alone. This link is used before with ordinary differential equations (see [25,34]) and partial differential equations (see [1]). Mathematica is used powerfully in obtaining the different approximations. Different case studies and figures are illustrated to examine the method of analysis.
2. Analysis ADM method is used first to obtain the approximate truncated series solution then the Pade´ approximants is used to obtain an equivalent rational function approximation or the closed form solution (see [1]). 2.1. Applying ADM on Burgers’ equation Following the analysis of Adomian [4], Eq. (1) can be re-written in an operator form as the following: Lu Ru þ Nu ¼ 0;
x 2 R;
uðx; 0Þ ¼ f ðxÞ;
ð5Þ
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where the differential operator L and R are o ; ot 2 o R¼ 2; ox
L¼
ð6Þ ð7Þ
and the nonlinear term is ð8Þ
Nu ¼ uux .
The Adomian decomposition method assumes that the unknown function u(x, t) can be expressed by an infinite series of the form 1 X uðx; tÞ ¼ un ðx; tÞ; ð9Þ n¼0
and the nonlinear operator Nu can be decomposed by an infinite series of polynomials given by 1 X An ; Nu ¼
ð10Þ
n¼0
where the components un(x, t) will be determined recurrently and An are the so-called Adomian polynomials of u0, u1, . . ., un defined by " !# 1 X 1 dn i An ¼ N k ui ; n ¼ 0; 1; 2; 3; . . . ; ð11Þ n! dkn i¼0 k¼0
where N(u) = uux, these polynomials can be constructed for all nonlinearity according to algorithms set by Adomian [4]. The inverse operator L1 is an integral operator given by Z t ðÞ dt. ð12Þ L1 ðÞ ¼ 0
Applying L1 on Eq. (5) and using the constrain we find uðx; tÞ ¼ f ðxÞ þ L1 ðNu þ RuÞ.
ð13Þ
Substituting Eqs. (9) and (10) into Eq. (13) gives 1 X
un ðx; tÞ ¼ f ðxÞ þ L1
n¼0
1 X
An þ R
n¼0
1 X
!! un
ð14Þ
;
n¼0
where An are Adomian polynomials that represent the nonlinear term uux and given by A0 ¼ u0x u0 ; A1 ¼ u0x u1 þ u1x u0 ; A2 ¼ u0x u2 þ u1x u1 þ u2x u0 ; ð15Þ
A3 ¼ u0x u3 þ u1x u2 þ u1 u2x þ u3x u0 ; A4 ¼ u0x u4 þ u1x u3 þ u2x u2 þ u3x u1 þ u4x u0 ; .. . other polynomials can be generated in a like manner. The component of un(x, t) follows immediately upon setting: u0 ðx; tÞ ¼ f ðxÞ;
unþ1 ðx; tÞ ¼ L1 ðAn þ Run Þ;
n P 0.
ð16Þ
From these results we obtain the truncated power series, which we use to obtain [L/M] Pade´ approximants. 2.2. Applying ADM on Boussinesq equation Following the analysis of Adomian [4], Eqs. (2) and (3) can be re-written in an operator form as the following: Lu Ru Nu ¼ 0;
x 2 R;
uðx; 0Þ ¼ f ðxÞ;
ut ðx; 0Þ ¼ gðxÞ;
ð17Þ
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where the differential operator L and R are o2 ; ot2 o2 o4 R¼ 2 4; ox ox and the nonlinear term is Nu ¼ ðu2 Þxx . L¼
ð18Þ ð19Þ ð20Þ
1
The inverse operator L is an integral operator given by Z tZ t ðÞ dt dt. L1 ðÞ ¼ 0
ð21Þ
0
Applying L1 on Eq. (17) and using the constrain we find uðx; tÞ ¼ f ðxÞ þ gðxÞt þ L1 ðNu þ RuÞ.
ð22Þ
Substituting Eqs. (9) and (10) into Eq. (22) gives 1 X
1
un ðx; tÞ ¼ f ðxÞ þ gðxÞt þ L
n¼0
1 X n¼0
An þ R
1 X
!! un
ð23Þ
;
n¼0
where An are Adomian polynomials that represent the nonlinear term (u2)xx defined by " !# 1 X 1 dn i An ¼ N k u ; n ¼ 0; 1; 2; 3; . . . ; i n! dkn i¼0
ð24Þ
k¼0
where N(u) = (u2)xx, these polynomials can be constructed for all nonlinearity according to algorithms set by Adomian [4], and given by A0 ¼ 2ðu0x Þ2 þ 2u0 u0xx ; A1 ¼ 4u0x u1x þ 2u1 u0xx þ 2u0 u1xx ; A2 ¼ 2ðu21x þ 2u0x u2x þ u0xx u2 þ u1 u1xx þ u2xx u0 Þ;
ð25Þ
A3 ¼ 2ð2u1x u2x þ 2u0x u3x þ u0xx u3 þ u1xx u2 þ u2xx u1 þ u3xx u0 Þ; .. . other polynomials can be generated in a like manner. The component of un(x, t) follows immediately upon setting: u0 ðx; tÞ ¼ f ðxÞ þ gðxÞt;
unþ1 ðx; tÞ ¼ L1 ðAn þ Run Þ;
n P 0.
ð26Þ
From these results we obtain the truncated power series solution, which we use to obtain [L/M] Pade´ approximants. 2.3. Applying Pade´ approximants on the truncated series solution As we see in the previous two section we will obtain the truncated series solution of order at least (L + M) in t that we will use it to obtain Pade´ [L/M](x, t) approximate solution for u(x, t). We denote the L, M Pade´ approximants to A(x) by L P L ðxÞ ¼ ; ð27Þ M QM ðxÞ where PL(x) is a polynomial of degree at most L and QM(x) is a polynomial of degree at most M. The formal power series 1 X AðxÞ ¼ ai xi ; ð28Þ i¼1
P L ðxÞ ¼ OðxLþMþ1 Þ; AðxÞ QM ðxÞ determines the coefficients of PL(x) and QM(x) by the equation.
ð29Þ
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Since we can obviously multiply the numerator and denominator by constant and leave [L/M] unchanged, we impose the normalization condition QM ð0Þ ¼ 1:0. Finally we require that PL(x) and QM(x) have no common factors. If we write the coefficients of PL(x) and QM(x) as P L ðxÞ ¼ p0 þ p1 x þ p2 x2 þ þ pL xL ; QM ðxÞ ¼ q0 þ q1 x þ q2 x2 þ þ qM xM .
ð30Þ
ð31Þ
Then by (30) and (31) we may multiply (29) by QM(x), which linearizes the coefficient equations. We can write out (29) in more detail as 9 aLþ1 þ aL q1 þ þ aLMþ1 qM ¼ 0; > > > > aLþ2 þ aLþ1 q1 þ þ aLMþ2 qM ¼ 0; > = ð32Þ .. > > . > > > ; aLþM þ aLþm1 q1 þ þ aL qM ¼ 0; 9 a0 ¼ p0 ; > > > > > a1 þ a0 q1 ¼ p1 ; > > > = a2 þ a1 q1 þ a0 q2 ¼ p2 ; ð33Þ > > > .. > > . > > > ; aL þ aL1 q1 þ þ a0 qL ¼ pL . To solve these equations, starts with Eqs. (32), which is a set of linear equations for all the unknown q’s. Once the q’s are known, then Eqs. (33) gives an explicit formula for the unknown p’s, which complete the solution. If Eqs. (32) and (33) are nonsingular, then we can solve them directly and obtain Eq. (34) [6], where Eq. (34) holds and, if the lower index on a sum exceeds the upper, the sum is replaced by zero: aLMþ1 aLMþ2 aLþ1 .. .. .. .. . . . . det a a a L Lþ1 LþM PL j j PL ajM xj PL a x a x L j¼M j¼M 1 jMþ1 j¼0 j ð34Þ ; ¼ aLMþ1 aLMþ2 aLþ1 M .. .. .. .. . . . . det aL a a Lþ1 LþM xM xM1 1 To obtain diagonal Pade´ approximants of different order like, [2/2], [4/4] or [6/6] we can use Mathematica.
3. Case studies In the following, two test case studies are solved to illustrate the efficiency of the ADM–Pade´ technique. 3.1. Case study 1 Consider Eq. (1) with the following constraint: uðx; 0Þ ¼
k kx 1 tanh . 2 4
ð35Þ
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Substituting Eq. (35) in Eq. (16), we obtain the following resulting components: k kx u0 ðx; tÞ ¼ 1 tanh ; 2 4 k3 kx u1 ðx; tÞ ¼ sech2 t; 16 4 k5 kx kx 2 sech2 u2 ðx; tÞ ¼ tanh t ; 128 4 4 k7 kx kx 2 þ cosh t3 ; u3 ðx; tÞ ¼ sech4 4 2 3072 k9 kx kx 2kx 11 sinh þ sinh t4 ; u4 ðx; tÞ ¼ sech5 4 4 4 98304 k 11 kx kx 33 26 cosh þ cosh½kx t5 ; u5 ðx; tÞ ¼ sech6 4 2 3932160 k 13 kx kx 3kx 5kx sech7 302 sinh 75 sinh þ sinh t6 ; u6 ðx; tÞ ¼ 188743680 4 4 4 4 k 15 kx kx 3kx u7 ðx; tÞ ¼ sech8 1208 þ 1191 cosh 120 cosh ½kx þ cosh t7 ; 1569646080 4 2 2 k 17 kx kx 3kx 5kx 7kx u8 ðx; tÞ ¼ sech9 15619 sinh þ 4293 sinh 247 sinh þ sinh t8 . 676457349120 4 4 4 4 4 Considering these components, the solution can be approximated as n X um ðx; tÞ. /n ðx; tÞ ¼
1013
ð36Þ
ð37Þ
m¼0
It is known that the exact solution of this problem is [13] k k k 1 tanh x t . uðx; tÞ ¼ 2 4 2
ð38Þ
For k = 1, ADM truncated series solution (Eq. (37) with n = 8) gives good approximation in small interval of convergence. This is illustrated in Fig. 1 which shows ADM truncated series solution /8(x, t) compared with the exact solution at x = 0. The error between the exact solution and the ADM truncated series solution /8(x, t) at x = 0 is shown in Fig. 2, where we can notice high errors beyond the small interval. Using ADM–Pade´ approximation at x = 0, the rational approximations [2/2] and [4/4] take the form of Eqs. (39) and (40), respectively:
Fig. 1. The truncated ADM series solution /8(x, t) and the exact solution u(x, t) for Burgers’ equation at k = 1 and x = 0.
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Fig. 2. The error between the exact solution u(x, t) and the truncated series solution /8(x, t) for Burgers’ equation at k = 1 and x = 0.
2 0:5 þ 0:625t þ 0:00260417t2 ; ð0; tÞ ¼ 1 þ 0:00520833t2 2 4 0:5 þ 0:0625t þ 0:00334821t2 þ 0:000093t3 þ 1:16257 106 t4 ð0; tÞ ¼ . 4 1 þ 0:00669643t2 þ 2:32515 106 t4
ð39Þ ð40Þ
Fig. 3. The ADM–Pade´ rational approximation [2/2](x, t) and [4/4](x, t) for Burgers’ equation at k = 1 and x = 0.
Fig. 4. The error in ADM–Pade´ between the exact solution u(x, t) and the rational approximation [2/2](x, t) and [4/4](x, t) for Burgers’ equation at k = 1 and x = 0.
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The results are shown in Fig. 3 which illustrate how much the results are improved at x = 0. The error between the exact solution u(x, t) and [2/2](x, t) and [4/4](x, t) at x = 0 is also illustrated in Fig. 4 where we can notice that [4/4](x, t) is better than [2/2](x, t) and both are more reliable than ADM alone. Repeating the previous calculations at x = 1, we obtain the following rational approximations [2/2](x, t) and [4/4](x, t):
Fig. 5. The truncated ADM series solution /8(x, t) and the exact solution u(x, t) for Burgers’ equation at k = 1 and x = 1.
Fig. 6. The error between the exact solution u(x, t) and the truncated series solution /8(x, t) for Burgers’ equation at k = 1 and x = 1.
Fig. 7. The ADM–Pade´ rational approximation [2/2](x, t) and [4/4](x, t) for Burgers’ equation at k = 1 and x = 1.
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T.A. Abassy et al. / Chaos, Solitons and Fractals 32 (2007) 1008–1026
2 0:37754 þ 0:0471926t þ 0:00196636t2 ; ð1; tÞ ¼ 1 0:0306148t þ 0:00520833t2 2 4 0:37754 þ 0:04719t 0:00253t2 0:00007023t3 þ 8:778 107 t4 . ð1; tÞ ¼ 4 1 0:0306148t þ 0:0066964t2 0:00004556t3 þ 2:3252 106 t4
ð41Þ ð42Þ
Figs. 5–8 show how much the results improved after using ADM–Pade´ approximation. Also, Figs. 9–12 show the three-dimensional plot of /8(x, t), o/ot(/8(x, t)), the Pade´ [4/4](x, t) and o/ot([4/4](x, t)) in the interval indicated in the figures, respectively. Figs. 9 and 10 show the deterioration in the ADM solution beyond the interval of convergence. Figs. 11 and 12 show the improvement in results after using ADM–Pade´ approximation. It is clear that the interval of convergence has increased. Fig. 12 shows solitary wave of Burgers’ equation. Figs. 13 and 14 shows The approximate solution [4/4](x, t) and o/ot([4/4](x, t)), respectively, for Burgers’ equation at different time level.
Fig. 8. The error in ADM–Pade´ between the exact solution u(x, t) and the rational approximations [2/2](x, t) and [4/4](x, t) for Burgers’ equation at k = 1 and x = 1.
Fig. 9. The surface generated from the truncated series solution /8(x, t) for Burgers’ equation in the interval 15 < t < 15 and 15 < x < 15 at k = 1.
T.A. Abassy et al. / Chaos, Solitons and Fractals 32 (2007) 1008–1026
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3.2. Case study 2 Consider ‘‘good’’ Boussinesq equation (3) with the following constraint pffiffiffiffiffiffiffiffiffiffiffiffiffi h i hcxi hcxi 3c2 3c3 1 þ c2 2 cx ¼ f ðxÞ; ut ðx; 0Þ ¼ tanh ¼ gðxÞ. sech sech2 uðx; 0Þ ¼ 2 2 2 2 2
ð43Þ
Fig. 10. The surface generated from o/ot(/8(x, t)) for Burgers’ equation in the interval 5 < t < 5 and 5 < x < 5 at k = 1.
Fig. 11. The surface generated from the approximate solution [4/4](x, t) for Burgers’ equation in the interval 20 < t < 20 and 20 < x < 20 at k = 1.
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Substituting Eq. (43) in Eq. (26), we obtain the components un(x, t) then the solution can be approximated as n X um ðx; tÞ. /n ðx; tÞ ¼
ð44Þ
m¼0
Eq. (44) with n = 4 gives ADM truncated series solution /4(x, t) hcxi 3 pffiffiffiffiffiffiffiffiffiffiffiffi hcxi hcxi 3 hcxi 3 /4 ðx;tÞ ¼ c2 sech2 þ c3 1 c2 sech2 tanh t þ c4 ð1 þ c2 Þð2 þ cosh ½cxÞsech4 t2 2 2 2 2 2 8 2 hcxi hcxi 3 1 3cx 1 6 c5 ð1 c2 Þ2 sech5 11sinh þ sinh t3 c ð1 þ c2 Þ2 ð33 26 cosh ½cx þ 16 2 2 2 128 hcxi hcxi hcxi 5 1 3cx 5cx t4 þ c7 ð1 c2 Þ2 sech7 302sinh 57 sinh þ sinh t5 þ cosh ½2cxÞsech6 2 1280 2 2 2 2 hcxi 1 c8 ð1 þ c2 Þ3 ð1208 þ 1191 cosh ½cx 120cosh ½2cx þ cosh ½3cxÞsech8 t6 þ 15360 2 hcxi hcxi 7 1 3cx 5cx 7cx c9 ð1 c2 Þ2 sech9 15619sinh þ 4293sinh 247sinh þ sinh t7 215040 2 2 2 2 2 1 þ c10 ð1 þ c2 Þ4 ð78095 88234cosh ½cx þ 14608cosh ½2cx 502cosh ½3cx 3440640 hcxi hcxi hcxi 9 1 3cx t8 þ c11 ð1 c2 Þ2 sech11 1310354 sinh 455192sinh þ cosh ½4cxÞsech10 2 61931520 2 2 2 5cx 7cx 9cx 1013sinh þ sinh t9 þ O½t10 . þ47840sinh ð45Þ 2 2 2 It is known that the exact solution of this problem is [24] hc pffiffiffiffiffiffiffiffiffiffiffiffiffi i 3c2 sech2 ðx þ 1 c2 tÞ . uðx; tÞ ¼ 2 2
ð46Þ
Following the same procedure as we do in the Burgers’ equation on the Boussinesq equation, with c = 0.5 and ADM truncated series solution (Eq. (44) with n = 4), we obtained the following results and figures. The results and Figs. 15–25 enhance the conclusion that the ADM–Pade´ technique gives very good results than using ADM alone. Using ADM–Pade´ approximation at x = 0, the rational approximations [2/2] and [4/4] take the form of Eqs. (47) and (48), respectively:
Fig. 12. The surface generated from o/ot([4/4](x, t)) for Burgers’ equation in the interval 18 < t < 18 and 18 < x < 18 at k = 1.
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Fig. 13. The approximate solution [4/4](x, t) for Burgers’ equation at k = 1 and different time level.
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Fig. 14. Plot of o/ot([4/4](x, t)) for Burgers’ equation at k = 1 and different time level.
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Fig. 15. The truncated ADM series solution /4(x, t) and the exact solution u(x, t) for good Boussinesq equation at c = 0.5 and x = 0.
Fig. 16. The error between the exact solution u(x, t) and the truncated series solution /4(x, t) for good Boussinesq equation at c = 0.5 and x = 0.
2 0:375 þ 0:00585938t2 ð0; tÞ ¼ ; 2 1 þ 0:03125t2 4 0:375 þ 0:003069t2 þ 0:000011335t4 ð0; tÞ ¼ . 4 1 þ 0:0386905t2 þ 0:000378999t4
ð47Þ ð48Þ
The results are shown in Fig. 17 which illustrate how much the results are improved at x = 0. The error between the exact solution u(x, t) and [2/2](x, t) and [4/4](x, t) at x = 0 is also illustrated in Fig. 18 where we can notice that [4/4](x, t) is better than [2/2](x, t) and both are more reliable than ADM alone. Repeating the previous calculations at x = 1, we obtain the following rational approximations [2/2](x, t) and [4/4](x, t): 2 0:35251 þ 0:013708t þ 0:00459358t2 ð1; tÞ ¼ ; ð49Þ 2 1 þ 0:0671662t þ 0:0325315t2 4 0:35251 þ 0:0071633t þ 0:002694t2 0:000053t3 8:6 106 t4 . ð50Þ ð1; tÞ ¼ 1 þ 0:0857319t þ 0:039889t2 þ 0:0016437t3 þ 0:00036964t4 4 Figs. 19–22 show how much the results improved after using ADM–Pade´ approximation. Also, Figs. 23 and 24 show the three-dimensional plot of /4(x, t) and the Pade´ [4/4](x, t) in the interval indicated in figures. Fig. 23 shows the deterioration in the ADM solution beyond the interval of convergence. Fig. 24 shows the improvement in results after using ADM–Pade´ approximation. It is clear that the interval of convergence has increased. Fig. 25 shows the solution at deferent time level.
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Fig. 17. The ADM–Pade´ rational approximation [2/2](x, t) and [4/4](x, t) for good Boussinesq equation at c = 0.5 and x = 0.
Fig. 18. The error in ADM–Pade´ between the exact solution u(x, t) and the rational approximation [2/2](x, t) and [4/4](x, t) for good Boussinesq equation at c = 0.5 and x = 0.
Fig. 19. The truncated ADM series solution /4(x, t) and the exact solution u(x, t) for good Boussinesq equation at c = 0.5 and x = 1.
Note. If we use the ‘‘bad’’ Boussinesq equation with the constrain pffiffiffiffiffiffiffiffiffiffiffiffiffi h i hcxi hcxi 3c2 3c2 1 þ c2 2 cx sech uðx; 0Þ ¼ sech2 ¼ f ðxÞ; ut ðx; 0Þ ¼ tanh ¼ gðxÞ. 2 2 2 2 2
ð51Þ
We will obtain the same results as in ‘‘good’’ Boussinesq equation but with no limit on the range of the constant ‘‘c’’.
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Fig. 20. The error between the exact solution u(x, t) and the truncated series solution /4(x, t) for good Boussinesq equation at c = 0.5 and x = 1.
Fig. 21. The ADM–Pade´ rational approximation [2/2](x, t) and [4/4](x, t) for good Boussinesq equation at c = 0.5 and x = 1.
Fig. 22. The error in ADM–Pade´ between the exact solution u(x, t) and the rational approximation [2/2](x, t) and [4/4](x, t) for good Boussinesq equation at c = 0.5 and x = 1.
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Fig. 23. The surface generated from the truncated series solution /4(x, t) for good Boussinesq equation in the interval 3 < t < 3 and 3 < x < 3 at c = 0.5.
Fig. 24. Shows the surface generated from the approximate solution [4/4](x, t) for good Boussinesq equation in the interval 10 < t < 10 and 20 < x < 20 at c = 0.5.
4. Conclusion From the previous results and the results of Abassy [1] we conclude that using ADM–Pade´ technique in solving solitary wave equations is very efficient than using ADM alone. For obtaining more accurate results and larger intervals of convergence we increase the order of Pade´ approximants we use like [6/6] and [8/8]. The ADM–Pade´ technique can be
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Fig. 25. The approximate solution [4/4](x, t) for good Boussinesq equation at c = 0.5 and different time level.
used in solving large scale of nonlinear partial differential equations. The use of Mathematica facilitates the calculations of the ADM–Pade´ technique.
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