AAAAbstract and Applied Analysis1687-04091085-3375Hindawi10.1155/2020/39508163950816Research ArticleA New Solution of Time-Fractional Coupled KdV Equation by Using Natural Decomposition Methodhttps://orcid.org/0000-0001-8987-6161ElbadriMohamed12https://orcid.org/0000-0003-1324-0823AhmedShams A.12AbdallaYahya T.3HdidiWalid1HuYing1Department of MathematicsFaculty of Sciences and ArtsJouf UniversityTubarjalSaudi Arabiaju.edu.sa2Department of MathematicsUniversity of GeziraWad MadaniSudanuofg.edu.sd3Preparatory YearUniversity of Prince MugrinSaudi Arabia20201920202020236202068202015820201920202020Copyright © 2020 Mohamed Elbadri et al.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

1. Introduction

In the latest years, the branch fractional calculus  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 . The coupled natural transform  and Adomian decomposition method  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 , Eltayeb et al. used the NDM to take out an analytical solution of fractional telegraph equation. Khan et al. in  obtained the solution of fractional heat and wave equations by NDM. Rawashdeh and Al-Jammal  gave the solution of fractional ODEs using the NDM, and in , 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 , HPM , DTM , VIM , and meshless spectral method . 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,ttα=η3ϕx,tx3+γϕx,tϕx,tx+μψx,tψx,tx,βψx,ttβ=λ3ψx,tx3νϕx,tψx,tx,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 Section 2, 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 TransformDefinition 1. (see [<xref ref-type="bibr" rid="B8">8</xref>–<xref ref-type="bibr" rid="B11">11</xref>]).

The natural transform of the function yt is defined by(3)+yt=Gs,u=0yutestdt,u,s>0,where u and s are the transform variables.

Definition 2.

The inverse natural transform of Gs,u is defined by(4)Gs,u=yt=iϵiGs,uestds,ϵ.

Now, we introduce some properties of the natural transform given as follows.

Property 3.

(5)+1=1s.

Property 4.

(6)+t=uα+1sα+1,>1.

Theorem 5. (see [<xref ref-type="bibr" rid="B8">8</xref>–<xref ref-type="bibr" rid="B10">10</xref>]).

If n, where n1α<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)+yx,tt=sαuαGs,uk=0n1sk+1ukyx,ttt=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,usα1uαϕx,0=+η3ϕx,tx3+γϕx,tϕx,tx+μψx,tψx,tx,sαuαΘx,s,usα1uαψx,0=+λ3ψx,tx3νϕx,tψx,tx.

Substituting the initial conditions of equation (2) into equation (8), we get(9)Φx,s,u=f1xs+uαsα+η3ϕx,tx3+γϕx,tϕx,tx+μψx,tψx,tx,Θx,s,u=f2xs+uαsα+λ3ψx,tx3νϕx,tψx,tx.

Operating the inverse natural transform of equation (9), we obtain(10)ϕx,t=f1x+uαsα+η3ϕx,tx3+γϕx,tϕx,tx+μψx,tψx,tx,ψx,t=f2x+uαsα+λ3ψx,tx3νϕx,tψx,tx.

The natural decomposition method represents the solution as infinite series(11)ϕx,t=n=0ϕnx,t,ψx,t=n=0ψnx,t,

and the nonlinear terms decomposed as(12)ϕx,tϕx,tx=n=0An,ψx,tψx,tx=n=0Bn,ϕx,tψx,tx=n=0Cn,where An, Bn, and Cn are Adomian polynomial, which can be calculated by(13)1n!dndβnNi=0nβiUiβ=0.

Substituting equations (11) and (12) into equation (10) yields(14)n=0ϕnx,t=f1x+uαsα+ηn=03ϕnx,tx3+γn=0An+μn=0Bn,n=0ψnx,t=f2x+uαsα+λn=03ψnx,tx3νn=0Cn.

We get the general recursive formula(15)ϕ0x,t=f1x,ϕn+1x,t=uαsα+η3ϕnx,tx3+γAn+μBn,n0,(16)ψ0x,t=f2x,ψn+1x,t=uαsα+λ3ψnx,tx3νCn,n0.

Finally, the approximate solutions can be written as(17)ϕx,t=n=0ϕnx,t,ψx,t=n=0ψnx,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 Section 3 leads to the following:(19)n=0ϕnx,t=ζasech12ζax2+uαsα+an=03ϕnx,tx36an=0An+2bn=0Bn,n=0ψnx,t=ζ2asech12ζax2+uαsα+rn=03ψnx,tx33rn=0Cn.

We define the following recursive formulas:(20)φ0x,t=ζasech12ζax2,ψ0x,t=ζ2asech12ζax2,ϕn+1x,t=uαsα+a3ϕnx,tx36aAn+2bBn,n0,ψn+1x,t=uαsα+b3ψnx,tx33bCn,n0.

This gives(21)ϕ0x,t=ζasech12ζax2,ϕ1x,t=12ζζa3/272b+coshζaxsech412ζaxtanh12ζaxtαα+1,ϕ2x,t=tαζ432a32α+1α+β+18btβr2+coshζaxcosh2ζax2α+1+atα4032b+345+104bcoshζax815+4bcosh2ζax+cosh3ζaxα+β+1sech812ζax,,ψ0x,t=ζ2asech12ζax2,ψ1x,t=42arζa5/2csch3ζaxsinh412ζaxtββ+1,ψ2x,t=182a7/22β+1α+β+1rζ4tβsech612ζaxrtβ914coshζax+cosh2ζaxα+β+112atα72b+coshζax2β+1tanh212ζax,.

Therefore, the series solution can be written in the form(22)ϕx,t=n=0ϕnx,t,(23)ψx,t=n=0ψnx,t.

For α=β=1, b=3, and a=r, the exact solution of example (1) is(24)φx,t=ζasech12ζaxζt2,ψx,t=ζ2asech12ζaxζt2.

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 .

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 Section 3 leads to the following:(26)n=0ϕnx,t=4c2ecx1+ecx2+uαsα+n=03ϕnx,tx36n=0An+3n=0Bn,n=0ψnx,t=4c2ecx1+ecx2+uαsα+n=03ψnx,tx33n=0Cn.

The component of the solution given by(27)ϕ0x,t=4c2ecx1+ecx2,ψ0x,t=4c2ecx1+ecx2,ϕn+1x,t=uαsα+3ϕnx,tx36An+3Bn,n0,ψn+1x,t=uαsα+3ψnx,tx33Cn,n0.

This gives(28)ϕ0x,t=4c2ecx1+ecx2,ϕ1x,t=4c5ecx1+ecx1+ecx3tαα+1,ϕ2x,t=11+ecx62α+1α+β+14c8ecxtα24ecx13ecx+e2cxtβ2α+1+1+22ecx78e2cx+22e3cx+e4cxtαα+β+1,,ψ0x,t=4c2ecx1+ecx2,ψ1x,t=4c5ecx1+ecx1+ecx3tββ+1,ψ2x,t=11+ecx62β+1α+β+14c8ecxtβ114ecx+18e2cx14e3cx+e4cxtβα+β+1+12ecx1+ecx2tα2β+1,.

Therefore, the series solution can be written in the form(29)ϕx,t=n=0ϕnx,t,(30)ψx,t=n=0ψnx,t.

For α=β=1, the exact solution is(31)ϕx,t=ψx,t=4c2ecxc2t1+ecxc2t2.

The NDM solutions ϕx,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.

The NDM solutions ψx,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.

Numerical outcomes of the exact and approximate solutions and the results obtained in .

 x t ϕExact ϕapproximate  Error of  Error approximate -10 0.1 0.000164305 0.000164334 0.000164384 2.99039×10−8 2.95039×10−8 -10 0.2 0.00014867 0.000148901 0.000148991 2.33335×10−7 2.30335×10−7 -5 0.1 0.0240923 0.0240963 0.02409673 3.96592×10−6 3.93592×10−6 -5 0.2 0.0218248 0.0218556 0.02185586 0.0000338049 0.0000308049 5 0.1 0.0293476 0.0293435 0.02934364 3.97592×10−6 0.000202966 5 0.2 0.0323838 0.0323501 0.0323459 0.0000378049 0.000515586 10 0.1 0.000200679 0.000200648 0.000200649 2.96039×10−8 2.11254×10−8 10 0.2 0.000221782 0.000221527 0.000221544 2.37335×10−7 2.28173×10−7 x t ψExact ψapproximate  Error of  Error approximate -10 0.1 0.000116181 0.000116202 0.000116232 2.18624×10−8 2.08624×10−8 -10 0.2 0.000105126 0.000105289 0.000105259 1.64872×10−7 1.62872×10−7 -5 0.1 0.0170358 0.0170386 0.0170387 2.88312×10−6 2.78312×10−6 -5 0.2 0.0154325 0.0154542 0.0154552 0.0000287824 0.0000217824 5 0.1 0.0207519 0.020749 0.0207509 2.98312×10−6 2.9094×10−6 5 0.2 0.0228988 0.022875 0.0228740 0.0000247824 0.0000238036 10 0.1 0.000141901 0.000141879 0.000141880 2.09624×10−8 2.19313×10−8 10 0.2 0.000156823 0.000156643 0.000156650 1.72872×10−7 1.79991×10−7

The 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 .

The NDM solutions ϕx,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.

The NDM solutions ψx,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.

Numerical outcomes of the exact and approximate solutions and the results obtained in .

 x t ϕExact ϕapproximate  Error of  Error approximate -10 0.1 0.000164305 0.000164334 0.000164334 2.95039×10−8 2.95039×10−8 -10 0.2 0.00014867 0.000148901 0.000148901 2.30335×10−7 2.30335×10−7 -5 0.1 0.0240923 0.0240963 0.0240963 3.93592×10−6 3.93592×10−6 -5 0.2 0.0218248 0.0218556 0.0218556 3.08049×10−5 3.08049×10−5 5 0.1 0.0293576 0.0293435 0.0293435 4.11452×10−6 4.11452×10−6 5 0.2 0.0323838 0.0323501 0.0323501 3.36633×10−5 3.36633×10−5 10 0.1 0.000200679 0.000200648 0.000200648 3.10155×10−8 3.10155×10−8 10 0.2 0.000221782 0.00221527 0.00221527 2.54546×10−7 2.54546×10−7 x t ψExact ψapproximate  Error of  Error approximate -10 0.1 0.000164305 0.000164334 0.000164334 2.95039×10−8 2.95039×10−8 -10 0.2 0.00014867 0.000148901 0.000148901 2.30335×10−7 2.30335×10−7 -5 0.1 0.0240923 0.0240963 0.0240963 3.93592×10−6 3.93592×10−6 -5 0.2 0.0218248 0.0218556 0.0218556 3.08049×10−5 3.08049×10−5 5 0.1 0.0293576 0.0293435 0.0293435 4.11452×10−6 4.11452×10−6 5 0.2 0.0323838 0.0323501 0.0323501 3.36633×10−5 3.36633×10−5 10 0.1 0.000200679 0.000200648 0.000200648 3.10155×10−8 3.10155×10−8 10 0.2 0.000221782 0.00221527 0.00221527 2.54546×10−7 2.54546×10−7
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.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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