# A New Solution of Time-Fractional Coupled KdV Equation by Using Natural Decomposition Method

Authors: Mohamed Elbadri; Shams A. Ahmed; Yahya T. Abdalla; Walid Hdidi
Journal: Abstract and Applied Analysis
 (2020)

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## Abstract

In this article, we applied a new technique for solving the time-fractional coupled Korteweg-de Vries (KdV) equation. This method is a combination of the natural transform method with the Adomian decomposition method called the natural decomposition method (NDM). The solutions have been made in a convergent series form. To demonstrate the performances of the technique, two examples are provided.


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## Body

## 1. Introduction

In the latest years, the branch fractional calculus [1–5] has played a significant role in applied mathematics; this is evident when it is used in phenomena of physical science and engineering, which are described by fractional differential equations. Fractional partial differential equations (FPDEs) have lately been studied and solved in several ways [6, 7]. Many transforms coupled with other techniques were used to solve differential equations [8–10]. The coupled natural transform [11–14] and Adomian decomposition method [15–17] called the natural decomposition method (NDM) is introduced in [18, 19] to solve differential equations, and it presents the approximate solution in the series form. The natural decomposition method has been used by many researchers to find approximate analytical solutions; it has shown reliable and closely converged results of the solution. In [20], Eltayeb et al. used the NDM to take out an analytical solution of fractional telegraph equation. Khan et al. in [21] obtained the solution of fractional heat and wave equations by NDM. Rawashdeh and Al-Jammal [22] gave the solution of fractional ODEs using the NDM, and in [23], Shah et al. obtained the solution of fractional partial differential equations with proportional delay by using the NDM. Many analytical and numerical methods were used to solve the fractional coupled KdV equation, such as spectral collection method [24], HPM [25], DTM [26], VIM [27], and meshless spectral method [28]. In this paper, we provided the application of the NDM to find the approximate solutions of nonlinear time-fractional coupled KdV equations given by(1)∂αϕx,t∂tα=η∂3ϕx,t∂x3+γϕx,t∂ϕx,t∂x+μψx,t∂ψx,t∂x,∂βψx,t∂tβ=λ∂3ψx,t∂x3−νϕx,t∂ψx,t∂x,subject to initial conditions(2)ϕx,0=f1x,ψx,0=f2x,where η,λ<0; γ,μ, and ν are constant parameters.The KdV equation arose in many physical phenomena and application in the study of shallow-water waves, and it has been studied by many researchers. This work is organized as follows. In Section2, we give definitions and properties of the natural transform. In Section 3, the NDM is made. Section 4 discusses the new technique and compares it with two different techniques by two examples and presents tables and graphs to offer the validation of the NDM. Discussion and conclusion are included.


## 2. Natural Transform

Definition 1. (see [8–11]).
The natural transform of the functionyt is defined by(3)ℕ+yt=Gs,u=∫0∞yute−stdt,u,s>0,where u and s are the transform variables.Definition 2.
The inverse natural transform ofGs,u is defined by(4)ℕ−Gs,u=yt=∫∈−i∞ϵ−i∞Gs,uestds,ϵ∈ℝ.Now, we introduce some properties of the natural transform given as follows.Property 3.
(5)ℕ+1=1s.Property 4.
(6)ℕ+t∝=uαℾ∝+1sα+1,∝>−1.Theorem 5. (see [8–10]).
Ifn∈ℕ, where n−1≤α<n and Gs,u is the natural transform of the function yt, then the natural transform of the Caputo fractional derivative ∂∝yx,t/∂t∝ is given by(7)ℕ+∂∝yx,t∂t∝=sαuαGs,u−∑k=0n−1s∝−k+1u∝−k∂∝yx,t∂t∝t=0.


## 3. Analysis of Method

We explain the algorithm of the NDM by considering the fractional coupled KdV equations (1) and (2).Applying natural transform to equation (1), we have(8)sαuαΦx,s,u−sα−1uαϕx,0=ℕ+η∂3ϕx,t∂x3+γϕx,t∂ϕx,t∂x+μψx,t∂ψx,t∂x,sαuαΘx,s,u−sα−1uαψx,0=ℕ+λ∂3ψx,t∂x3−νϕx,t∂ψx,t∂x.Substituting the initial conditions of equation (2) into equation (8), we get(9)Φx,s,u=f1xs+uαsαℕ+η∂3ϕx,t∂x3+γϕx,t∂ϕx,t∂x+μψx,t∂ψx,t∂x,Θx,s,u=f2xs+uαsαℕ+λ∂3ψx,t∂x3−νϕx,t∂ψx,t∂x.Operating the inverse natural transform of equation (9), we obtain(10)ϕx,t=f1x+ℕ−uαsαℕ+η∂3ϕx,t∂x3+γϕx,t∂ϕx,t∂x+μψx,t∂ψx,t∂x,ψx,t=f2x+ℕ−uαsαℕ+λ∂3ψx,t∂x3−νϕx,t∂ψx,t∂x.The natural decomposition method represents the solution as infinite series(11)ϕx,t=∑n=0∞ϕnx,t,ψx,t=∑n=0∞ψnx,t,and the nonlinear terms decomposed as(12)ϕx,t∂ϕx,t∂x=∑n=0∞An,ψx,t∂ψx,t∂x=∑n=0∞Bn,ϕx,t∂ψx,t∂x=∑n=0∞Cn,where An, Bn, and Cn are Adomian polynomial, which can be calculated by(13)1n!dndβnN∑i=0nβiUiβ=0.Substituting equations (11) and (12) into equation (10) yields(14)∑n=0∞ϕnx,t=f1x+ℕ−uαsαℕ+η∑n=0∞∂3ϕnx,t∂x3+γ∑n=0∞An+μ∑n=0∞Bn,∑n=0∞ψnx,t=f2x+ℕ−uαsαℕ+λ∑n=0∞∂3ψnx,t∂x3−ν∑n=0∞Cn.We get the general recursive formula(15)ϕ0x,t=f1x,ϕn+1x,t=ℕ−uαsαℕ+η∂3ϕnx,t∂x3+γAn+μBn,n≥0,(16)ψ0x,t=f2x,ψn+1x,t=ℕ−uαsαℕ+λ∂3ψnx,t∂x3−νCn,n≥0.Finally, the approximate solutions can be written as(17)ϕx,t=∑n=0∞ϕnx,t,ψx,t=∑n=0∞ψnx,t.


## 4. Applications

Now, we explain the appropriateness of the technique by the following examples.Example 1.
Consider the fractional coupled KdV equation (1) with η=−a, γ=−6a, μ=2b, λ=−r, and ν=3r subject to(18)ϕx,0=ζasech12ζax2,ψx,0=ζ2asech12ζax2.Solution 1.
Applying the method formulated in Section3 leads to the following:(19)∑n=0∞ϕnx,t=ζasech12ζax2+ℕ−uαsαℕ+−a∑n=0∞∂3ϕnx,t∂x3−6a∑n=0∞An+2b∑n=0∞Bn,∑n=0∞ψnx,t=ζ2asech12ζax2+ℕ−uαsαℕ+−r∑n=0∞∂3ψnx,t∂x3−3r∑n=0∞Cn.
We define the following recursive formulas:(20)φ0x,t=ζasech12ζax2,ψ0x,t=ζ2asech12ζax2,ϕn+1x,t=ℕ−uαsαℕ+−a∂3ϕnx,t∂x3−6aAn+2bBn,n≥0,ψn+1x,t=ℕ−uαsαℕ+−b∂3ψnx,t∂x3−3bCn,n≥0.
This gives(21)ϕ0x,t=ζasech12ζax2,ϕ1x,t=12ζζa3/27−2b+coshζaxsech412ζaxtanh12ζaxtαℾα+1,ϕ2x,t=tαζ432a3ℾ2α+1ℾα+β+18btβr2+coshζax−cosh2ζaxℾ2α+1+atα40−32b+−345+104bcoshζax−8−15+4bcosh2ζax+cosh3ζaxℾα+β+1sech812ζax,⋯,ψ0x,t=ζ2asech12ζax2,ψ1x,t=42arζa5/2csch3ζaxsinh412ζaxtβℾβ+1,ψ2x,t=−182a7/2ℾ2β+1ℾα+β+1rζ4tβsech612ζax−rtβ9−14coshζax+cosh2ζaxℾα+β+1−12atα7−2b+coshζaxℾ2β+1tanh212ζax,⋯.
Therefore, the series solution can be written in the form(22)ϕx,t=∑n=0∞ϕnx,t,(23)ψx,t=∑n=0∞ψnx,t.
Forα=β=1, b=3, and a=r, the exact solution of example (1) is(24)φx,t=ζasech12ζax−ζt2,ψx,t=ζ2asech12ζax−ζt2.The approximate solutions (22) and (23) for the special cases are shown in Figures 1 and 2; the numerical results in Table 1 where α=β=1, a=1, b=3, ζ=1, and r=1 show that the solutions obtained by the NDM are nearly identical with the exact solution. Also, it can be seen from Table 1 that the solutions obtained by the NDM are more accurate than those obtained in [27].Example 2.
Consider the fractional coupled KdV in the form equation (1) with η=−1, γ=−6, μ=3, λ=−1, and ν=3 subject to(25)ϕx,0=4c2ecx1+ecx2,ψx,0=4c2ecx1+ecx2.Solution 2.
Applying the method formulated in Section3 leads to the following:(26)∑n=0∞ϕnx,t=4c2ecx1+ecx2+ℕ−uαsαℕ+−∑n=0∞∂3ϕnx,t∂x3−6∑n=0∞An+3∑n=0∞Bn,∑n=0∞ψnx,t=4c2ecx1+ecx2+ℕ−uαsαℕ+−∑n=0∞∂3ψnx,t∂x3−3∑n=0∞Cn.
The component of the solution given by(27)ϕ0x,t=4c2ecx1+ecx2,ψ0x,t=4c2ecx1+ecx2,ϕn+1x,t=ℕ−uαsαℕ+−∂3ϕnx,t∂x3−6An+3Bn,n≥0,ψn+1x,t=ℕ−uαsαℕ+−∂3ψnx,t∂x3−3Cn,n≥0.
This gives(28)ϕ0x,t=4c2ecx1+ecx2,ϕ1x,t=4c5ecx−1+ecx1+ecx3tαℾα+1,ϕ2x,t=11+ecx6ℾ2α+1ℾα+β+14c8ecxtα−24ecx1−3ecx+e2cxtβℾ2α+1+1+22ecx−78e2cx+22e3cx+e4cxtαℾα+β+1,⋯,ψ0x,t=4c2ecx1+ecx2,ψ1x,t=4c5ecx−1+ecx1+ecx3tβℾβ+1,ψ2x,t=11+ecx6ℾ2β+1ℾα+β+14c8ecxtβ1−14ecx+18e2cx−14e3cx+e4cxtβℾα+β+1+12ecx1+ecx2tαℾ2β+1,⋯.
Therefore, the series solution can be written in the form(29)ϕx,t=∑n=0∞ϕnx,t,(30)ψx,t=∑n=0∞ψnx,t.
Forα=β=1, the exact solution is(31)ϕx,t=ψx,t=4c2ecx−c2t1+ecx−c2t2.Figure 1
The NDM solutionsϕx,t for Example 1 when a=1, b=3, ζ=1, and r=1: (a) α=1,β=1; (b) α=0.75,β=0.75; (c) α=0.5,β=0.5.
(a)(b)(c)Figure 2
The NDM solutionsψx,t for Example 1 when a=1, b=3, ζ=1, and r=1: (a) α=1,β=1; (b) α=0.75,β=0.75; (c) α=0.5,β=0.5.
(a)(b)(c)Table 1
Numerical outcomes of the exact and approximate solutions and the results obtained in [27].
xtϕExactϕapproximate[24]Error of [24]Error approximate-100.10.0001643050.0001643340.0001643842.99039×10−82.95039×10−8-100.20.000148670.0001489010.0001489912.33335×10−72.30335×10−7-50.10.02409230.02409630.024096733.96592×10−63.93592×10−6-50.20.02182480.02185560.021855860.00003380490.000030804950.10.02934760.02934350.029343643.97592×10−60.00020296650.20.03238380.03235010.03234590.00003780490.000515586100.10.0002006790.0002006480.0002006492.96039×10−82.11254×10−8100.20.0002217820.0002215270.0002215442.37335×10−72.28173×10−7xtψExactψapproximate[24]Error of [24]Error approximate-100.10.0001161810.0001162020.0001162322.18624×10−82.08624×10−8-100.20.0001051260.0001052890.0001052591.64872×10−71.62872×10−7-50.10.01703580.01703860.01703872.88312×10−62.78312×10−6-50.20.01543250.01545420.01545520.00002878240.000021782450.10.02075190.0207490.02075092.98312×10−62.9094×10−650.20.02289880.0228750.02287400.00002478240.0000238036100.10.0001419010.0001418790.0001418802.09624×10−82.19313×10−8100.20.0001568230.0001566430.0001566501.72872×10−71.79991×10−7The approximate solutions (29) and (30) for the special cases are shown in Figures 3 and 4; the numerical results in Table 2 where α=β=c=1 show that the solutions obtained by the NDM are nearly identical with the exact solution. Also, it can be seen from Table 2 that the solutions obtained by the NDM are the same with those obtained in [26].Figure 3
The NDM solutionsϕx,t for Example 2 when c=1: (a) α=1,β=1; (b) α=0.75,β=0.5; (c) α=1,β=0.90.
(a)(b)(c)Figure 4
The NDM solutionsψx,t for Example 2 when c=1: (a) α=1,β=1; (b) α=0.75,β=0.5; (c) α=1,β=0.90.
(a)(b)(c)Table 2
Numerical outcomes of the exact and approximate solutions and the results obtained in [26].
xtϕExactϕapproximate[23]Error of [23]Error approximate-100.10.0001643050.0001643340.0001643342.95039×10−82.95039×10−8-100.20.000148670.0001489010.0001489012.30335×10−72.30335×10−7-50.10.02409230.02409630.02409633.93592×10−63.93592×10−6-50.20.02182480.02185560.02185563.08049×10−53.08049×10−550.10.02935760.02934350.02934354.11452×10−64.11452×10−650.20.03238380.03235010.03235013.36633×10−53.36633×10−5100.10.0002006790.0002006480.0002006483.10155×10−83.10155×10−8100.20.0002217820.002215270.002215272.54546×10−72.54546×10−7xtψExactψapproximate[23]Error of [23]Error approximate-100.10.0001643050.0001643340.0001643342.95039×10−82.95039×10−8-100.20.000148670.0001489010.0001489012.30335×10−72.30335×10−7-50.10.02409230.02409630.02409633.93592×10−63.93592×10−6-50.20.02182480.02185560.02185563.08049×10−53.08049×10−550.10.02935760.02934350.02934354.11452×10−64.11452×10−650.20.03238380.03235010.03235013.36633×10−53.36633×10−5100.10.0002006790.0002006480.0002006483.10155×10−83.10155×10−8100.20.0002217820.002215270.002215272.54546×10−72.54546×10−7


## 5. Conclusion

In this paper, the approximate solution of the time-fractional coupled KdV equation has been successfully done by using the NDM. We have tested the method on two examples, which revealed that the method is highly efficacious by comparing the approximate solutions with two different methods and exact solutions.



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