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Solitary-wave solutions for compound KdV-type and compound KdV–Burgers-type equations with nonlinear terms of any order
Applied Mathematics and Computation 152 (2004) 709–720 www.elsevier.com/locate/amc
Solitary-wave solutions for compound KdV-type and compound KdV–Burgers-type equations with nonlinear terms of any order Do gan Kaya Department of Mathematics, Firat University, Elazig 23119, Turkey
Abstract In this Letter, based on the idea of AdomianÕs decomposition method and with help of Mathematica, we obtain the exact solutions for compound KdV-type and compound KdV–Burgers-type equations with nonlinear terms of any order. In this paper by considering the decomposition scheme, we first obtain the exact solutions of these two equation for the initial conditions without using any classical transformations and then their numerical solutions are constructed without using any discretization technique. The numerical solutions are compared with the known analytical solutions. Their remarkable accuracy is finally demonstrated in the study of some values p P 1 of the compound KdV-type and compound KdV–Burgers-type equations. 2003 Elsevier Inc. All rights reserved. Keywords: The decomposition method; Compound KdV-type equation; Compound KdV–Burgerstype equation; Solitary-wave solution; Mathematica
1. Introduction In this study we will consider two nonlinear KdV equations first to find the known explicit exact solution in a series form and then we construct the numerical solution from this series form. Nonlinear phenomena play a crucial role in applied mathematics and physics. The nonlinear problem are solved E-mail address: dkaya@firat.edu.tr (D. Kaya). 0096-3003/$ - see front matter 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(03)00589-7
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D. Kaya / Appl. Math. Comput. 152 (2004) 709–720
easily and elegantly without linearizing the problem by using the AdomianÕs decomposition method (ADM) [1,2]. Explicit solutions to the nonlinear equations are of fundamental importance. Various methods for obtaining explicit solutions to nonlinear evolution equations have been proposed [3–13]. Among them are the homogenous balance method (HB) [3–11,13], the autoB€ acklund transformations [5,8,9], and the similarity reductions [8,9] the B€ acklund transformation in mathematical physics. The nonlinear iterative principle from B€ acklund transformations converts the problem of solving nonlinear PDE to purely algebraic calculations [14,15]. In recent paper Li et al. [18], the authors have further extended the HB method so that it can deal with the other cases whose balance constant is fraction or negative integer. A feature common to all these methods is that they are using the transformations to reduce the equation into more simple equation then solve it. Unlike classical techniques, the nonlinear equations are solved easily and elegantly without transforming the equation by using the ADM. The technique has many advantages over the classical techniques, mainly, it avoids linearization and perturbation in order to find solutions of a given nonlinear equations. It is providing an efficient explicit solution with high accuracy, minimal calculation, avoidance of physically unrealistic assumptions. The aim of this work is to implement ADM and to illustrate this method, we consider the compound KdV-type equation (cKdV) in order to show the effectiveness of the current method. We consider the solitary-wave solution uðx; tÞ of the cKdV equation with nonlinear terms any order ut þ aup ux þ bu2p ux þ duxxx ¼ 0;
ð1Þ
and the Compound KdV–Burgers-type equation (cKdV–B) equation with nonlinear terms any order ut þ aup ux þ bu2p ux þ cuxx þ duxxx ¼ 0;
ð2Þ
where a, b, c, d, p ¼ constants, p > 0 [18]. These equations include a number of equations which have been studied by many authors [15–18]. The cKdV equation (1) have some application in quantum field theory, plasma physics and solid-state physics. For example, the kink soliton can be used to calculate energy and momentum flow and topological charge in the quantum field. In [16,17], Dey and Coffey considered the kink-profile solitary-wave solutions for Eq. (1) with p ¼ 1 and p ¼ 2. Kaya et al. considered for finding the exact and numerical soliton solutions by using ADM for Eq. (1) with p ¼ 1 [19]. Zhang et al. studied the solitary-wave solutions for Eq. (1) [17]. The cKdV–B equation (2) with p P 1 is a model for long-wave propagation in nonlinear media with dispersion and dissipation [15–18]. In [17], Zhang et al. obtained kink-profile solitary-wave solutions for Eq. (2). The stability of traveling-wave solutions of Eq. (2) with b ¼ 0, c P 0 and p P 1 are studied by Pego et al. [15].
D. Kaya / Appl. Math. Comput. 152 (2004) 709–720
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We implemented the ADM for solving explicit solution of the nonlinear partial differential equations. In this paper, we will consider various cKdV and cKdV–B equations (i.e., p ¼ 1, 2, 4, 6) to find explicit solutions and numerical solutions of these equations by using the ADM rather than the traditional methods. The decomposition scheme will be illustrated by studying cKdV and cKdV–B equations to compute explicit and numerical solutions. The method is useful for obtaining both approximate and numerical approximations of linear or nonlinear differential equations and it is also quite straightforward to write computer codes in any symbolic languages. If the numerical solutions are necessary to compute, the rapid convergence is obvious. Furthermore, as the ADM does not require discretization of the variables, it is not effected by computation roundoff errors and one is not faced with necessity of large computer memory and time. There are some rather significant advantages over methods which must assume linearity, ‘‘smallness’’, deterministic behavior, etc. The method has features in common with many other methods, but it is distinctly different on close examination, and one should not be mislead by apparent simplicity into superficial conclusions [1,2].
2. Method and its applications In the preceding section we have discussed particular devices of the general type of cKdV and cKdV–B equations. For purposes of illustration of the ADM, in this study we shall consider Eq. (1) with the explicit exact solution [18] ( uðx;tÞ ¼
##!)1=p " " að1 þ 2pÞ að1 þ 2pÞ a2 ð1 þ 2pÞ t ; 1 þ tanh xþ 2bð2 þ pÞ bbð2 þ pÞ bð2 þ pÞ2 ð1 þ pÞ ð3Þ
of which is to be obtained subject to the initial condition uðx; 0Þ ¼ f ðxÞ, where ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b¼
ð1þpÞð1þ2pÞd bp2
and a, b, d, p 6¼ 0 are real constants. We shall also consider
Eq. (2) with explicit exact solution [18] uðx; tÞ ¼
að1 þ 2pÞd þ bpbc ½1 þ tanh½Kðx ktÞ 2bð2 þ pÞd
1=p ;
ð4Þ
of which is to be obtained subject to the initial condition uðx; 0Þ ¼ gðxÞ, where K¼
bpbc þ að1 þ 2pÞd ; 2bbdð2 þ pÞ
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D. Kaya / Appl. Math. Comput. 152 (2004) 709–720
k¼
ðbpð1 þ pÞbc að1 þ 2pÞdÞðbpbc þ að1 þ 2pÞdÞ 2
bð2 þ pÞ ð1 þ pÞð1 þ 2pÞd2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 þ pÞð1 þ 2pÞd b¼ bp2
;
and a, b, b, c, d, p 6¼ 0 are real constants. Following [1,2] we define for the above equation the linear operator Łt ¼ o=ot and the definite integration inverse operator Ł1 t . Therefore the solution can be written as for Eq. (2) 1 1 1 UðuÞ þ b WðuÞ þ cuxx þ duxxx ; uðx; tÞ ¼ uðx; 0Þ Lt a 1þp 1þr x x ð5Þ pþ1
rþ1
where UðuÞ ¼ u , WðuÞ ¼ u , r ¼ 2p and p > 0. Following ADM [4,5], we expect the decomposition of the solution into a sum of components to be defined by the decomposition series form 1 X uðx; tÞ ¼ un ðx; tÞ: ð6Þ n¼0
Substituting the initial condition into (3) identifying the zeroth component u0 ¼ uðx; 0Þ by terms arising from initial condition, then we obtain the subsequent components by the following recursive relationship by 1 1 A B a þ cu þ du unþ1 ¼ L1 þ b n P 0; n n nxx nxxx ; t 1þp 1þr x x P1
ð7Þ
P1
where UðuÞ ¼ n¼0 An ðu0 ; u1 ; . . . ; un Þ and WðuÞ ¼ n¼0 Bn ðu0 ; u1 ; . . . ; un Þ are called the Adomian polynomials, these polynomials can be calculated for all forms of nonlinearity according to specific algorithms constructed by Adomian [1,2]. In this specific nonlinearity, we use the general form of formula [20] for An polynomials as " !# 1 X 1 dn k U k uk ; n P 0: ð8Þ An ¼ n! dkn k¼0 k¼0
This formulae is easy to set computer code to get as many polynomial as we need in the calculation of the numerical as well as explicit solutions. To follow of the reader, some of the first Adomian polynomials An are computed according to (6) with UðuÞ ¼ upþ1 and this gives A0 ¼ upþ1 0 ;
A1 ¼ ðp þ 1Þup0 u1 ;
A3 ¼ pðp þ 1Þðp 1Þu0p2
A2 ¼ pðp þ 1Þup1 0
1 2 u þ ðp þ 1Þup0 u2 ; 2! 1
1 3 u þ pðp þ 1Þu0p1 u1 u2 þ ðp þ 1Þup0 u3 ; 3! 1
D. Kaya / Appl. Math. Comput. 152 (2004) 709–720
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and so on, the rest of the polynomials can be constructed in a similar manner. The Adomian polynomials Bn can be constructed same as formulae (8). A slight modification to the ADM was proposed by Wazwaz [21] that gives some flexibility in the choice of the zeroth component u0 to be any simple term and modify the term u1 accordingly and since the computations in (5) depends heavily on u0 the whole computations to find the solution will be simplified considerably. For example an alternative scheme to (5) might be 1 1 A B u0 ¼ 0; u1 ¼ uðx; 0Þ L1 a þ cu þ du þ b 0 0 0xx 0xxx ; t 1þp 1þr x x 1 1 A B a þ cu þ du unþ1 ¼ L1 þ b n n nxx nxxx ; n P 1: t 1þp 1þr x x ð9Þ Table 1 The numerical results for juðx; tÞ /2 ðx; tÞj when a ¼ 1, b ¼ 1, d ¼ 1 and c ¼ 1, for the solitarywave solutions (14) of the Eq. (2) ðxi ; ti Þ (0.1,0.1)
(0.2,0.2)
(0.3,0.3)
(0.4,0.4)
(0.5,0.5)
p¼1 0.0000287359 0.0000288311 0.0000289253 0.0000290186 0.000029111
0.0000574719 0.0000576622 0.0000578507 0.0000580374 0.0000582223
0.0000862079 0.0000864935 0.0000867764 0.0000870565 0.0000873338
0.000114944 0.000115325 0.000115702 0.000116076 0.000116446
0.00014368 0.000144156 0.000144628 0.000145095 0.000145558
p¼2 0.000165647 0.000166019 0.00016637 0.0001667 0.000167008
0.000331325 0.00033207 0.000332773 0.000333432 0.000334048
0.000497033 0.000498152 0.000499206 0.000500197 0.000501122
0.000662772 0.000664265 0.000665672 0.000666993 0.000668228
0.000828542 0.000830409 0.00083217 0.000833822 0.000835367
p¼4 0.000290829 0.000290745 0.000290575 0.000290319 0.000289977
0.000581716 0.000581549 0.00058121 0.000580698 0.000580015
0.000872663 0.000872413 0.000871904 0.000871137 0.000870113
0.00116367 0.00116334 0.00116266 0.00116164 0.00116027
0.00145473 0.00145432 0.00145347 0.0014522 0.00145049
p¼6 0.000297283 0.000296657 0.00029591 0.000295045 0.000294061
0.000594611 0.000593359 0.000591867 0.000590135 0.000588169
0.000891984 0.000890107 0.000887868 0.000885272 0.000882322
0.0011894 0.0011869 0.00118392 0.00118045 0.00117652
0.00148686 0.00148374 0.00148001 0.00147568 0.00147077
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D. Kaya / Appl. Math. Comput. 152 (2004) 709–720
Finally based on the ADM, we constructed the solution uðx; tÞ as lim /n ¼ uðx; tÞ;
n!1
where /n ðx; tÞ ¼
n X
uk ðx; tÞ;
ð10Þ
nP0
k¼0
and the recurrence relation is given as in (7). Moreover, the decomposition series (8) solutions are generally converged very rapidly in real physical problems [2]. The convergence of the decomposition series have investigated by several authors. The theoretical treatment of convergence of the decomposition method has been considered in the literature [22–27]. They obtained some results about the speed of convergence of this method providing us to solve linear and nonlinear functional equations. In recent work of Abbaoui et al. [27] have proposed a new approach of convergence of the decomposition series. The authors have given a new condition for obtaining convergence of the decomposition series to the classical presentation of the ADM in [27]. In this work, we Table 2 The numerical results for juðx; tÞ /5 ðx; tÞj when a ¼ 1, b ¼ 1, d ¼ 1 and c ¼ 1, for the solitarywave solutions (14) of the Eq. (2) ðxi ; ti Þ (0.1,0.1)
(0.2,0.2)
(0.3,0.3)
(0.4,0.4)
(0.5,0.5)
p¼1 8.59915 · 109 8.60308 · 109 8.60603 · 109 8.60802 · 109 8.60902 · 109
3.43947 · 108 3.44103 · 108 3.4422 · 108 3.44297 · 108 3.44336 · 108
7.73839 · 108 7.74185 · 108 7.74445 · 108 7.74617 · 108 7.74701 · 108
1.37564 · 107 1.37625 · 107 1.3767 · 107 1.377 · 107 1.37715 · 107
2.14932 · 107 2.15026 · 107 2.15096 · 107 2.15142 · 107 2.15164 · 107
p¼2 0.0000272923 0.0000275842 0.0000278741 0.0000281619 0.0000284475
0.0000543881 0.0000549702 0.0000555483 0.0000561224 0.000056692
0.0000812865 0.0000821569 0.0000830216 0.0000838803 0.0000847325
0.000107987 0.000109143 0.000110293 0.000111435 0.000112568
0.000134487 0.000135929 0.000137362 0.000138785 0.000140198
p¼4 0.000291223 0.000291135 0.00029096 0.000290698 0.000290351
0.000583297 0.000583112 0.000582753 0.00058222 0.000581514
0.000876228 0.000875938 0.000875385 0.000874569 0.000873493
0.00117002 0.00116962 0.00116886 0.00116775 0.00116629
0.00146469 0.00146416 0.00146318 0.00146177 0.00145992
p¼6 0.000297828 0.000297188 0.000296428 0.000295547 0.000294549
0.000596794 0.000595488 0.000593938 0.000592147 0.000590118
0.000896904 0.000894903 0.000892533 0.0008898 0.000886708
0.00119816 0.00119544 0.00119222 0.00118851 0.00118432
0.00150058 0.0014971 0.00149299 0.00148827 0.00148295
D. Kaya / Appl. Math. Comput. 152 (2004) 709–720
715
demonstrate how approximate solutions of the cKdV and cKdV–B equations are close to exact solutions. In the first example, we will consider Eq. (1) for the special case associated the initial condition uðx; 0Þ ¼ W ð1 þ tanhðKxÞÞ; where W ¼
að1 þ 2pÞd þ bpbc ; 2bð2 þ pÞd
K¼
bpbc þ að1 þ 2pÞd 2bbdð2 þ pÞ
and p ¼ 1 is positive constant. To find the solution of the initial value problem (9) with (8) for p ¼ 1 and performing the integration we obtain the following u0 ¼ 0;
u1 ¼ W ð1 þ tanhðKxÞÞ;
ð11Þ
u2 ¼ K 2 tW sech4 ðKxÞ½4dK 2dK coshð2KxÞ þ c sinhð2KxÞ;
ð12Þ
Table 3 The numerical results for juðx; tÞ /7 ðx; tÞj when a ¼ 1, b ¼ 1, d ¼ 1 and c ¼ 1, for the solitarywave solutions (14) of the Eq. (2) ðxi ; ti Þ (0.1,0.1)
(0.2,0.2)
(0.3,0.3)
(0.4,0.4)
(0.5,0.5)
p¼1 1.54757 · 1012 1.54374 · 1012 1.53952 · 1012 1.53484 · 1012 1.52971 · 1012
1.23419 · 1011 1.23116 · 1011 1.22779 · 1011 1.22407 · 1011 1.22 · 1011
4.15239 · 1011 4.14222 · 1011 4.13091 · 1011 4.11843 · 1011 4.10478 · 1011
9.81192 · 1011 9.78797 · 1011 9.76133 · 1011 9.73194 · 1011 9.6998 · 1011
1.91039 · 1010 1.90575 · 1010 1.90058 · 1010 1.89487 · 1010 1.88863 · 1010
p¼2 6.2543 · 108 6.27652 · 108 6.29616 · 108 6.31317 · 108 6.3275 · 108
2.49053 · 107 2.49954 · 107 2.50753 · 107 2.51448 · 107 2.52037 · 107
5.57839 · 107 5.59895 · 107 5.61723 · 107 5.63319 · 107 5.6468 · 107
9.87196 · 107 9.90901 · 107 9.94206 · 107 9.97104 · 107 9.99587 · 107
1.5354 · 106 1.54127 · 106 1.54652 · 106 1.55114 · 106 1.55513 · 106
p¼4 0.0000700311 0.0000709808 0.0000719129 0.0000728264 0.0000737204
0.000140886 0.000142778 0.000144634 0.000146451 0.000148229
0.000212558 0.000215385 0.000218157 0.000220871 0.000223522
0.000285041 0.000288798 0.000292479 0.00029608 0.000299596
0.000358331 0.000363011 0.000367594 0.000372075 0.000376447
p¼6 0.000297828 0.000297188 0.000296428 0.000295547 0.000294549
0.000596793 0.000595487 0.000593937 0.000592146 0.000590117
0.0008969 0.000894899 0.00089253 0.000889797 0.000886705
0.00119815 0.00119543 0.00119221 0.0011885 0.00118431
0.00150055 0.00149707 0.00149296 0.00148825 0.00148293
716
D. Kaya / Appl. Math. Comput. 152 (2004) 709–720
u3 ¼ sech3 ðKxÞ½aKtW 2 coshðKxÞ þ aKW 2 t sinhðKxÞ K 4 t2 W sech7 ðKxÞ ½160dcK coshðKxÞ þ 100dcK coshð3KxÞ 4 4dcK coshð5KxÞ 10c2 sinhðKxÞ þ 1208d2 K 2 sinhðKxÞ
9c2 sinhð3KxÞ 228d2 K 2 sinhð3KxÞ þ c2 sinhð5KxÞ þ 4d2 K 2 sinhð5KxÞ; ð13Þ in this manner the components of the decomposition series (10) are obtained as far as we like. This series is exact to the last term, as one can verify, of the Taylor series of the exact closed form solution (2) for the special case of the p ¼ 1 as uðx; tÞ ¼ W f1 þ tanh½Kðx ktÞg;
ð14Þ
Table 4 The numerical results for juðx; tÞ /9 ðx; tÞj when a ¼ 1, b ¼ 1, d ¼ 1 and c ¼ 1, for the solitarywave solutions (14) of the Eq. (2) ðxi ; ti Þ (0.1,0.1)
(0.2,0.2)
(0.3,0.3)
(0.4,0.4)
(0.5,0.5)
p¼1 1.29202 · 1014 1.30451 · 1014 1.32117 · 1014 1.33366 · 1014 1.34615 · 1014
9.82825 · 1014 9.95037 · 1014 1.00739 · 1013 1.01988 · 1013 1.03181 · 1013
3.14845 · 1013 3.19342 · 1013 3.2388 · 1013 3.28376 · 1013 3.32845 · 1013
7.0674 · 1013 7.18245 · 1013 7.29833 · 1013 7.41393 · 1013 7.52953 · 1013
1. 30386 · 1012 1.32799 · 1012 1.35222 · 1012 1.37657 · 1012 1.40092 · 1012
p¼2 6.75975 · 109 6.85603 · 109 6.95076 · 109 7.04383 · 109 7.13513 · 109
2.63079 · 108 2.67039 · 108 2.70947 · 108 2.74797 · 108 2.78585 · 108
5.75369 · 108 5.84529 · 108 5.93592 · 108 6.02546 · 108 6.1138 · 108
9.93244 · 108 1.00998 · 107 1.02658 · 107 1.04303 · 107 1.0593 · 107
1.50534 · 107 1.53221 · 107 1.55893 · 107 1. 58546 · 107 1.61178 · 107
p¼4 0.000070034 0.0000709835 0.0000719154 0.0000728287 0.0000737225
0.000140909 0.000142799 0.000144654 0.00014647 0.000148246
0.000212635 0.000215458 0.000218225 0.000220934 0.00022358
0.000285225 0.000288971 0.00029264 0.000296228 0.000299733
0.000358689 0.000363347 0.000367907 0.000372364 0.000376713
p¼6 0.0000840544 0.0000853079 0.0000865292 0.0000877162 0.0000888673
0.00016928 0.000171774 0.000174201 0.000176557 0.00017884
0.000255674 0.000259396 0.000263014 0.000266524 0.000269919
0.000343232 0.000348171 0.000352968 0.000357616 0.000362107
0.000431952 0.0004381 0.000444064 0.000449836 0.000455407
D. Kaya / Appl. Math. Comput. 152 (2004) 709–720
717
where bpbc þ að1 þ 2pÞd ; 2bbdð2 þ pÞ ðbpð1 þ pÞbc að1 þ 2pÞdÞðbpbc þ að1 þ 2pÞdÞ
K¼ k¼
2
bð2 þ pÞ ð1 þ pÞð1 þ 2pÞd2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 þ pÞð1 þ 2pÞd b¼ bp2
;
and a, b, b, c, d, p ¼ 1 are real constants. In the some other examples, we will consider for the cKdV equation (1) when c ¼ 0 in Eq. (2) with the general form of the initial condition (3) for different values of the constant p. In a similar way, performing the calculations in (9) with (8) using Mathematica and substituting into (10) gives the exact Table 5 The numerical results for juðx; tÞ /2 ðx; tÞj when a ¼ 1, b ¼ 10, and d ¼ 1, for the solitary-wave solutions (3) of the Eq. (1) ðxi ; ti Þ (0.1,0.1)
(0.2,0.2)
(0.3,0.3)
(0.4,0.4)
(0.5,0.5)
p¼1 0.0000322588 0.0000322104 0.00003213 0.0000320177 0.0000318738
0.0000645176 0.000064421 0.0000642602 0.0000640356 0.0000637479
0.0000967764 0.0000966316 0.0000963905 0.0000960538 0.0000956223
0.000129035 0.000128842 0.000128521 0.000128072 0.000127497
0.000161294 0.000161053 0.000160652 0.000160091 0.000159372
p¼2 0.00015442 0.000149039 0.000143356 0.000137394 0.000131181
0.000308843 0.000298082 0.000286714 0.000274791 0.000262365
0.000463268 0.000447127 0.000430077 0.000412192 0.000393553
0.000617697 0.000596176 0.000573442 0.000549596 0.000524745
0.000772129 0.000745228 0.00071681 0.000687003 0.000655939
p¼4 0.000201247 0.000179832 0.000157938 0.000135724 0.000113352
0.000402497 0.000359667 0.000315878 0.00027145 0.000226706
0.000603748 0.000539504 0.00047382 0.000407179 0.000340062
0.000805001 0.000719342 0.000631765 0.000542909 0.00045342
0.00100626 0.000899183 0.000789711 0.000678642 0.00056678
p¼6 0.000156556 0.000127407 0.0000979896 0.0000685682 0.0000394029
0.000313113 0.000254815 0.00019598 0.000137137 0.0000788067
0.000469671 0.000382224 0.000293971 0.000205707 0.000118211
0.00062623 0.000509634 0.000391964 0.000274278 0.000157617
0.000782789 0.000637044 0.000489957 0.00034285 0.000197023
718
D. Kaya / Appl. Math. Comput. 152 (2004) 709–720
solution in a series form (3). A numerical discussions will be given in the following section.
3. Experimental results for the cKdV and cKdV–B equations In order to verify numerically whether the proposed methodology lead to accurate solutions, we will evaluate the decomposition series solutions using the n-terms (n ¼ 2, 5, 7, and 9) approximation for some examples of the cKdV– B equations solved in the previous section. The differences between the n-terms solution and the exact solution for some values of the constant p ¼ 1, 2, 4, 6 are shown in Table 1–4. The numerical values in all the tables show that we achieved a very good approximation to the actual solution of the equations by using small values of n-terms of the decomposition series solution derived Table 6 The numerical results for juðx; tÞ /5 ðx; tÞj when a ¼ 1, b ¼ 10, and d ¼ 1, for the solitary-wave solutions (3) of the Eq. (1) ðxi ; ti Þ (0.1,0.1)
(0.2,0.2)
(0.3,0.3)
(0.4,0.4)
(0.5,0.5)
p¼1 0.0000160888 0.0000160541 0.0000159982 0.0000159212 0.0000158234
0.000032099 0.0000320249 0.0000319087 0.0000317506 0.0000315511
0.0000480269 0.0000479086 0.0000477276 0.0000474843 0.0000471795
0.0000638684 0.0000637013 0.000063451 0.0000631186 0.000062705
0.0000796197 0.0000793989 0.0000790753 0.0000786499 0.0000781241
p¼2 7.74788 · 106 0.0000146063 0.000021599 0.0000286966 0.0000358694
0.000017903 0.0000317606 0.0000458637 0.0000601528 0.0000745677
0.0000306496 0.0000516191 0.0000729194 0.0000944612 0.000116154
0.0000461722 0.0000743376 0.000102891 0.000131715 0.000160688
0.000064655 0.000100072 0.000135905 0.000172006 0.000208229
p¼4 0.000193418 0.000171879 0.000149977 0.000127871 0.000105714
0.000370567 0.000327627 0.000284199 0.000240576 0.000197043
0.000530529 0.000466907 0.00040291 0.000338925 0.00027532
0.000672386 0.000589382 0.000506357 0.000423726 0.000341878
0.000795218 0.000694714 0.000594785 0.000495789 0.000398048
p¼6 0.000143295 0.000114278 0.0000852293 0.0000563945 0.0000280091
0.000259513 0.000202786 0.000146433 0.0000908594 0.0000364463
0.000347817 0.000266256 0.000185853 0.00010702 0.0000301338
0.000407371 0.000305419 0.00020573 0.000108502 0.0000138939
0.00043734 0.000321006 0.000208307 0.0000989301 7.45107 · 106
D. Kaya / Appl. Math. Comput. 152 (2004) 709–720
719
above. It is evident that the overall errors can be made smaller by adding new terms of the decomposition series. We also evaluate the decomposition series solutions using the n-terms (n ¼ 2, 5) approximation for some examples of the cKdV equations solved as same as in Eq. (2). It is surprisingly, numerical efficiency not as good as in the decomposition series solution of cKdV–B equation. The numerical comparison of the Eq. (1) is given for some values of the constant p ¼ 1, 2, 4, 6 are shown in Table 5 and 6. The solutions are very rapidly convergent by utilizing the ADM. The numerical results we obtained justify the advantage of this methodology. Furthermore, as the decomposition method does not require discretization of the variables, i.e. time and space, it is not effected by computation roundoff errors and necessity of large computer memory and time. Clearly, the series solution methodology can be applied to various type of linear or nonlinear system of partial differential equations [28,29] and single partial differential equations [30–36] as well.
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[14] R.L. Pego, Oscillatory instability of traveling waves for a KdV–Burgers equation, Physica D 67 (1993) 45–65. [15] B. Dey, Domain wall solutions of KdV-like equations with higher order nonlinearity, J. Phys. A 19 (1986) L9–L12. [16] M.W. Coffey, On series expansions giving closed-form solutions of Korteweg–de Vries-like equations, SIAM J. Appl. Math. 50 (1990) 1580–1592. [17] W.G. Zhang, Q.S. Chang, B.G. Jiang, Explicit exact solitary-wave solutions for compound KdV-type and compound KdV–Burgers-type equations with nonlinear terms of any order, Chaos Solitons Fractals 13 (2002) 311–319. [18] B. Li, Y. Chen, H. Zhang, Auto-B€acklund transformation and exact solutions for compound KdV-type and compound KdV–Burgers-type equations with nonlinear terms of any order, Phys. Lett. A 305 (2002) 377–382. [19] D. Kaya, I.E. Inan, A numerical application of the decomposition method for the combined KdV–MKdV equation, Appl. Math. Comput., (in press). [20] V. Seng, K. Abbaoui, Y. Cherruault, AdomianÕs polynomials for nonlinear operators, Math. Comput. Modelling 24 (1996) 59–65. [21] A.M. Wazwaz, A reliable modification of Adomian decomposition method, Appl. Math. Comp. 102 (1999) 77–86. [22] Y. Cherruault, Convergence of AdomianÕs method, Kybernetes 18 (1989) 31–38. [23] A. Repaci, Nonlinear dynamical systems: on the accuracy of AdomianÕs decomposition method, Appl. Math. Lett. 3 (1990) 35–39. [24] Y. Cherruault, G. Adomian, Decomposition methods: a new proof of convergence, Math. Comput. Modelling 18 (1993) 103–106. [25] K. Abbaoui, Y. Cherruault, Convergence of AdomianÕs method applied to differential equations, Comp. Math. Appl. 28 (1994) 103–109. [26] K. Abbaoui, Y. Cherruault, New ideas for proving convergence of decomposition methods, Comp. Math. Appl. 29 (1995) 103–108. [27] K. Abbaoui, M.J. Pujol, Y. Cherruault, N. Himoun, P. Grimalt, A new formulation of Adomian method: convergence result, Kybernetes 30 (2001) 1183–1191. [28] D. Kaya, An explicit solution of coupled viscous BurgersÕ equation by the decomposition method, Int. J. Math. Math. Sci. 27 (2001) 675–680. [29] D. Kaya, Solitary wave solutions for a generalized Hirota–Satsuma coupled KdV equation, Appl. Math. Comp., in press. [30] D. Kaya, On the solution of a Korteweg–de Vries like equation by the decomposition method, Int. J. Comp. Math. 72 (1999) 531–539. [31] D. Kaya, An application of the decomposition method on second order wave equations, Int. J. Comp. Math. 75 (2000) 235–245. [32] A.M. Wazwaz, A study of nonlinear dispersive equations with solitary-wave solutions having compact support, Math. Comput. Simulation 56 (2001) 269–276. [33] D. Kaya, An application of the decomposition method on second order wave equations, Int. J. Comp. Math. 75 (2000) 235–245. [34] D. Kaya, Explicit solution of a generalized nonlinear Boussinesq equation, J. Appl. Math. 1 (2001) 29–37. [35] D. Kaya, M. Aassila, An application for a generalized KdV equation by decomposition method, Phys. Lett. A 299 (2002) 201–206. [36] D. Kaya, An application of the decomposition method for the KdVB equation, Appl. Math. Comput., (in press).
Solitary-wave solutions for compound KdV-type and compound KdV–Burgers-type equations with nonlinear terms of any order Do gan Kaya Department of Mathematics, Firat University, Elazig 23119, Turkey
Abstract In this Letter, based on the idea of AdomianÕs decomposition method and with help of Mathematica, we obtain the exact solutions for compound KdV-type and compound KdV–Burgers-type equations with nonlinear terms of any order. In this paper by considering the decomposition scheme, we first obtain the exact solutions of these two equation for the initial conditions without using any classical transformations and then their numerical solutions are constructed without using any discretization technique. The numerical solutions are compared with the known analytical solutions. Their remarkable accuracy is finally demonstrated in the study of some values p P 1 of the compound KdV-type and compound KdV–Burgers-type equations. 2003 Elsevier Inc. All rights reserved. Keywords: The decomposition method; Compound KdV-type equation; Compound KdV–Burgerstype equation; Solitary-wave solution; Mathematica
1. Introduction In this study we will consider two nonlinear KdV equations first to find the known explicit exact solution in a series form and then we construct the numerical solution from this series form. Nonlinear phenomena play a crucial role in applied mathematics and physics. The nonlinear problem are solved E-mail address: dkaya@firat.edu.tr (D. Kaya). 0096-3003/$ - see front matter 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(03)00589-7
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easily and elegantly without linearizing the problem by using the AdomianÕs decomposition method (ADM) [1,2]. Explicit solutions to the nonlinear equations are of fundamental importance. Various methods for obtaining explicit solutions to nonlinear evolution equations have been proposed [3–13]. Among them are the homogenous balance method (HB) [3–11,13], the autoB€ acklund transformations [5,8,9], and the similarity reductions [8,9] the B€ acklund transformation in mathematical physics. The nonlinear iterative principle from B€ acklund transformations converts the problem of solving nonlinear PDE to purely algebraic calculations [14,15]. In recent paper Li et al. [18], the authors have further extended the HB method so that it can deal with the other cases whose balance constant is fraction or negative integer. A feature common to all these methods is that they are using the transformations to reduce the equation into more simple equation then solve it. Unlike classical techniques, the nonlinear equations are solved easily and elegantly without transforming the equation by using the ADM. The technique has many advantages over the classical techniques, mainly, it avoids linearization and perturbation in order to find solutions of a given nonlinear equations. It is providing an efficient explicit solution with high accuracy, minimal calculation, avoidance of physically unrealistic assumptions. The aim of this work is to implement ADM and to illustrate this method, we consider the compound KdV-type equation (cKdV) in order to show the effectiveness of the current method. We consider the solitary-wave solution uðx; tÞ of the cKdV equation with nonlinear terms any order ut þ aup ux þ bu2p ux þ duxxx ¼ 0;
ð1Þ
and the Compound KdV–Burgers-type equation (cKdV–B) equation with nonlinear terms any order ut þ aup ux þ bu2p ux þ cuxx þ duxxx ¼ 0;
ð2Þ
where a, b, c, d, p ¼ constants, p > 0 [18]. These equations include a number of equations which have been studied by many authors [15–18]. The cKdV equation (1) have some application in quantum field theory, plasma physics and solid-state physics. For example, the kink soliton can be used to calculate energy and momentum flow and topological charge in the quantum field. In [16,17], Dey and Coffey considered the kink-profile solitary-wave solutions for Eq. (1) with p ¼ 1 and p ¼ 2. Kaya et al. considered for finding the exact and numerical soliton solutions by using ADM for Eq. (1) with p ¼ 1 [19]. Zhang et al. studied the solitary-wave solutions for Eq. (1) [17]. The cKdV–B equation (2) with p P 1 is a model for long-wave propagation in nonlinear media with dispersion and dissipation [15–18]. In [17], Zhang et al. obtained kink-profile solitary-wave solutions for Eq. (2). The stability of traveling-wave solutions of Eq. (2) with b ¼ 0, c P 0 and p P 1 are studied by Pego et al. [15].
D. Kaya / Appl. Math. Comput. 152 (2004) 709–720
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We implemented the ADM for solving explicit solution of the nonlinear partial differential equations. In this paper, we will consider various cKdV and cKdV–B equations (i.e., p ¼ 1, 2, 4, 6) to find explicit solutions and numerical solutions of these equations by using the ADM rather than the traditional methods. The decomposition scheme will be illustrated by studying cKdV and cKdV–B equations to compute explicit and numerical solutions. The method is useful for obtaining both approximate and numerical approximations of linear or nonlinear differential equations and it is also quite straightforward to write computer codes in any symbolic languages. If the numerical solutions are necessary to compute, the rapid convergence is obvious. Furthermore, as the ADM does not require discretization of the variables, it is not effected by computation roundoff errors and one is not faced with necessity of large computer memory and time. There are some rather significant advantages over methods which must assume linearity, ‘‘smallness’’, deterministic behavior, etc. The method has features in common with many other methods, but it is distinctly different on close examination, and one should not be mislead by apparent simplicity into superficial conclusions [1,2].
2. Method and its applications In the preceding section we have discussed particular devices of the general type of cKdV and cKdV–B equations. For purposes of illustration of the ADM, in this study we shall consider Eq. (1) with the explicit exact solution [18] ( uðx;tÞ ¼
##!)1=p " " að1 þ 2pÞ að1 þ 2pÞ a2 ð1 þ 2pÞ t ; 1 þ tanh xþ 2bð2 þ pÞ bbð2 þ pÞ bð2 þ pÞ2 ð1 þ pÞ ð3Þ
of which is to be obtained subject to the initial condition uðx; 0Þ ¼ f ðxÞ, where ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b¼
ð1þpÞð1þ2pÞd bp2
and a, b, d, p 6¼ 0 are real constants. We shall also consider
Eq. (2) with explicit exact solution [18] uðx; tÞ ¼
að1 þ 2pÞd þ bpbc ½1 þ tanh½Kðx ktÞ 2bð2 þ pÞd
1=p ;
ð4Þ
of which is to be obtained subject to the initial condition uðx; 0Þ ¼ gðxÞ, where K¼
bpbc þ að1 þ 2pÞd ; 2bbdð2 þ pÞ
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D. Kaya / Appl. Math. Comput. 152 (2004) 709–720
k¼
ðbpð1 þ pÞbc að1 þ 2pÞdÞðbpbc þ að1 þ 2pÞdÞ 2
bð2 þ pÞ ð1 þ pÞð1 þ 2pÞd2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 þ pÞð1 þ 2pÞd b¼ bp2
;
and a, b, b, c, d, p 6¼ 0 are real constants. Following [1,2] we define for the above equation the linear operator Łt ¼ o=ot and the definite integration inverse operator Ł1 t . Therefore the solution can be written as for Eq. (2) 1 1 1 UðuÞ þ b WðuÞ þ cuxx þ duxxx ; uðx; tÞ ¼ uðx; 0Þ Lt a 1þp 1þr x x ð5Þ pþ1
rþ1
where UðuÞ ¼ u , WðuÞ ¼ u , r ¼ 2p and p > 0. Following ADM [4,5], we expect the decomposition of the solution into a sum of components to be defined by the decomposition series form 1 X uðx; tÞ ¼ un ðx; tÞ: ð6Þ n¼0
Substituting the initial condition into (3) identifying the zeroth component u0 ¼ uðx; 0Þ by terms arising from initial condition, then we obtain the subsequent components by the following recursive relationship by 1 1 A B a þ cu þ du unþ1 ¼ L1 þ b n P 0; n n nxx nxxx ; t 1þp 1þr x x P1
ð7Þ
P1
where UðuÞ ¼ n¼0 An ðu0 ; u1 ; . . . ; un Þ and WðuÞ ¼ n¼0 Bn ðu0 ; u1 ; . . . ; un Þ are called the Adomian polynomials, these polynomials can be calculated for all forms of nonlinearity according to specific algorithms constructed by Adomian [1,2]. In this specific nonlinearity, we use the general form of formula [20] for An polynomials as " !# 1 X 1 dn k U k uk ; n P 0: ð8Þ An ¼ n! dkn k¼0 k¼0
This formulae is easy to set computer code to get as many polynomial as we need in the calculation of the numerical as well as explicit solutions. To follow of the reader, some of the first Adomian polynomials An are computed according to (6) with UðuÞ ¼ upþ1 and this gives A0 ¼ upþ1 0 ;
A1 ¼ ðp þ 1Þup0 u1 ;
A3 ¼ pðp þ 1Þðp 1Þu0p2
A2 ¼ pðp þ 1Þup1 0
1 2 u þ ðp þ 1Þup0 u2 ; 2! 1
1 3 u þ pðp þ 1Þu0p1 u1 u2 þ ðp þ 1Þup0 u3 ; 3! 1
D. Kaya / Appl. Math. Comput. 152 (2004) 709–720
713
and so on, the rest of the polynomials can be constructed in a similar manner. The Adomian polynomials Bn can be constructed same as formulae (8). A slight modification to the ADM was proposed by Wazwaz [21] that gives some flexibility in the choice of the zeroth component u0 to be any simple term and modify the term u1 accordingly and since the computations in (5) depends heavily on u0 the whole computations to find the solution will be simplified considerably. For example an alternative scheme to (5) might be 1 1 A B u0 ¼ 0; u1 ¼ uðx; 0Þ L1 a þ cu þ du þ b 0 0 0xx 0xxx ; t 1þp 1þr x x 1 1 A B a þ cu þ du unþ1 ¼ L1 þ b n n nxx nxxx ; n P 1: t 1þp 1þr x x ð9Þ Table 1 The numerical results for juðx; tÞ /2 ðx; tÞj when a ¼ 1, b ¼ 1, d ¼ 1 and c ¼ 1, for the solitarywave solutions (14) of the Eq. (2) ðxi ; ti Þ (0.1,0.1)
(0.2,0.2)
(0.3,0.3)
(0.4,0.4)
(0.5,0.5)
p¼1 0.0000287359 0.0000288311 0.0000289253 0.0000290186 0.000029111
0.0000574719 0.0000576622 0.0000578507 0.0000580374 0.0000582223
0.0000862079 0.0000864935 0.0000867764 0.0000870565 0.0000873338
0.000114944 0.000115325 0.000115702 0.000116076 0.000116446
0.00014368 0.000144156 0.000144628 0.000145095 0.000145558
p¼2 0.000165647 0.000166019 0.00016637 0.0001667 0.000167008
0.000331325 0.00033207 0.000332773 0.000333432 0.000334048
0.000497033 0.000498152 0.000499206 0.000500197 0.000501122
0.000662772 0.000664265 0.000665672 0.000666993 0.000668228
0.000828542 0.000830409 0.00083217 0.000833822 0.000835367
p¼4 0.000290829 0.000290745 0.000290575 0.000290319 0.000289977
0.000581716 0.000581549 0.00058121 0.000580698 0.000580015
0.000872663 0.000872413 0.000871904 0.000871137 0.000870113
0.00116367 0.00116334 0.00116266 0.00116164 0.00116027
0.00145473 0.00145432 0.00145347 0.0014522 0.00145049
p¼6 0.000297283 0.000296657 0.00029591 0.000295045 0.000294061
0.000594611 0.000593359 0.000591867 0.000590135 0.000588169
0.000891984 0.000890107 0.000887868 0.000885272 0.000882322
0.0011894 0.0011869 0.00118392 0.00118045 0.00117652
0.00148686 0.00148374 0.00148001 0.00147568 0.00147077
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D. Kaya / Appl. Math. Comput. 152 (2004) 709–720
Finally based on the ADM, we constructed the solution uðx; tÞ as lim /n ¼ uðx; tÞ;
n!1
where /n ðx; tÞ ¼
n X
uk ðx; tÞ;
ð10Þ
nP0
k¼0
and the recurrence relation is given as in (7). Moreover, the decomposition series (8) solutions are generally converged very rapidly in real physical problems [2]. The convergence of the decomposition series have investigated by several authors. The theoretical treatment of convergence of the decomposition method has been considered in the literature [22–27]. They obtained some results about the speed of convergence of this method providing us to solve linear and nonlinear functional equations. In recent work of Abbaoui et al. [27] have proposed a new approach of convergence of the decomposition series. The authors have given a new condition for obtaining convergence of the decomposition series to the classical presentation of the ADM in [27]. In this work, we Table 2 The numerical results for juðx; tÞ /5 ðx; tÞj when a ¼ 1, b ¼ 1, d ¼ 1 and c ¼ 1, for the solitarywave solutions (14) of the Eq. (2) ðxi ; ti Þ (0.1,0.1)
(0.2,0.2)
(0.3,0.3)
(0.4,0.4)
(0.5,0.5)
p¼1 8.59915 · 109 8.60308 · 109 8.60603 · 109 8.60802 · 109 8.60902 · 109
3.43947 · 108 3.44103 · 108 3.4422 · 108 3.44297 · 108 3.44336 · 108
7.73839 · 108 7.74185 · 108 7.74445 · 108 7.74617 · 108 7.74701 · 108
1.37564 · 107 1.37625 · 107 1.3767 · 107 1.377 · 107 1.37715 · 107
2.14932 · 107 2.15026 · 107 2.15096 · 107 2.15142 · 107 2.15164 · 107
p¼2 0.0000272923 0.0000275842 0.0000278741 0.0000281619 0.0000284475
0.0000543881 0.0000549702 0.0000555483 0.0000561224 0.000056692
0.0000812865 0.0000821569 0.0000830216 0.0000838803 0.0000847325
0.000107987 0.000109143 0.000110293 0.000111435 0.000112568
0.000134487 0.000135929 0.000137362 0.000138785 0.000140198
p¼4 0.000291223 0.000291135 0.00029096 0.000290698 0.000290351
0.000583297 0.000583112 0.000582753 0.00058222 0.000581514
0.000876228 0.000875938 0.000875385 0.000874569 0.000873493
0.00117002 0.00116962 0.00116886 0.00116775 0.00116629
0.00146469 0.00146416 0.00146318 0.00146177 0.00145992
p¼6 0.000297828 0.000297188 0.000296428 0.000295547 0.000294549
0.000596794 0.000595488 0.000593938 0.000592147 0.000590118
0.000896904 0.000894903 0.000892533 0.0008898 0.000886708
0.00119816 0.00119544 0.00119222 0.00118851 0.00118432
0.00150058 0.0014971 0.00149299 0.00148827 0.00148295
D. Kaya / Appl. Math. Comput. 152 (2004) 709–720
715
demonstrate how approximate solutions of the cKdV and cKdV–B equations are close to exact solutions. In the first example, we will consider Eq. (1) for the special case associated the initial condition uðx; 0Þ ¼ W ð1 þ tanhðKxÞÞ; where W ¼
að1 þ 2pÞd þ bpbc ; 2bð2 þ pÞd
K¼
bpbc þ að1 þ 2pÞd 2bbdð2 þ pÞ
and p ¼ 1 is positive constant. To find the solution of the initial value problem (9) with (8) for p ¼ 1 and performing the integration we obtain the following u0 ¼ 0;
u1 ¼ W ð1 þ tanhðKxÞÞ;
ð11Þ
u2 ¼ K 2 tW sech4 ðKxÞ½4dK 2dK coshð2KxÞ þ c sinhð2KxÞ;
ð12Þ
Table 3 The numerical results for juðx; tÞ /7 ðx; tÞj when a ¼ 1, b ¼ 1, d ¼ 1 and c ¼ 1, for the solitarywave solutions (14) of the Eq. (2) ðxi ; ti Þ (0.1,0.1)
(0.2,0.2)
(0.3,0.3)
(0.4,0.4)
(0.5,0.5)
p¼1 1.54757 · 1012 1.54374 · 1012 1.53952 · 1012 1.53484 · 1012 1.52971 · 1012
1.23419 · 1011 1.23116 · 1011 1.22779 · 1011 1.22407 · 1011 1.22 · 1011
4.15239 · 1011 4.14222 · 1011 4.13091 · 1011 4.11843 · 1011 4.10478 · 1011
9.81192 · 1011 9.78797 · 1011 9.76133 · 1011 9.73194 · 1011 9.6998 · 1011
1.91039 · 1010 1.90575 · 1010 1.90058 · 1010 1.89487 · 1010 1.88863 · 1010
p¼2 6.2543 · 108 6.27652 · 108 6.29616 · 108 6.31317 · 108 6.3275 · 108
2.49053 · 107 2.49954 · 107 2.50753 · 107 2.51448 · 107 2.52037 · 107
5.57839 · 107 5.59895 · 107 5.61723 · 107 5.63319 · 107 5.6468 · 107
9.87196 · 107 9.90901 · 107 9.94206 · 107 9.97104 · 107 9.99587 · 107
1.5354 · 106 1.54127 · 106 1.54652 · 106 1.55114 · 106 1.55513 · 106
p¼4 0.0000700311 0.0000709808 0.0000719129 0.0000728264 0.0000737204
0.000140886 0.000142778 0.000144634 0.000146451 0.000148229
0.000212558 0.000215385 0.000218157 0.000220871 0.000223522
0.000285041 0.000288798 0.000292479 0.00029608 0.000299596
0.000358331 0.000363011 0.000367594 0.000372075 0.000376447
p¼6 0.000297828 0.000297188 0.000296428 0.000295547 0.000294549
0.000596793 0.000595487 0.000593937 0.000592146 0.000590117
0.0008969 0.000894899 0.00089253 0.000889797 0.000886705
0.00119815 0.00119543 0.00119221 0.0011885 0.00118431
0.00150055 0.00149707 0.00149296 0.00148825 0.00148293
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D. Kaya / Appl. Math. Comput. 152 (2004) 709–720
u3 ¼ sech3 ðKxÞ½aKtW 2 coshðKxÞ þ aKW 2 t sinhðKxÞ K 4 t2 W sech7 ðKxÞ ½160dcK coshðKxÞ þ 100dcK coshð3KxÞ 4 4dcK coshð5KxÞ 10c2 sinhðKxÞ þ 1208d2 K 2 sinhðKxÞ
9c2 sinhð3KxÞ 228d2 K 2 sinhð3KxÞ þ c2 sinhð5KxÞ þ 4d2 K 2 sinhð5KxÞ; ð13Þ in this manner the components of the decomposition series (10) are obtained as far as we like. This series is exact to the last term, as one can verify, of the Taylor series of the exact closed form solution (2) for the special case of the p ¼ 1 as uðx; tÞ ¼ W f1 þ tanh½Kðx ktÞg;
ð14Þ
Table 4 The numerical results for juðx; tÞ /9 ðx; tÞj when a ¼ 1, b ¼ 1, d ¼ 1 and c ¼ 1, for the solitarywave solutions (14) of the Eq. (2) ðxi ; ti Þ (0.1,0.1)
(0.2,0.2)
(0.3,0.3)
(0.4,0.4)
(0.5,0.5)
p¼1 1.29202 · 1014 1.30451 · 1014 1.32117 · 1014 1.33366 · 1014 1.34615 · 1014
9.82825 · 1014 9.95037 · 1014 1.00739 · 1013 1.01988 · 1013 1.03181 · 1013
3.14845 · 1013 3.19342 · 1013 3.2388 · 1013 3.28376 · 1013 3.32845 · 1013
7.0674 · 1013 7.18245 · 1013 7.29833 · 1013 7.41393 · 1013 7.52953 · 1013
1. 30386 · 1012 1.32799 · 1012 1.35222 · 1012 1.37657 · 1012 1.40092 · 1012
p¼2 6.75975 · 109 6.85603 · 109 6.95076 · 109 7.04383 · 109 7.13513 · 109
2.63079 · 108 2.67039 · 108 2.70947 · 108 2.74797 · 108 2.78585 · 108
5.75369 · 108 5.84529 · 108 5.93592 · 108 6.02546 · 108 6.1138 · 108
9.93244 · 108 1.00998 · 107 1.02658 · 107 1.04303 · 107 1.0593 · 107
1.50534 · 107 1.53221 · 107 1.55893 · 107 1. 58546 · 107 1.61178 · 107
p¼4 0.000070034 0.0000709835 0.0000719154 0.0000728287 0.0000737225
0.000140909 0.000142799 0.000144654 0.00014647 0.000148246
0.000212635 0.000215458 0.000218225 0.000220934 0.00022358
0.000285225 0.000288971 0.00029264 0.000296228 0.000299733
0.000358689 0.000363347 0.000367907 0.000372364 0.000376713
p¼6 0.0000840544 0.0000853079 0.0000865292 0.0000877162 0.0000888673
0.00016928 0.000171774 0.000174201 0.000176557 0.00017884
0.000255674 0.000259396 0.000263014 0.000266524 0.000269919
0.000343232 0.000348171 0.000352968 0.000357616 0.000362107
0.000431952 0.0004381 0.000444064 0.000449836 0.000455407
D. Kaya / Appl. Math. Comput. 152 (2004) 709–720
717
where bpbc þ að1 þ 2pÞd ; 2bbdð2 þ pÞ ðbpð1 þ pÞbc að1 þ 2pÞdÞðbpbc þ að1 þ 2pÞdÞ
K¼ k¼
2
bð2 þ pÞ ð1 þ pÞð1 þ 2pÞd2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 þ pÞð1 þ 2pÞd b¼ bp2
;
and a, b, b, c, d, p ¼ 1 are real constants. In the some other examples, we will consider for the cKdV equation (1) when c ¼ 0 in Eq. (2) with the general form of the initial condition (3) for different values of the constant p. In a similar way, performing the calculations in (9) with (8) using Mathematica and substituting into (10) gives the exact Table 5 The numerical results for juðx; tÞ /2 ðx; tÞj when a ¼ 1, b ¼ 10, and d ¼ 1, for the solitary-wave solutions (3) of the Eq. (1) ðxi ; ti Þ (0.1,0.1)
(0.2,0.2)
(0.3,0.3)
(0.4,0.4)
(0.5,0.5)
p¼1 0.0000322588 0.0000322104 0.00003213 0.0000320177 0.0000318738
0.0000645176 0.000064421 0.0000642602 0.0000640356 0.0000637479
0.0000967764 0.0000966316 0.0000963905 0.0000960538 0.0000956223
0.000129035 0.000128842 0.000128521 0.000128072 0.000127497
0.000161294 0.000161053 0.000160652 0.000160091 0.000159372
p¼2 0.00015442 0.000149039 0.000143356 0.000137394 0.000131181
0.000308843 0.000298082 0.000286714 0.000274791 0.000262365
0.000463268 0.000447127 0.000430077 0.000412192 0.000393553
0.000617697 0.000596176 0.000573442 0.000549596 0.000524745
0.000772129 0.000745228 0.00071681 0.000687003 0.000655939
p¼4 0.000201247 0.000179832 0.000157938 0.000135724 0.000113352
0.000402497 0.000359667 0.000315878 0.00027145 0.000226706
0.000603748 0.000539504 0.00047382 0.000407179 0.000340062
0.000805001 0.000719342 0.000631765 0.000542909 0.00045342
0.00100626 0.000899183 0.000789711 0.000678642 0.00056678
p¼6 0.000156556 0.000127407 0.0000979896 0.0000685682 0.0000394029
0.000313113 0.000254815 0.00019598 0.000137137 0.0000788067
0.000469671 0.000382224 0.000293971 0.000205707 0.000118211
0.00062623 0.000509634 0.000391964 0.000274278 0.000157617
0.000782789 0.000637044 0.000489957 0.00034285 0.000197023
718
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solution in a series form (3). A numerical discussions will be given in the following section.
3. Experimental results for the cKdV and cKdV–B equations In order to verify numerically whether the proposed methodology lead to accurate solutions, we will evaluate the decomposition series solutions using the n-terms (n ¼ 2, 5, 7, and 9) approximation for some examples of the cKdV– B equations solved in the previous section. The differences between the n-terms solution and the exact solution for some values of the constant p ¼ 1, 2, 4, 6 are shown in Table 1–4. The numerical values in all the tables show that we achieved a very good approximation to the actual solution of the equations by using small values of n-terms of the decomposition series solution derived Table 6 The numerical results for juðx; tÞ /5 ðx; tÞj when a ¼ 1, b ¼ 10, and d ¼ 1, for the solitary-wave solutions (3) of the Eq. (1) ðxi ; ti Þ (0.1,0.1)
(0.2,0.2)
(0.3,0.3)
(0.4,0.4)
(0.5,0.5)
p¼1 0.0000160888 0.0000160541 0.0000159982 0.0000159212 0.0000158234
0.000032099 0.0000320249 0.0000319087 0.0000317506 0.0000315511
0.0000480269 0.0000479086 0.0000477276 0.0000474843 0.0000471795
0.0000638684 0.0000637013 0.000063451 0.0000631186 0.000062705
0.0000796197 0.0000793989 0.0000790753 0.0000786499 0.0000781241
p¼2 7.74788 · 106 0.0000146063 0.000021599 0.0000286966 0.0000358694
0.000017903 0.0000317606 0.0000458637 0.0000601528 0.0000745677
0.0000306496 0.0000516191 0.0000729194 0.0000944612 0.000116154
0.0000461722 0.0000743376 0.000102891 0.000131715 0.000160688
0.000064655 0.000100072 0.000135905 0.000172006 0.000208229
p¼4 0.000193418 0.000171879 0.000149977 0.000127871 0.000105714
0.000370567 0.000327627 0.000284199 0.000240576 0.000197043
0.000530529 0.000466907 0.00040291 0.000338925 0.00027532
0.000672386 0.000589382 0.000506357 0.000423726 0.000341878
0.000795218 0.000694714 0.000594785 0.000495789 0.000398048
p¼6 0.000143295 0.000114278 0.0000852293 0.0000563945 0.0000280091
0.000259513 0.000202786 0.000146433 0.0000908594 0.0000364463
0.000347817 0.000266256 0.000185853 0.00010702 0.0000301338
0.000407371 0.000305419 0.00020573 0.000108502 0.0000138939
0.00043734 0.000321006 0.000208307 0.0000989301 7.45107 · 106
D. Kaya / Appl. Math. Comput. 152 (2004) 709–720
719
above. It is evident that the overall errors can be made smaller by adding new terms of the decomposition series. We also evaluate the decomposition series solutions using the n-terms (n ¼ 2, 5) approximation for some examples of the cKdV equations solved as same as in Eq. (2). It is surprisingly, numerical efficiency not as good as in the decomposition series solution of cKdV–B equation. The numerical comparison of the Eq. (1) is given for some values of the constant p ¼ 1, 2, 4, 6 are shown in Table 5 and 6. The solutions are very rapidly convergent by utilizing the ADM. The numerical results we obtained justify the advantage of this methodology. Furthermore, as the decomposition method does not require discretization of the variables, i.e. time and space, it is not effected by computation roundoff errors and necessity of large computer memory and time. Clearly, the series solution methodology can be applied to various type of linear or nonlinear system of partial differential equations [28,29] and single partial differential equations [30–36] as well.
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