# Iterative Solutions of Hirota Satsuma Coupled KDV and Modified Coupled KDV Systems

Authors: Muhammad Jibran; Rashid Nawaz; Asfandyar Khan; Sajjad Afzal
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

In this article the approximate solutions of nonlinear Hirota Satsuma coupled Korteweg De- Vries (KDV) and modified coupled KDV equations have been obtained by using reliable algorithm of New Iterative Method (NIM). The results obtained give higher accuracy than that of homotopy analysis method (HAM). The obtained solutions show that NIM is effective, simpler, easier, and explicit and gives a suitable way to control the convergence of the approximate solution.


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## Body

## 1. Introduction

The nonlinear coupled system of partial differential equations has variety of applications in different fields of mechanics, biology, hydrodynamics, and plasma physics. The exact solution of nonlinear partial differential equations cannot be found easily. To find the approximate solutions of these types of equations many researchers have adopted different approaches. The well-known approaches for approximate solutions of coupled system of differential equations are numerical methods, iterative methods, perturbation methods, homotopy based methods, etc. Each approach has its own advantages and disadvantages. In numerical methods discretization is used which affect the accuracy. They also required most computational work and time. In the case of strong nonlinear problems the numerical methods do not give us more accurate result. Some numerical methods are collocation methods (CMs) [1, 2], finite difference method (FDM) [3, 4], finite element method (FEM) [5, 6], radial basis function (RBF) method [7, 8], etc. Perturbation techniques use perturbation quantities to transform a nonlinear problem into infinite number of linear subproblems. Perturbation methods have some limitations as well, such as nonavailability of perturbative quantity and strong nonlinearity. To avoid this difficulty, there do exist homotopy based methods as well [9–11]. Adomian Decomposition Method (ADM) [11] and Differential Transformation Method (DTM) [12] deal with the nonlinearity but their main drawback is that the convergence region of obtained solution is generally small. Iterative methods are mathematical techniques that produce a sequence of approximate solutions. They used initial guess to create sequential approximations. Since these methods involve repetition of the same process many times, computers can act well for finding solutions of equations iteratively. A definite way of operation of an iterative method is termination criteria. Iterative methods have important role in finding approximate solution of differential equations. Some of the well-known iterative methods are variational iteration method (VIM) [13, 14], modified variational iteration method (MVIM) [15, 16], optimal variational iteration method (OVIM) [17], etc.Recently Daftardar-Gejji and Jafari [18] have proposed an iterative method for solving linear and nonlinear functional equations called the New Iterative Method (NIM). This method has proven useful for solving a variety of nonlinear equations such as algebraic equations, ODEs, PDEs, evolution equations, and system of nonlinear dynamical systems [19–33].The KDV equation arises in the study of nonlinear dispersive waves [34]. It was derived by Korteweg and de Vries in 1895 for modeling of shallow water waves in canal [35]. This equation plays important role in diverse areas of engineering and scientific applications and, therefore, enormous amount of research work has been invested in the study of such equations. Wu et al. derived a new corresponding hierarchy of nonlinear evolution equations [35]. Fan worked on the extended tanh-function method and symbolic computation to obtain, respectively, four kinds of Soliton solutions for a new coupled MKDV and new generalized Hirota Satsuma coupled KDV equations [36]. Raslan et al. worked on the decomposition method to obtain the Soliton solutions for generalized Hirota Satsuma coupled KDV and coupled MKDV equations [37]. Zhong et al. applied the VIM to obtain approximate analytic solutions of coupled MKDV and generalized Hirota Satsuma coupled KDV equations [38]. Kangalgil et al. demonstrated the feasibility and validity of DTM for proposed equations [39]. Arife et al. presented the numerical solution of systems of Hirota Satsuma coupled KDV and coupled MKDV equations by means of HAM [9]. Ghoreishi et al. applied HAM for obtaining the approximate solution of MKDV system [40].Our aim in this paper is to extend the application of NIM for finding the approximate solutions of nonlinear Hirota Satsuma coupled KDV and modified coupled KDV equations. The results obtained are in close agreement with the exact solutions than that of HAM. The accuracy of NIM can further be improved by increasing the number of iterations.


## 2. The Governing Coupled System

Recently, by introducing a 4 × 4 matrix spectral problem with three potentials, Wu et al. derived a new corresponding hierarchy of nonlinear evolution equations [35]. Two typical equations in the hierarchy are new generalized Hirota Satsuma coupled KDV and new coupled MKDV equations.We considered the following new generalized Hirota Satsuma coupled KDV equation,(1)ut=12uxxx-3uux+3vwx+3wvx,vt=-vxxx+3uvx,wt=-wxxx+3uwx,and modified coupled KDV equation(2)ut=12uxxx-3u2ux+32vxx+3uvx+3vux-3λux,vt=-vxxx-3vvx-3uxvx+3u2vx+3λvx.Withw=v∗ (1)-(2) reduce to new complex coupled Hirota Satsuma KDV and modified coupled complex KDV equations, respectively [35, 41]. For u=0, (2) reduces to generalized KDV equation and, for v=0, it reduces to modified KDV equation.


### 2.1. Basic Idea of NIM

The basic mathematical theory of New Iterative Method is described as follows.Let us consider the nonlinear equation of the form of(3)ux,t=fx,t+Nux,t.wheref(x) is known function x=x1,x2,x3,…,xn and N denotes nonlinear operator from a Banach space B→B. According to the basic idea of NIM solution of the above equation has the series form(4)ux,t=∑k=0∞ukx,t,The nonlinear operator N can be decomposed as (5)N∑k=0∞uk=Nu0+∑k=1∞N∑j=0kuj-N∑j=0k-1uj.Hence general equation of (1) can be written as (6)ux=∑k=0∞ukx=f+Nu0+∑k=1∞N∑j=0kuj-N∑j=0k-1uj,From (6) we have (7)u0=f,G0=Nu0,G1=Nu0+u1-Nu0,G2=Nu0+u1+u2-Nu0+u1,and in general (8)Gm=Nu0+u1+…+um-Nu0+u1+…um-1.m=1,2,3…The k-term approximate solution of the general equation (1) is (9)u=u0+u1+…+uk-1.


### 2.2. Convergence Criteria of NIM

Theorem 1.
IfN is C∞ in a neighborhood of u0 and (10)Nmu0=sup⁡Nmu0k1,k2,…knkj≤1,1≤j≤m≤l,for any m and for some real l>0&uj≤M<1/ej=1,2,…….., then the series ∑m=0∞Gm is absolutely convergent and (11)um≤lMmem-1e-1.m=1,2,…Sufficient condition for convergence is as follows.Theorem 2.
IfN is C∞&Nm(u0)≤M≤e-1,∀m, then the series ∑m=0∞Gm is absolutely convergent.The reader is referred to [18, 22] to get more insight of convergence analysis.


## 2.1. Basic Idea of NIM

The basic mathematical theory of New Iterative Method is described as follows.Let us consider the nonlinear equation of the form of(3)ux,t=fx,t+Nux,t.wheref(x) is known function x=x1,x2,x3,…,xn and N denotes nonlinear operator from a Banach space B→B. According to the basic idea of NIM solution of the above equation has the series form(4)ux,t=∑k=0∞ukx,t,The nonlinear operator N can be decomposed as (5)N∑k=0∞uk=Nu0+∑k=1∞N∑j=0kuj-N∑j=0k-1uj.Hence general equation of (1) can be written as (6)ux=∑k=0∞ukx=f+Nu0+∑k=1∞N∑j=0kuj-N∑j=0k-1uj,From (6) we have (7)u0=f,G0=Nu0,G1=Nu0+u1-Nu0,G2=Nu0+u1+u2-Nu0+u1,and in general (8)Gm=Nu0+u1+…+um-Nu0+u1+…um-1.m=1,2,3…The k-term approximate solution of the general equation (1) is (9)u=u0+u1+…+uk-1.


## 2.2. Convergence Criteria of NIM

Theorem 1.
IfN is C∞ in a neighborhood of u0 and (10)Nmu0=sup⁡Nmu0k1,k2,…knkj≤1,1≤j≤m≤l,for any m and for some real l>0&uj≤M<1/ej=1,2,…….., then the series ∑m=0∞Gm is absolutely convergent and (11)um≤lMmem-1e-1.m=1,2,…Sufficient condition for convergence is as follows.Theorem 2.
IfN is C∞&Nm(u0)≤M≤e-1,∀m, then the series ∑m=0∞Gm is absolutely convergent.The reader is referred to [18, 22] to get more insight of convergence analysis.


## 3. Implementation of NIM to Hirota Satsuma Coupled KDV and Coupled MKDV Equations

Problem 3.
Consider the coupled Hirota Satsuma KDV equation given by (1) together with following initial conditions [35].(12)ux,0=13β-8k2+4k2tanh2⁡kx,vx,0=-43k4F0-2βk2F2+4k4F23F22+4k2F2tanh2⁡kx,wx,0=F0+F2tanh2⁡kx,where k=0.1, λ=1. By using the initial conditions, the given equation (1) is equivalent to the following integral equations:(13)ux,t=13β-8k2+4k2tanh2⁡kx+∫12uxxx-3uux+3vwx+3wvxdt,vx,t=-43k4F0-2βk2F2+4k4F23F22+4k2F2tanh2⁡kx+∫-vxxx+3uvxdt,wx,t=F0+F2tanh2⁡kx+∫-wxxx+3uwxdt.Appling the basic idea of NIM and using the recursive relation (6) we have the following.Zeroth Order Solutions (14) u 0 x , t = 1 3 β - 8 k 2 + 4 k 2 tanh 2 ⁡ k x , (15) v 0 x , t = - 4 3 k 4 F 0 - 2 β k 2 F 2 + 4 k 4 F 2 3 F 2 2 + 4 k 2 F 2 tanh 2 ⁡ k x , (16) w 0 x , t = F 0 + F 2 tanh 2 ⁡ k x .   First Order Solutions (17) u 1 x , t = t - 24 k 3 sech 2 ⁡ k x tanh ⁡ k x 1 3 - 8 k 2 + β + 4 k 2 tanh 2 ⁡ k x + 6 k sech 2 ⁡ k x F 2 tanh ⁡ k x - 4 3 k 4 F 0 + 4 k 4 F 2 - 2 k 2 β F 2 3 F 2 2 + 4 k 2 tanh 2 ⁡ k x F 2 + 24 k 3 sech 2 ⁡ k x tanh ⁡ k x F 0 + F 2 tanh 2 ⁡ k x F 2 + 2 k 2 - 16 k 3 sech 4 ⁡ k x tanh ⁡ k x + 8 k 3 sech 2 ⁡ k x tanh 3 ⁡ k x , (18) v 1 x , t = t 24 k 3 sech 2 ⁡ k x tanh ⁡ k x 1 / 3 - 8 k 2 + β + 4 k 2 tanh 2 ⁡ k x F 2 - 4 k 2 - 16 k 3 sech 4 ⁡ k x tanh ⁡ k x F 2 + 8 k 3 sech 2 ⁡ k x tanh 3 ⁡ k x F 2 , (19) w 1 x , t = t 6 k sech 2 ⁡ k x F 2 tanh ⁡ k x 1 3 - 8 k 2 + β + 4 k 2 tanh 2 ⁡ k x - F 2 - 16 k 3 sech 4 ⁡ k x tanh ⁡ k x + 8 k 3 sech 2 ⁡ k x 2 tanh 3 ⁡ k x .   Second Order Solutions  (20)
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1
32 The NIM yields the 2nd order solutions u(x,t),v(x,t),w(x,t) as follows: (23)ux,t=u0x,t+u1x,t+u2x,t+⋯,vx,t=v0x,t+v1x,t+v2x,t+⋯,wx,t=w0x,t+w1x,t+w2x,t+⋯.(24)ux,t=13-8k2+β+4k2tanh2⁡kx+t43k4F0+4k4F2-2k2βF23F22-24k3sech2⁡kxtanh⁡kx13-8k2+β+4k2tanh2⁡kx+6ksech2⁡kxF2tanh⁡kx-43k4F0+4k4F2-2k2βF23F22+4k2tanh2⁡kxF2+24k3sech2⁡kxF2+tanh⁡kxF2F0+F2tanh2⁡kx+2k2-16k3sech4⁡kxtanh⁡kx+8k3sech2⁡kxtanh3⁡kx43k4F0+4k4F2-2k2βF23F22+1F228k5t2sech2⁡kx72k2-1+k22t-2+cosh⁡2kxsech4⁡kxF02tanh⁡kx+3k-1+k2F0F22+sech2⁡kx32kt-6+6k2-β+48ktsech2⁡kx12-12k2+β+8-1+k2sech2⁡kx-15tanh⁡kxtanh⁡kx+F222k-6+6k2-β+sech2⁡kx3k62-62k2+5β+15ksech2⁡kx-32+32k2-β-21-1+k2sech2⁡kx-4t2-72k6+β2+24k46+β-2k26+β2+3sech2⁡kx216k6-β2-48k49+β+2k26+β18+β+8k2-1+k2sech2⁡kx49-9k2+β+15-1+k2sech2⁡kxtanh⁡kx-6+6k2-β-6+6k2-β-1+k22,13(25)vx,t=13F224k2-3k2F0+3-4k2+2β+3sech2⁡kx-1-2k2t2β2+3k2t2sech2⁡kx-48k4-16k2-3+β+β2+8k2β+-6+6k2+βcosh⁡2kxsech4⁡kxF2+6ktβsech2⁡kxtanh⁡kx1+4k4t2-2+cosh⁡2kx6-6k2+β+-6+6k2+βcosh⁡2kx×sech6⁡kxF2+8k3tsech2⁡kx2ktβ-2+3sech2⁡kx+3tanh⁡kx.13F22(26)wx,t=F0-132ktsech9⁡kx-384k3-1+k2tcosh2⁡kxsinh⁡kx-8ktβ+4ktβcosh⁡2kx-3sinh⁡2kxF0-kt576k2-576k4+25β2cosh⁡kx+9kt-96k2+96k4+β2cosh⁡3kx+288k3tcosh⁡5kx-288k5tcosh⁡5kx-ktβ2cosh⁡5kx-ktβ2cosh⁡7kx+5βsinh⁡kx-6144k4t2βsinh⁡kx+6144k6t2βsinh⁡kx+256k4t2β2sinh⁡kx+9βsinh⁡3kx+2688k4t2βsinh⁡3kx-2688k6t2βsinh⁡3kx+192k4t2β2sinh⁡3kx+5βsinh⁡5kx-384k4t2βsinh⁡5kx+384k6t2βsinh⁡5kx-64k4t2β2sinh⁡5kx+βsinh⁡7kxF2+F2tanh2⁡kx,132Problem 4.
Consider the modified coupled KDV equation given by (2) together with following initial conditions [35].(27)ux,0=ktanh⁡kx,vx,0=124k2+λ-2k2tanh2⁡kx.Zeroth Order Solutions(28)u0x,t=ktanh⁡kx,(29)v0t,x=124k2+λ-2k2tanh2⁡kx.First Order Solutions(30)u1x,t=t-3k2λsech2⁡kx-15k4sech2⁡kxtanh2⁡kx+3k2sech2⁡kx124k2+λ-2k2tanh2⁡kx-3k22k2sech4⁡kx-4k2sech2⁡kxtanh2⁡kx+12k-2k3sech4⁡kx+4k3sech2⁡kxtanh2⁡kx,(31)v1x,t=t-12k3λsech2⁡kxtanh⁡kx+12k5sech4⁡kxtanh⁡kx-12k5sech2⁡kxtanh3⁡kx+12k3sech2⁡kxtanh⁡kx124k2+λ-2k2tanh2⁡kx+2k2-16k3sech4⁡kxtanh⁡kx+8k3sech2⁡kxtanh3⁡kx.Second Order Solutions(32)u2x,t=-164k2sech8⁡kx20k2t-2752k8t3+30tλ-4416k6t3λ-432k4t3λ2+15t2k2+3λcosh⁡2kx+8k4t3154k2-9λ2k2+3λcosh⁡2kx+6t2k2+3λcosh⁡4kx-8k4t310k2-9λ2k2+3λcosh⁡4kx+2k2tcosh⁡6kx+3tλcosh⁡6kx-4022k5t2sinh⁡2kx+48k11t4sinh⁡2kx-546k3t2λsinh⁡2kx+216k9t4λsinh⁡2kx+452kt2λ2sinh⁡2kx+324k7t4λ2sinh⁡2kx+162k5t4λ3sinh⁡2kx+584k5t2sinh⁡4kx-264k3t2λsinh⁡4kx+18kt2λ2sinh⁡4kx+2k5t2sinh⁡6kx+6k3t2λsinh⁡6kx+92kt2λ2sinh⁡6kx-t-3k2λsech2⁡kx-15k4sech2⁡kxtanh2⁡kx+3k2sech2⁡kx124k2+λ-2k2tanh2⁡kx-3k22k2sech4⁡kx-4k2sech2⁡kxtanh2⁡kx+12k-2k3sech4⁡kx+4k3sech2⁡kxtanh2⁡kx,(33)v2x,t=18k4t2sech2⁡kx82k2-3λ2-90k6tsech8⁡kxt2k2+3λ2+64ktanh⁡kx+3k4sech6⁡kx3-736+t210k2-3λ2k2+3λ2+64kt50k2-9λtanh⁡kx+4sech2⁡kx-3212k4-84k2λ+9λ2+64k5t2k2-3λtanh⁡kx+2k2sech4⁡kx-468λ+3k21480-t22k2-3λ2k2+3λ2-4kt580k4-276k2λ+9λ2tanh⁡kx,the NIM yields the solutions u(x,t), v(x,t) as follows:(34)ux,t=u0x,t+u1x,t+u2x,t+⋯,vx,t=v0x,t+v1x,t+v2x,t+⋯.(35)ux,t=-164k2sech8⁡kx20k2t-2752k8t3+30tλ-4416k6t3λ-432k4t3λ2+15t2k2+3λcosh⁡2kx+8k4t3154k2-9λ2k2+3λcosh⁡2kx+6t2k2+3λcosh⁡4kx-8k4t310k2-9λ2k2+3λcosh⁡4kx+2k2tcosh⁡6kx+3tλcosh⁡6kx-4022k5t2sinh⁡2kx+48k11t4sinh⁡2kx-546k3t2λsinh⁡2kx+216k9t4λsinh⁡2kx+452kt2λ2sinh⁡2kx+324k7t4λ2sinh⁡2kx+162k5t4λ3sinh⁡2kx+584k5t2sinh⁡4kx-264k3t2λsinh⁡4kx+18kt2λ2sinh⁡4kx+2k5t2sinh⁡6kx+6k3t2λsinh⁡6kx+92kt2λ2sinh⁡6kx+ktanh⁡kx,(36)vx,t=184λ+k2sech2⁡kx82+k2t22k2-3λ2+ktsech⁡kx162k2-3λsinh⁡kx-90k7t2sech7⁡kxt2k2+3λ2+64ktanh⁡kx+4ksech⁡kx-3t212k4-84k2λ+9λ2+8k-3+8k4t22k2-3λtanh⁡kx+3k5tsech5⁡kx3-736+t210k2-3λ2k2+3λ2+64kt50k2-9λtanh⁡kx-2k3tsech3⁡kx468λ+3k2-1480+t22k2-3λ2k2+32λ+4kt580k4-276k2λ+9λ2tanh⁡kx.Problem 5.
Consider the coupled Hirota Satsuma KDV equation given by (1) together with following initial conditions [35].(37)ux,0=13β-2k2+2k2tanh2⁡kx,vx,0=-4k2F0β+k23F22+4k2β+k23F2tanh⁡kx,wx,0=F0+F2tanh⁡kx.Zeroth Order Solutions(38)u0x,t=13β-2k2+2k2tanh2⁡kx,(39)v0x,t=-4k2F0β+k23F22+4k2β+k23F2tanh⁡kx,(40)w0x,t=F0+F2tanh⁡kx.First Order Solutions(41)u1x,t=t3ksech2⁡kxF2-4k2k2+βF03F22+4k2k2+βtanh⁡kx3F2+4k3k2+βsech2⁡kxF0+F2tanh⁡kxF2-12k3sech2⁡kxtanh⁡kx13-2k2+β+2k2tanh2⁡kx+k2-16k3sech4⁡kxtanh⁡kx+8k3sech2⁡kxtanh3⁡kx,(42)v1x,t=t4k3k2+βsech2⁡kx13-2k2+β+2k2tanh2⁡kxF2-4k2k2+β-2k3sech4⁡kx3F2+4k3sech2⁡kxtanh2⁡kx3F2.(43)w1x,t=t3ksech2⁡kxF213-2k2+β+2k2tanh2⁡kx-F2-2k3sech4⁡kx+4k3sech2⁡kxtanh2⁡kx.Second Order Solutions(44)u2x,t=-2k5sech6⁡kx-3kt2β+2kt2βcosh⁡2kx-kt2βcosh⁡4kx-4tsinh⁡2kx+43k25t3β2sinh⁡2kx+43t3β3sinh⁡2kx+tsinh⁡4kx-t3ksech2⁡kxF2-4k2k2+βF03F22+4k2k2+βtanh⁡kx3F2+4k3k2+βsech2⁡kxF0+F2tanh⁡kxF2-12k3sech2⁡kxtanh⁡kx13-2k2+β+2k2tanh2⁡kx+k2-16k3sech4⁡kxtanh⁡kx+8k3sech4⁡kxtanh3⁡kx,(45)v2x,t=-16F2k3βk2+βsech6⁡kx-3t-32k4t3β-4tcosh⁡2kx+32k4t3βcosh⁡2kx-tcosh⁡4kx+kt2βsinh⁡4kx-t4k3k2+βsech2⁡kx1/3-2k2+β+2k2tanh2⁡kxF2-4k2k2+β-2k3sech4⁡kx+4k3sech2⁡kxtanh2⁡kx3F2+2kt2βsinh⁡2kx,(46)w2x,t=18kβsech6⁡kx3t+32k4t3β+4tcosh⁡2kx-32k4t3βcosh⁡2kx+tcosh⁡4kx-40k3t2sinh⁡2kx+4k3t2sinh⁡4kxF2-t3ksech2⁡kxF213-2k2+β+2k2tanh2⁡kx-F2-2k3sech4⁡kx+4k3sech2⁡kxtanh2⁡kx.The NIM yields the solutions u(x,t), v(x,t), w(x,t) as follows:(47)ux,t=u0x,t+u1x,t+u2x,t,vx,t=v0x,t+v1x,t+v2x,t,wx,t=w0x,t+w1x,t+w2x,t.(48)ux,t=13-2k2+β-2k5sech6⁡kx-3kt2β+2kt2βcosh⁡2kx-kt2βcosh⁡4kx-4tsinh⁡2kx+43k2t3β2sinh⁡2kx+43t3β3sinh⁡2kx+tsinh⁡4kx+2k2tanh2⁡kx,(49)vx,t=-4k2k2+βF03F22-16F2k3βk2+βsech6⁡kx-3t-32k4t3β-4tcosh⁡2kx+32k4t3βcosh⁡2kx-tcosh⁡4kx+2kt2βsinh⁡2kx+kt2βsinh⁡4kx+4k2k2+βtanh⁡kx3F2,(50)wx,t=F0-18kβsech6⁡kx-3t-32k4t3β-4tcosh⁡2kx+32k4t3βcosh⁡2kx-tcosh⁡4kx+2kt2βsinh⁡2kx+kt2βsinh⁡4kxF2+F2tanh⁡kx,Problem 6.
Consider the coupled modified KDV equation given by (2) together with following initial condition [35].(51)ux,0=b12k+ktanh⁡kx,vx,0=λ21+kb1+b1tanh⁡kx.Zeroth Order Solutions(52)u0x,t=b12k+ktanh⁡kx,(53)v0x,t=λ21+kb1+b1tanh⁡kx.First Order Solutions(54)u1x,t=t-3k2λsech2⁡kx-3k2sech2⁡kxb1tanh⁡kx+3ksech2⁡kxb1b12k+ktanh⁡kx-3k2sech2⁡kxb12k+ktanh⁡kx2+3k2sech2⁡kx12λ1+kb1+b1tanh⁡kx+12k-2k3sech4⁡kx+4k3sech2⁡kxtanh2⁡kx,(55)v1x,t=t3kλsech2⁡kxb1-3k3sech4⁡kxb1+3ksech2⁡kxb1b12k+ktanh⁡kx2-3ksech2⁡kxb112λ1+kb1+b1tanh⁡kx-b1-2k3sech4⁡kx+4k3sech2⁡kxtanh2⁡kx.Second Order Problem(56)u2x,t=116b12kt2sech2⁡kx12b12k2b14k4+6λk2-3b13-6λk3-k3t-2b12k4+3λk2+3b13+6λk32sech4⁡kx-4b1kb14k4+6λk2-3b13-6λk3tanh⁡kxsech2⁡kxb1t-3b12+4k4-6λk2+3k32λt+1+12b12k2-2b12k4+3λk2+3b13+6λk3sech2⁡kx+29b13λk3-9b14λ-2k2-12b1λk5k2-3λ+1+2b12k42k2+3λ4k2-3λ+2-9b16-18λ2k6tanh⁡kx,(57)v2x,t=1128k2b12t2sech⁡kx324k3tsech⁡kx3b114k3λ+26k4-7k2λb1-9b13-6k3λ+4k4+6k2λb1-3b13-8sinh⁡kxb1-6k3λ+-4k4+6k2λb1+3b132-64k3sech⁡kxb1-36k6tλ2+b172k5tλ2+b116k8t-9k2λ-36k4tλ2+9kλb1-24k4tb12+9tb14+3t2sech⁡kx36k3λ+4k4-6k2λb1-3b136k4λ-22k5+3k3λb1+3kb132tanh⁡kx+192k3λsech⁡kxk-b1b126k32+tλ+tb14k4-6k2λ-3b12tanh⁡kx,NIM yields the solutions u(x,t), v(x,t) as follows:(58)ux,t=u0x,t+u1x,t+u2x,t+⋯,vx,t=v0x,t+v1x,t+v2x,t+⋯.(59)ux,t=t3k2sech2⁡kx12λkb1+1+b1tanh⁡kx-3k2sech2⁡kxb12k+ktanh⁡kx2-3b1k2tanh⁡kxsech2⁡kx+3b1ksech2⁡kxb12k+ktanh⁡kx+12k4k3tanh2⁡kxsech2⁡kx-2k3sech4⁡kx-3k2λsech2⁡kx+116b12kt2sech2⁡kx12b12k2b14k4+6λk2-3b13-6λk3-k3t-2b12k4+3λk2+3b13+6λk32sech4⁡kx-4b1kb14k4+6λk2-3b13-6λk3tanh⁡kxsech2⁡kxb1t-3b12+4k4-6λk2+3k32λt+1+12b12k2-2b12k4+3λk2+3b13+6λk3sech2⁡kx+29b13λk3-9b14λ-2k2-12b1λk5k2-3λ+1+2b12k42k2+3λ4k2-3λ+2-9b16-18λ2k6tanh⁡kx+b12k+ktanh⁡kx,(60)vx,t=1128k2b1264k2λb1k+b1-32ktsech2⁡kxb126k3λ+4k4-6k2λb1-3b13+24k3t3sech6⁡kxb114k3λ+26k4-7k2λb1-9b13-6k3λ+4k4+6k2λb1-3b13-64k3t2sech4⁡kxb1-36k6tλ2+b172k5tλ2+b116k8t-9k2λ-36k4tλ2+9kλb1-24k4tb12+9tb14+128k2b13tanh⁡kx-8t2sech⁡kxb1-6k3λ+-4k4+6k2λb1+3b132tanh⁡kx+3t4sech6⁡kx6k3λ+4k4-6k2λb1-3b136k4λ-22k5+3k3λb1+3kb132tanh⁡kx+192k3t2λsech4⁡kxk-b1b126k32+tλ+tb14k4-6k2λ-3b12tanh⁡kx.


## 4. Results and Discussion

NIM formulation is tested upon the nonlinear Hirota Satsuma coupled KDV and modified coupled KDV equations. We have used Mathematica 7 for most of our computational work. Table1 shows the comparison of 2nd order NIM and HAM solution for u(x,t), v(x,t), and w(x,t) components of Hirata Satsuma coupled KDV equation with initial condition given by (12), while Table 2 shows the absolute errors of NIM at F0=1, F2=1, k=0.1, β=1, and t=1. Table 5 shows the comparison of 2nd order NIM and HAM solution for u(x,t), v(x,t), and w(x,t) components of Hirata Satsuma coupled KDV equation with initial condition (37), while Table 6 shows the absolute errors of NIM at F0=1, F2=1, k=0.1, t=2, and β=1.Table 1
Comparison of 2nd order NIM solution with 2nd order HAM solution for nonlinear coupled Hirota Satsuma KDV equation usingF0=1, F2=1, k=0.1, β=1, and t=1.
x u e x a c t u H A M [9] u N I M v e x a c t v H A M [9] v N I M w e x a c t w H A M [9] w N I M -50 0.346658 0.346659 0.346659 0.0657245 0.065726 0.0657245 1.99978 1.99982 1.99978 -40 0.346601 0.346613 0.346613 0.0656678 0.0656793 0.0656679 1.99836 1.99865 1.99836 -30 0.346185 0.346273 0.346269 0.0652518 0.0653356 0.0652524 1.98796 1.99006 1.98798 -20 0.343242 0.343845 0.343831 0.0623089 0.0628901 0.0623147 1.91439 1.92892 1.91457 -10 0.32719 0.329861 0.329974 0.0462566 0.0489609 0.0462879 1.51308 1.58069 1.51464 0 0.307064 0.306667 0.306682 0.0261307 0.0257347 0.0261333 1.00993 1.00003 1.01 10 0.332299 0.329874 0.329913 0.0513653 0.489077 0.0513998 1.6408 1.57936 1.64234 20 0.344338 0.343836 0.343839 0.0634044 0.0629245 0.0634037 1.94178 1.92978 1.94179 30 0.346343 0.346271 0.346269 0.0654099 0.0653417 0.0654095 1.99192 1.99021 1.9919 40 0.346623 0.346613 0.346613 0.0656894 0.0656801 0.0656893 1.9989 1.99867 1.9989 50 0.346661 0346659 0.346659 0.0657274 0.0657261 0.0657274 1.99985 1.99982 1.99985Table 2
Absolute errors of 2nd order NIM solution for Hirota Satsuma KDV equation usingF0=1, F2=1, k=0.1, β=1, and t=1.
x u N I M v N I M w N I M 50 1.55045×10−6 1.02035×10−8 2.55355×10−7 40 1.14454×10−5 7.60484×10−8 1.91571×10−6 30 8.39765×10−5 5.96212×10−7 1.56830×10−5 20 5.88572×10−4 5.77582×10−6 1.81550×10−4 10 2.78424×10−3 3.12612×10−5 1.55799×10−3 0 3.81828×10−4 2.65163×10−6 6.62908×10−5 -10 2.38551×10−3 3.45628×10−5 1.54605×10−3 -20 4.98990×10−4 6.68171×10−7 1.22730×10−5 -30 7.38904×10−5 4.38566×10−7 1.03679×10−5 -40 1.01336 ×10−5 6.69141×10−8 1.66176×10−6 -50 1.37393×10−6 9.19754×10−9 2.29735×10−7Tables3 and 4 show the comparison of 2nd order NIM solution with HAM solution for u(x,t) and v(x,t) components of modified coupled KDV equation with initial condition (27) at k=0.1λ=1 and t=0.5. Table 7 shows the comparison of 2nd order NIM with exact solution for u(x,t) and v(x,t) of coupled modified KDV equation with initial condition (51) at k=0.1, b1=0.1, λ=0.1, and t=0.5.Table 3
Comparison of 2nd order NIM solution with 2nd order HAM solution foru(x,t) part of nonlinear coupled modified KDV equation at k=0.1λ=1, t=0.5.
x u e x a c t u H A M [9] u N I M u Absolute errors -50 -0.0999922 -0.0999895 -0.0999922 5.01465×10−9 -40 -0.0999423 -0.0999228 -0.0999423 3.68284×10−8 -30 -0.0995746 -0.0994310 -0.0995744 2.59984×10−7 -20 -0.0968991 -0.0958693 -0.0968978 1.31416×10−6 -10 -0.0791524 -0.0729886 -0.0791603 7.82324×10−6 0 -0.00753569 -0.0075500 -0.0075491 1.34504×10−5 10 0.0728019 0.0793302 0.0728181 1.61449×10−5 20 0.098286 0.0969362 0.0958309 2.28210×10−6 30 0.0994251 0.0995800 0.0994254 2.97728×10−7 40 0.099922 00999431 0.099922 4.00387×10−8 50 0.0999894 0.0999923 0.0999894 5.41386×10−9Table 4
Comparison of 2nd order NIM solution with 2nd order HAM solution forv(x,t) of nonlinear coupled modified KDV equation at k=0.1λ=1, t=0.5.
x v e x a c t v H A M [9] v N I M v A b s o l u t e   e r r o r s -50 0.500003 0.50003 0.500004 1.04894×10−6 -40 0.500023 0.500019 0.500031 8.34759×10−6 -30 0.50017 0.500141 0.500229 5.89469×10−5 -20 0.501221 0.501021 0.50163 4.09288×10−4 -10 0.50747 0.506530 0.509398 1.92802×10−3 0 0.519886 0.519889 0.519887 2.36659×10−7 10 0.5094 0.510342 0.507475 1.92439×10−3 20 0.501634 0.501833 0.501224 4.09834×10−4 30 0.500229 0.500258 0.50017 5.90785×10−5 40 0.500031 0.500035 0.500023 8.06431×10−6 50 0.500004 0.500005 0.500003 1.09265×10−6Table 5
Comparison of 2nd order NIM solution with the exact solution foru(x,t), v(x,t), and w(x,t) of nonlinear coupled Hirota Satsuma KDV equation using F0=1, F2=1, k=0.1, β=1, and t=2.
x u e x a c t u N I M v e x a c t v N I M w e x a c t w N I M -50 0.346661 0.346662 -0.0269315 -0.0269315 0.00013 0.00013 -40 0.346627 0.346629 -0.0269199 -0.0269200 0.00100 0.00099 -30 0.346373 0.346391 -0.0268341 -0.0268349 0.00736 0.00731 -20 0.344595 0.344710 -0.0262170 -0.0262220 0.05319 0.05319 -10 0.335486 0.335776 -0.0224089 -0.0224203 0.33596 0.33596 0 0.327446 0.326635 -0.0108087 -0.0107733 1.19738 1.20000 10 0.340566 0.340779 -0.0022401 -0.0022525 1.83365 1.83273 20 0.345708 0.345795 -0.0003266 -0.0003308 1.97574 1.97543 30 0.346534 0.346548 -0.0000447 -0.0000453 1.99668 1.99664 40 0.346649 0.346651 -6.05507x10−6 -6.14233x10−6 1.99955 1.99954 50 0.346664 0.346664 -8.19623x10−7 -8.19623x10−7 1.99994 1.99994Table 6
Absolute errors of 2nd order NIM solution of Hirota Satsuma KDV equation usingF0=1, F2=1, k=0.1, t=2, and β=1.
x u N I M v N I M w N I M 50 3.27577×10−7 1.44533×10−8 1.07327×10−6 40 2.4149×10−6 1.06534×10−7 7.91091×10−6 30 1.75413×10−5 7.72958×10−7 5.73978×10−5 20 1.14092×10−4 4.98263×10−6 3.69997×10−4 10 2.90545×10−4 1.12311×10−5 8.33992×10−4 0 8.1114×10−4 3.53457×10−5 2.62468×10−3 -10 2.14213×10−4 1.24574×10−5 9.25057×10−4 -20 8.72541×10−5 4.2114×10−6 3.12728×10−4 -30 1.33693×10−5 6.35469×10−7 4.71883×10−5 -40 1.83949×10−6 8.72611×10−8 6.47979×10−6 -50 2.49504×10−7 1.18327×10−8 8.78668×10−7Table 7
Comparison of 2nd order NIM solution with the exact solution for nonlinear coupled Hirota Satsuma MKDV equations with initial conditions (59) using k=0.1, b1=0.1, λ=0.1, and t=0.5.
x Exact u N I M u A b s o l u t e   e r r o r s Exact v N I M v A b s o l u t e   e r r o r s -50 0.40001 0.40001 2.48113×10−8 9.77691x10−6 9.77629x10−6 6.2454×10−10 -40 0.400072 0.400072 1.83213×10−7 0.0000722196 0.000072215 4.60471×10−9 -30 0.400532 0.400531 1.34729×10−6 0.000532406 0.000532373 3.34797×10−8 -20 0.403868 0.403859 9.60358×10−6 0.0038819 0.00386797 2.19196×10−7 -10 0.425439 0.425385 5.33734×10−5 0.0254388 0.0254382 5.70005×10−7 0 0.503698 0.503672 2.60625×10−5 0.103698 0.1037 1.75596×10−6 10 0.57767 0.57774 6.95746×10−5 0.17767 0.17767 5.80831×10−7 20 0.596655 0.596671 1.61881×10−5 0.196655 0.196655 2.12492×10−7 30 0.599541 0.599543 2.36055×10−6 0199541 0.199541 3.22892×10−8 40 0.599938 0.599938 3.22719×10−7 0.199938 0.199938 4.43792×10−9 50 0.599992 0.599992 4.37352×10−8 0.199992 0.199992 6.01862×10−10Figures1, 3, and 5 show the 3D plots of 2nd order NIM solution, respectively, for u(x,t), v(x,t), and w(x,t) parts of Hirata Satsuma coupled KDV equation with initial condition (12) at F0=1, F2=1, k=0.1, β=1, and t=1., while Figures 11, 13, and 15 show the 3D plots of approximate solution by NIM with initial condition (37) at F0=1, F2=1, k=0.1, β=1, and t=2. Figures 2, 4, and 6 show the comparison of 2D plots of 2nd order NIM solution with exact solution for u(x,t), v(x,t), and w(x,t) parts of nonlinear Hirota Satsuma coupled KDV equation with initial condition given by (12) at F0=1, F2=1, k=0.1, β=1, and t=1, while Figures 12, 14, and 16 show the comparison of 2D plots of 2nd order approximate solution by NIM with exact solution for u(x,t), v(x,t), and w(x,t) with initial condition given by (37) using F0=1, F2=1, k=0.1, β=1, and t=2.Figure 1
3D graph foru(x,t) of nonlinear coupled Hirota Satsuma KDV equation by NIM at F0=1, F2=1, k=0.1, and β=1.Figure 2
Comparison of NIM solution with exact solution foru(x,t) of nonlinear coupled Hirota Satsuma KDV equation at F0=1, F2=1, k=0.1, β=1, and t=1.Figure 3
3D graph forv(x,t) of nonlinear coupled Hirota Satsuma KDV equation by NIM at F0=1, F2=1, k=0.1, and β=1.Figure 4
Comparison of NIM solution with exact solution forv(x,t) nonlinear coupled Hirota Satsuma KDV equation at F0=1, F2=1, k=0.1, β=1, and t=1.Figure 5
3D graph forw(x,t) of nonlinear coupled Hirota Satsuma KDV equation by NIM at F0=1, F2=1, k=0.1, and β=1.Figure 6
Comparison of NIM solution with exact solution forw(x,t) of nonlinear coupled Hirota Satsuma KDV equation at F0=1, F2=1, k=0.1, β=1, and t=1.The 3D plots for modified coupled KDV equation with initial condition (27) at k=0.1,λ=1 are given in Figures 7 and 9, while Figures 17 and 19 show the 3D graphs for the same equation with initial condition (42) at k=0.1, b1=0.1, and λ=0.1. Figures 8 and 10 show the comparison of 2D plots of 2nd order NIM solution with exact solution for u(x,t) and v(x,t) parts of modified coupled KDV equation of Problem 4 using k=0.1, t=0.5, b1=0.1, and λ=1, while Figures 18 and 20 show the comparison of 2D plots of 2nd order approximate solution by NIM with exact solution for u(x,t), v(x,t), and w(x,t) parts modified coupled KDV equation using k=0.1, t=0.5, b1=0.1, and λ=0.1.Figure 7
3D graph foru(x,t) of nonlinear coupled modified KDV equation by NIM using k=0.1 and λ=1.Figure 8
Comparison of NIM solution with exact solution foru(x,t) of nonlinear coupled modified KDV equation at k=0.1, t=0.5, and λ=1.Figure 9
3D graph forv(x,t) of nonlinear coupled modified KDV equation by NIM using k=0.1 and λ=1.Figure 10
Comparison of NIM solution forv(x,t) of nonlinear coupled modified KDV equation with exact solution using k=0.1, t=0.5, and λ=1.Figure 11
3D graph foru(x,t) part of Problem 5 by NIM using F0=1, F2=1, k=0.1, β=1.Figure 12
Comparing of numerical solution ofu(x,t) part of Problem 5 with exact solution by NIM using F0=1, F2=1, k=0.1, t=2, and β=1.Figure 13
3D graph forv(x,t) part of Problem 5 by NIM using F0=1, F2=1, k=0.1, and β=1.Figure 14
Comparison of NIM solution with exact solution forv(x,t) part of Hirota Satsuma KDV equation using F0=1, F2=1, k=0.1, t=2, and β=1.Figure 15
3D graph forw(x,t) part of Hirota Satsuma KDV equation by NIM using F0=1, F2=1, k=0.1, and β=1.Figure 16
Comparison of NIM solution with exact solution forw(x,t) part of Hirota Satsuma KDV equation using F0=1, F2=1, k=0.1, t=2, and β=1.Figure 17
3D graph foru(x,t) part of coupled modified KDV equation by NIM using k=0.1, b1=0.1, and λ=0.1.Figure 18
Comparison of NIM solution with exact solution foru(x,t) part of coupled modified KDV equation using k=0.1, t=0.5, b1=0.1, and λ=0.1.Figure 19
3D graph forv(x,t) part of coupled modified KDV equation by NIM using k=0.1, b1=0.1, and λ=0.1.Figure 20
Comparison of NIM solution forv(x,t) part of coupled modified KDV equation with exact solution using k=0.1, t=0.5, b1=0.1, and λ=0.1.From the presented problems it is observed that in a few iterations the proposed method yields the approximate solutions which are in close agreement with exact solution.


## 5. Conclusion

NIM has been successfully implemented for finding the approximate solution of the nonlinear Hirota Satsuma coupled KDV equations and coupled MKDV equations. From obtained results it is concluded that NIM is very effective, simple, and fast convergent and is independent of the assumption of the unrealistic small parameters. The results obtained by NIM are very consistent in comparison with HAM. NIM is easy to understand and easy to implement using computer packages such as Mathematica.



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*Source: 9042039-2018-12-03.xml*