In this paper, we study the (1+1)-dimensional dispersive long wave equations which describe the evolution of horizontal velocity component u

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 International Journal of Advanced and Applied Sciences, 4(8) 2017, Pages: 98-103       Contents lists available at Science-Gate  International Journal of Advanced and Applied Sciences Journal homepage: http://www.science-gate.com/IJAAS.html   98  Successive approximation method for solving (1+1)-dimensional dispersive long wave equations   Saad A. Manaa *, Fadhil H. Easif, Bewar Y. Ali  Department of Mathematics, Faculty of Science, University of Zakho, Kurdistan Region, Iraq  A R T I C L E  I N F O   A B S T R A C T  Article history: Received 30 March 2017 Received in revised form 8 July 2017 Accepted 17 July 2017 In this paper, we study the (1+1)-dimensional dispersive long wave equations which describe the evolution of horizontal velocity component 𝑢(𝑥, 𝑡) of water waves of height 𝑣(𝑥, 𝑡), and solved it numerically by successive approximation method (SAM) to compare with Adomian’s decomposition method (ADM), we found that SAM is suitable for this kind of problems also its effective and more accure than ADM. Mathematica has been used for computations. Keywords: Successive approximation method Adomain decomposition method (1+1)-dimensional dispersive long wave equations © 2017 The Authors. Published by IASE. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).   1. Introduction *The celebrated (1+1)-dimensional dispersive long wave equations (Ablowitz and Clarkson, 1991; Broer, 1975)  𝑢𝑡 + 𝑣𝑥 +12(𝑢2)𝑥 = 0,                     (1) 𝑣𝑡 + (𝑢𝑣 + 𝑢 + 𝑢𝑥𝑥)𝑥 = 0.   Play important roles in nonlinear physics, which describe the evolution of horizontal velocity component 𝑢(𝑥, 𝑡) of water waves of height 𝑣(𝑥, 𝑡) propagating in both directions in an infinite narrow channel of finite constant depth. Many properties of (1) have been reported (Broer, 1975; Kaup, 1975; Kupershmidt, 1985).  It is interesting to study the extensions of (1) in higher-dimensional spaces. To date, there exist two prototypical extensions of (1) to cover the situation of wide channel or open seas. Boiti et al. (1987) presented the following (2+1)-dimensional extension related to (1)  𝑢𝑦𝑡 + 𝑣𝑥𝑥 +12(𝑢2)𝑥𝑦 = 0,                     (2) 𝑣𝑡 + (𝑢𝑣 + 𝑢 + 𝑢𝑥𝑦)𝑥= 0,   in the one-dimensional reduction 𝑢 =  𝑢(𝑥 + 𝑦, 𝑡),  𝑣 =  𝑣(𝑥 + 𝑦, 𝑡), system (2) reduces to system (1) that describes the travel of the shallow water wave.                                                   * Corresponding Author.  Email Address: abdnawras@gmail.com (S. A. Manaa) https://doi.org/10.21833/ijaas.2017.08.014 2313-626X/© 2017 The Authors. Published by IASE.  This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Eckhaus (1985) presented another different two-dimensional extension of (1)  𝑢𝑡 + 𝑣𝑥 +12(𝑢2)𝑥 = 0,                     (3) 𝑣𝑡𝑥 + (𝑢𝑣 + 𝑢 + 𝑢𝑥𝑥)𝑥𝑥  + 𝑢𝑦𝑦 = 0,   which was obtained in the appropriate approximation from the basic equations of hydrodynamics. It is easy to see that if one makes the transformation either 𝑢 =  𝑢(𝑥 +  𝑦, 𝑡)  and  𝑣 = 𝑣(𝑥 +  𝑦, 𝑡) –  1 or 𝑢 =  𝑢(𝑥 +  𝑦, 𝑡) and 𝑣 = 𝑣(𝑥 + 𝑦, 𝑡), then (3) can also reduce to (1). Therefore, it follows that these two systems (2) and (3) can both reduce to the same system (1) under the proper transformations. But as Boiti et al. (1987) pointed out, system (2) is different from system (3).  2. Solitary wave solution The exact solitary wave solution of the (2 + 1)-dimensional dispersive long wave equations (2) are:   𝑢(𝑥, 𝑦, 𝑡) = 𝛼 + 𝛼 tanh (𝛼 𝑥+𝛽𝑦−𝛼2 𝑡+𝛿2),                   (4) 𝑣(𝑥, 𝑦, 𝑡) = −1 +𝛼𝛽2sech2 (𝛼 𝑥+𝛽𝑦−𝛼2 𝑡+𝛿2)                    (5)  where 𝛼, 𝛽 and 𝛿 are arbitrary constants. In particular, taking 𝛽 = 𝛼 and setting 𝑥 +  𝑦 = 𝑧 in system (1), we obtain a special solitary wave solution of (1 + 1)-dimensional dispersive long wave equations (1),  𝑢(𝑧, 𝑡) = 𝛼 + 𝛼 tanh (𝛼 𝑧−𝛼2 𝑡+𝛿2),                    (6) 𝑣(𝑧, 𝑡) = −1 +𝛼22sech2 (𝛼 𝑧−𝛼2 𝑡+𝛿2),                                       (7)  which was obtained by Bai et al. (2006) and Wang et al. (1996). Manaa et al/ International Journal of Advanced and Applied Sciences, 4(8) 2017, Pages: 98-103 99 3. Basic idea of successive approximationmethod (SAM)The method of SAM (Jerri, 1999; Saeed, 2006), provides a method that can, in principle, be used to solve any initial value problem 𝜕𝑢𝜕𝑡= 𝑓(𝑢, 𝑡),    𝑢(𝑧, 𝑡0) = 𝑢0.  (8) It starts by observing that any solution to (8) must also be a solution to 𝑢(𝑧, 𝑡) = 𝑢0 + ∫ 𝑓(𝑢(𝑧, 𝑠), 𝑠)𝑡𝑡0 𝑑𝑠,  (9) and then iteratively constructs a sequence of solutions that can get closer and closer to the actual (exact) solutions of (9). The SAM is based on the integral equation (9), as follows: 𝑢0(𝑧, 𝑡) = 𝑢0,  𝑢1(𝑧, 𝑡) = 𝑢0 + ∫ 𝑓(𝑢0(𝑧, 𝑠), 𝑠)𝑡𝑡0 𝑑𝑠, 𝑢2(𝑧, 𝑡) = 𝑢0 + ∫ 𝑓(𝑢1(𝑧, 𝑠), 𝑠)𝑡𝑡0 𝑑𝑠, 𝑢3(𝑧, 𝑡) = 𝑢0 + ∫ 𝑓(𝑢2(𝑧, 𝑠), 𝑠)𝑡𝑡0 𝑑𝑠.  This process can be continued to obtain the nth approximation, 𝑢𝑛(𝑧, 𝑡) = 𝑢0 + ∫ 𝑓(𝑢𝑛−1(𝑧, 𝑠), 𝑠)𝑡𝑡0 𝑑𝑠,   𝑛 = 1, 2,…  .  Then determine whether 𝑢𝑛(𝑧, 𝑡) approaches the solution 𝑢(𝑧, 𝑡) as n increases. This is done by proving the following: The sequence {𝑢𝑛(𝑧, 𝑡)} converges to a limit 𝑢(𝑧, 𝑡), that is: lim𝑛→∞𝑢𝑛(𝑧, 𝑡) = 𝑢(𝑧, 𝑡) ,   𝑡0 ≤ 𝑧 ≤ 𝑡 . The limiting function 𝑢(𝑧, 𝑡) is a solution of (9) on the interval 𝑡0 ≤ 𝑧 ≤ 𝑡. The solution 𝑢(𝑧, 𝑡) of (9) is unique. A proof of these results can be constructed along the lines of the corresponding proof for ordinary differential equations (Coddington, 1995). 4. SAM applied to (1+1)-dimensional dispersive long wave equationsIn this section, we solve the (1+1)-dimensional dispersive long wave equations: 𝜕𝑢(𝑧,𝑠)𝜕𝑠= −𝜕𝑣(𝑧,𝑠)𝜕𝑧−12𝜕𝑢2(𝑧,𝑡)𝜕𝑧 , 𝜕𝑣(𝑧,𝑠)𝜕𝑠= −𝜕𝜕𝑧(𝑢(𝑧, 𝑠) + 𝑢(𝑧, 𝑠) 𝑣(𝑧, 𝑠)) −𝜕3𝑢(𝑧,𝑠)𝜕𝑧3, }   (10) with initial solutions 𝑢(𝑧, 0) = 𝑢0 and 𝑣(𝑧, 0) = 𝑣0, by using SAM as follows: Integrating both sides of equation (10) with respect to 𝑠, from 0 to 𝑡, we get 𝑢(𝑧, 𝑡) = 𝑢(𝑧, 0) − ∫ (𝜕𝑣(𝑧,𝑠)𝜕𝑧+12𝜕𝑢2(𝑧,𝑠)𝜕𝑧) 𝑑𝑠,𝑡0  𝑣(𝑧, 𝑡) = 𝑣(𝑧, 0) − ∫ (𝜕𝜕𝑧(𝑢(𝑧, 𝑠) + 𝑢(𝑧, 𝑠) 𝑣(𝑧, 𝑠))+𝜕3𝑢(𝑧,𝑠)𝜕𝑧3)𝑡0𝑑𝑠.} (11) Using the initial conditions in (11) we get: 𝑢(𝑧, 𝑡) = 𝑢0 − ∫ (𝜕𝑣(𝑧,𝑠)𝜕𝑧+12𝜕𝑢2(𝑧,𝑠)𝜕𝑧) 𝑑𝑠𝑡0,  𝑣(𝑧, 𝑡) = 𝑣0 − ∫ (𝜕𝜕𝑧(𝑢(𝑧, 𝑠) + 𝑢(𝑧, 𝑠) 𝑣(𝑧, 𝑠))+𝜕3𝑢(𝑧,𝑠)𝜕𝑧3)𝑡0𝑑𝑠.} (12) Start with substituting initial approximation 𝑢0(𝑧, 𝑡) and 𝑣0(𝑧, 𝑡)  in the integral equation (12) to obtain a first approximation 𝑢1(𝑧, 𝑡) and 𝑣1(𝑧, 𝑡)  𝑢1(𝑧, 𝑡) = 𝑢0 − ∫ (𝜕𝑣0(𝑧,𝑠)𝜕𝑧+12𝜕𝑢02(𝑧,𝑠)𝜕𝑧) 𝑑𝑠𝑡0,  𝑣1(𝑧, 𝑡) = 𝑣0 − ∫ (𝜕𝜕𝑧(𝑢0(𝑧, 𝑠) + 𝑢0(𝑧, 𝑠) 𝑣0(𝑧, 𝑠))+𝜕3𝑢0(𝑧,𝑠)𝜕𝑧3)𝑡0𝑑𝑠.} (13) Then this 𝑢1(𝑧, 𝑡) and 𝑣1(𝑧, 𝑡) is substituted again in the integral of (12) after replacing t by s to obtain a second approximation 𝑢2(𝑧, 𝑡) and 𝑣2(𝑧, 𝑡) , 𝑢2(𝑧, 𝑡) = 𝑢0 − ∫ (𝜕𝑣1(𝑧,𝑠)𝜕𝑧+12𝜕𝑢12(𝑧,𝑠)𝜕𝑧) 𝑑𝑠,𝑡0  𝑣2(𝑧, 𝑡) = 𝑣0 − ∫ (𝜕𝜕𝑧(𝑢1(𝑧, 𝑠) + 𝑢1(𝑧, 𝑠) 𝑣1(𝑧, 𝑠))+𝜕3𝑢1(𝑧,𝑠)𝜕𝑧3)𝑡0𝑑𝑠.This process can be continued to obtain the 𝑛𝑡ℎ approximation  𝑢𝑛(𝑧, 𝑡) = 𝑢0 − ∫ (𝜕𝑣𝑛−1(𝑧,𝑠)𝜕𝑧+12𝜕𝑢𝑛−12 (𝑧,𝑠)𝜕𝑧) 𝑑𝑠,𝑡0  𝑣𝑛(𝑧, 𝑡) = 𝑣0 − ∫ (𝜕𝜕𝑧(𝑢𝑛−1(𝑧, 𝑠)+𝑢𝑛−1(𝑧, 𝑠) 𝑣𝑛−1(𝑧, 𝑠))+𝜕3𝑢𝑛−1(𝑧,𝑠)𝜕𝑧3)𝑡0𝑑𝑠,} (14) For 𝑛 =1,2,…. To solve the equation (14), we use the initial approximation 𝑢0(𝑧, 𝑡) and 𝑣0(𝑧, 𝑡)  which are given in equation (6) and (7), respectively. Using the iteration formula (14), we can obtain 𝑢1(𝑧, 𝑡), 𝑢2(𝑧, 𝑡), 𝑢3(𝑧, 𝑡), 𝑣1(𝑧, 𝑡), 𝑣2(𝑧, 𝑡) and 𝑣3(𝑧, 𝑡) as follows: 𝑢1(𝑧, 𝑡) = 𝛼 + 𝛼 tanh (𝛼 𝑧+𝛿2)  −𝛼32sech2 (𝛼 𝑧+𝛿2) 𝑡,  𝑢2(𝑧, 𝑡) =  𝛼 + 𝛼 tanh (𝛼 𝑧+𝛿2) −𝛼32sech2 (𝛼 𝑧+𝛿2) 𝑡 −𝛼54sech2 (𝛼 𝑧+𝛿2) tanh (𝛼 𝑧+𝛿2) 𝑡2 +𝛼712sech4 (𝛼 𝑧+𝛿2) tanh (𝛼 𝑧+𝛿2) 𝑡3, 𝑢3(𝑧, 𝑡) =  𝛼 + 𝛼 tanh (𝛼 𝑧+𝛿2) −𝛼32sech2 (𝛼 𝑧+𝛿2) 𝑡 −𝛼54sech2 (𝛼 𝑧+𝛿2) tanh (𝛼 𝑧+𝛿2) 𝑡2 +𝛼724sech2 (𝛼 𝑧+𝛿2) (1 −3 tanh2 (𝛼 𝑧+𝛿2)) 𝑡3 −5 𝛼9192sech4 (𝛼 𝑧+𝛿2) (1 −2 tanh (𝛼 𝑧+𝛿2) − 5 tanh2 (𝛼 𝑧+𝛿2) +518tanh3 (𝛼 𝑧+𝛿2)) 𝑡4 +Manaa et al/ International Journal of Advanced and Applied Sciences, 4(8) 2017, Pages: 98-103 100  𝛼11240sech4 (𝛼 𝑧+𝛿2) (1 −32tanh (𝛼 𝑧+𝛿2) − 8 tanh2 (𝛼 𝑧+𝛿2) +92tanh3 (𝛼 𝑧+𝛿2) + 7 tanh4 (𝛼 𝑧+𝛿2)) 𝑡5 +𝛼13288sech6 (𝛼 𝑧+𝛿2) tanh (𝛼 𝑧+𝛿2) (1 − 4 tanh2 (𝛼 𝑧+𝛿2)) 𝑡6 −𝛼152016sech8 (𝛼 𝑧+𝛿2) tanh (𝛼 𝑧+𝛿2) (1 − 5 tanh2 (𝛼 𝑧+𝛿2)) 𝑡7 , 𝑣1(𝑧, 𝑡) = −1 +𝛼22sech2 (𝛼 𝑧+𝛿2) +𝛼42sech2 (𝛼 𝑧+𝛿2) tanh (𝛼 𝑧+𝛿2) 𝑡  𝑣2(𝑧, 𝑡) = −1 +𝛼22sech2 (𝛼 𝑧+𝛿2) +𝛼42sech2 (𝛼 𝑧+𝛿2) tanh (𝛼 𝑧+𝛿2) 𝑡 −𝛼68sech2 (𝛼 𝑧+𝛿2) (1 −3 tanh2 (𝛼 𝑧+𝛿2)) 𝑡2 +𝛼824sech4 (𝛼 𝑧+𝛿2) (1 −5 tanh2 (𝛼 𝑧+𝛿2)) 𝑡3,  𝑣3(𝑧, 𝑡) =  −1 +𝛼22sech2 (𝛼 𝑧+𝛿2) +𝛼42sech2 (𝛼 𝑧+𝛿2) tanh (𝛼 𝑧+𝛿2) 𝑡 −𝛼68sech2 (𝛼 𝑧+𝛿2) (1 −3 tanh2 (𝛼 𝑧+𝛿2)) 𝑡2 −𝛼812sech2 (𝛼 𝑧+𝛿2) tanh (𝛼 𝑧+𝛿2) (2 −3 tanh2 (𝛼 𝑧+𝛿2)) 𝑡3 +𝛼10192sech4 (𝛼 𝑧+𝛿2) (5 +35 tanh (𝛼 𝑧+𝛿2) − 52 tanh2 (𝛼 𝑧+𝛿2) − 75 tanh3 (𝛼 𝑧+𝛿2) +63 tanh4 (𝛼 𝑧+𝛿2)) 𝑡4 −3 𝛼12320sech4 (𝛼 𝑧+𝛿2) (3 +40 tanh (𝛼 𝑧+𝛿2) − 42 tanh2 (𝛼 𝑧+𝛿2) − 152 tanh3 (𝛼 𝑧+𝛿2) +63 tanh4 (𝛼 𝑧+𝛿2) + 112 tanh5 (𝛼 𝑧+𝛿2)) 𝑡5 +𝛼14576sech6 (𝛼 𝑧+𝛿2) (1 − 19 tanh2 (𝛼 𝑧+𝛿2) +36 tanh4 (𝛼 𝑧+𝛿2)) 𝑡6 −𝛼164032sech8 (𝛼 𝑧+𝛿2) (1 −24 tanh2 (𝛼 𝑧+𝛿2) + 55 tanh4 (𝛼 𝑧+𝛿2)) 𝑡7.  5. Results and discussion We solved the (1+1)-dimensional dispersive long wave equations by SAM. Compared between 3rd-order approximate solutions of SAM and ADM (Mohamed, 2010) with exact solitary solutions. The results it shown by data and surfaces see in Tables 1a, 1b, 2a, and 2b and Figs. 1, 2, and 3. And we found the effects for 𝛼 and 𝛿 on the solution of (1+1)-dimensional dispersive long wave equations as shown in Figs. 4a, 4b, 5a, and 5b. It shown that 𝛼 has effects on long and short in wave and 𝛿 has effects on move wave only.   Table 1a: Exact solution, 3rd-order approximation solutions ADM (Mohamed, 2010) and SAM of 𝑢(𝑧, 𝑡) when 𝛼 = 2 , 𝛿 = 0 and 𝑧 = −0.01 t Exact ADM SAM 0 1.98000066664 1.98000066664 1.98000066664 0.01 1.9400179935224 1.9400179965071 1.9400178614665 0.02 1.9000832500842 1.9000833114428 1.9000811646242 0.03 1.8602282193671 1.860228598649 1.8602178003544 0.04 1.8204844305057 1.8204858453275 1.8204519384 0.05 1.7808830595711 1.7808870386801 1.7808047973543 0.06 1.7414548327879 1.7414641659086 1.7412947478293 0.07 1.7022299327534 1.7022492142149 1.7019374154519 0.08 1.6632379082584 1.6632741708006 1.662745783697 0.09 1.6245075882634 1.6245710228678 1.6237302965644 0.1 1.5860670005411 1.5861717576181 1.584898961109  Table 1b: Exact solution, 3rd-order approximation solutions ADM (Mohamed, 2010) and SAM of 𝑣(𝑧, 𝑡) when = 2 , 𝛿 = 0 and 𝑧 = −0.01 t Exact ADM SAM 0 0.99980001333258 0.99980001333258 0.99980001333258 0.01 0.99820107944946 0.99820086649064 0.99820111308927 0.02 0.99500832154314 0.99500491883283 0.99500884803683 0.03 0.99023192466936 0.9902147296339 0.99023454277574 0.04 0.98388708015456 0.98383285816862 0.98389524331417 0.05 0.97599388320855 0.97586186371178 0.97601363254549 0.06 0.96657719825563 0.96630430553815 0.96661796371 0.07 0.95566649352597 0.95516274292249 0.95574201102401 0.08 0.9432956467829 0.94243973513958 0.94342503665934 0.09 0.92950272436413 0.9281378414642 0.92971177325638 0.1 0.91432973597948 0.9122596211711 0.91465242115369  Table 2a: Absolute error between exact solution and 3rd-order approximation solutions ADM (Mohamed, 2010) and SAM of 𝑢(𝑧, 𝑡) when 𝛼 = 2 , 𝛿 = 0 and 𝑧 = −0.01 t | Exact - ADM | | Exact - SAM | 0 0 0 0.01 2.9847E-09 1.32056E-07 0.02 6.13586E-08 2.08546E-06 0.03 3.79282E-07 1.0419E-05 0.04 1.41482E-06 3.24921E-05 0.05 3.97911E-06 7.82622E-05 0.06 9.33312E-06 0.000160085 0.07 1.92815E-05 0.000292517 0.08 3.62625E-05 0.000492125 0.09 6.34346E-05 0.000777292 0.1 0.000104757 0.001168039 Total Error 0.000238906 0.003013449  Table 2b: Absolute error between exact solution and 3rd-order approximation solutions ADM (Mohamed, 2010) and SAM of 𝑣(𝑧, 𝑡) when 𝛼 = 2 , 𝛿 = 0 and 𝑧 = −0.01 t | Exact - ADM | | Exact - SAM | 0 0 0 0.01 2.12959E-07 3.36398E-08 0.02 3.40271E-06 5.26494E-07 0.03 1.7195E-05 2.61811E-06 0.04 5.4222E-05 8.16316E-06 0.05 0.000132019 1.97493E-05 0.06 0.000272893 4.07655E-05 0.07 0.000503751 7.55175E-05 0.08 0.000855912 0.00012939 0.09 0.001364883 0.000209049 0.1 0.002070115 0.000322685 Total Error 0.005274605 0.000808498  Manaa et al/ International Journal of Advanced and Applied Sciences, 4(8) 2017, Pages: 98-103 101   Exact 𝒖 Exact 𝒗   (a) (b) Fig. 1: The surfaces of exact solutions for 𝑢(𝑧, 𝑡) and 𝑣(𝑧, 𝑡), when 𝛼 = 2 , 𝛿 = 0 and  𝑧 ∈ [−10,10] at  0 ≤ 𝑡 ≤ 0.1  ADM 𝑢 ADM 𝑣   (a) (b) Fig. 2: The surfaces of ADM solutions for 𝑢(𝑧, 𝑡) and 𝑣(𝑧, 𝑡), when 𝛼 = 2 , 𝛿 = 0 and  𝑧 ∈ [−10,10] at  0 ≤ 𝑡 ≤ 0.1  SAM 𝑢 SAM 𝑣   (a) (b) Fig. 3: The surfaces of SAM solutions for 𝑢(𝑧, 𝑡) and 𝑣(𝑧, 𝑡), when 𝛼 = 2 ,𝛿 = 0 and  𝑧 ∈ [−10,10] at  0 ≤ 𝑡 ≤ 0.1  Effects of 𝜶 in 𝒖(𝒛, 𝒕)  𝜶 = 𝟎. 𝟓 𝛼 = 1 𝛼 = 2 𝛼 = 3 Fig. 4a: The surfaces of exact solution of 𝑢(𝑧, 𝑡), when 𝛼 = 0.5, 1,2, 3 at 𝛿 = 0, 𝑧 ∈ [−10,10], 0 ≤ 𝑡 ≤ 0.1 Manaa et al/ International Journal of Advanced and Applied Sciences, 4(8) 2017, Pages: 98-103 102  Effects of 𝜹 in 𝒖(𝒛, 𝒕)  𝜹 = −𝟒 𝛿 = 0 𝛿 = 4 𝛿 = 8 Fig. 4b: The surfaces of exact solution of 𝑢(𝑧, 𝑡), when 𝛿 = −4, 0, 4, 8 at 𝛼 = 1, 𝑧 ∈ [−10,10], 0 ≤ 𝑡 ≤ 0.1   Effects of 𝜶 in 𝒗(𝒛, 𝒕)  𝜶 = 𝟎. 𝟓 𝛼 = 1 𝛼 = 2 𝛼 = 3 Fig. 5a: The surfaces of exact solution of 𝑣(𝑧, 𝑡), when 𝛼 = 0.5, 1,2, 3 at 𝛿 = 0, 𝑧 ∈ [−10,10], 0 ≤ 𝑡 ≤ 0.1   Effects of 𝜹 in 𝒗(𝒛, 𝒕)  𝜹 = −𝟒 𝛿 = 0 𝛿 = 4 𝛿 = 8 Fig. 5b: The surfaces of exact solution of 𝑣(𝑧, 𝑡), when 𝛿 = −4, 0, 4, 8 at  𝛼 = 1, 𝑧 ∈ [−10,10], 0 ≤ 𝑡 ≤ 0.1  6. Conclusion In this paper, the SAM successfully applied to obtain approximate solutions of the (1+1)-dimensional dispersive long wave. We showed that successive approximation method easier, faster and more accurate than Adomain’s decomposition method as shown especially in Tables 1b and 2b and Figs. 2b and 3b. Finally, it has become clear that 𝛼 and 𝛿 have effects on solutions. Acknowledgment  We thank the editor and the referee for their comments. List of symbols  SAM Successive Approximate Method ADM Adomain Decomposition Method Greek symbols  𝛼, 𝛽 𝑎𝑛𝑑 𝛿 Constants  References  Ablowitz MJ and Clarkson PA (1991). Solitons, nonlinear evolution equations and inverse scattering. Cambridge University Press, Cambridge, UK. Bai CL, Zhao H, and Han JG (2006). Application of an extended homogeneous balance method to new exact solutions of nonlinear evolution equations. Czechoslovak Journal of Physics, 56(3): 237-242. Boiti M, Leon JJP, and Pempinelli F (1987). 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Comparison differential transformation technique with adomian decomposition method for dispersive long-wave equations in (2+1)-dimensions. Applications and Applied Mathematics: An International Journal (AAM), 5(1): 148-166. Saeed RK (2006). Computational methods for solving system of linear volterra integral and integro-differential equations. Ph.D. Dissertation, University of Salahaddin/Erbil-Collage of Science, Erbil, Iraqi Kurdistan. Wang M, Zhou Y, and Li Z (1996). Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics. Physics Letters A, 216(1–5): 67-75.   

Successive approximation method for solving (1+1)-dimensional dispersive long wave equations

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Successive approximation method for solving (1+1)-dimensional dispersive long wave equations

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