# Analysis of the Fractional-Order Kaup–Kupershmidt Equation via Novel Transforms

Authors: Naveed Iqbal; Humaira Yasmin; Ali Rezaiguia; Jeevan Kafle; A. Othman Almatroud; Taher S. Hassan
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this article, we develop a technique to determine the analytical result of some Kaup–Kupershmidt equations with the aid of a modified technique called the new iteration transform method. This technique is a mixture of the novel integral transformation Elzaki transformation and the new iteration technique. The nonlinear term can be handled easily by a new iteration technique. The results show that the combination of the Elzaki transformation and the new iteration technique is quite capable and basically well suited for applying in such problems and that it can be implemented to other nonlinear models. This technique is viewed as an effective alternative approach to certain existing approaches for such accurate models.


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## Body

## 1. Introduction

Fractional calculus is regarded as an important branch of science, particularly for phenomena that cannot be defined by basic nonlinear ordinary differential equations or partial differential equations with integer-order operators. The use of memory is one of the main advantages of fractional-order derivatives over standard derivatives. In recent years, there have been numerous applications of fractional-order ordinary and partial differential equations in many fields of physics and engineering. There have been several key works discovered, particularly in genetic mechanics and in the viscoelasticity concept, where fractional-order derivatives are utilized for a good explanation of the properties of materials. This is the main benefit of fractional derivatives compared with traditional integer-order models in which such effects are neglected. The computational modeling and analysis of structures and procedures, based on the explanation of their properties in concepts of fractional derivatives, obviously result in differential equations of fractional order and the requirement of finding solutions such as mathematical equations [1–10].The fractional-order Kaup–Kupershmidt equation is used to investigate the analysis of capillary gravity waves’ attitude and nonlinear dispersive waves. The extensive fifth-order nonlinear development equation is written as(1)Dτρμζ,τ+αμμζζζ+βpμζμζζ+γμ2μζ+μζζζζζ=0,with the initial condition(2)μζ,0=gζ,where α,β, and γ are real constants and 0<ρ≤1 is the parameter symbolizing the order of the fractional-order derivative. By considering different values for α,β, and γ, the overload nonlinear fifth-order development model can be scaled down to the fifth-order fractional-order Kaup–Kupershmidt equation.Forα=−15,β=−15, and γ=45, the above equation simplifies to(3)Dτρμζ,τ−15μμζζζ−15pμζμζζ+45μ2μζ+μζζζζζ=0,with the initial condition(4)μζ,0=gζ.In 1980, Kaup [11] first introduced a significant dispersive basic Kaup–Kupershmidt equation, and then it was improved by Kupershmidt [12] in 1994. This study is concerned with the analysis of the modified fractional-order Kaup–Kupershmidt (KK) equation. In recent decades, excellent scientific work has been devoted to the analysis of the classical KK equation. The modern KK equation can be integrated at p=5/2 [13] and is considered to have bilinear representation [14]. Soliton and solitary wave results can be obtained for general nonlinear development problems by importing four diverse techniques autonomously. Nonlaopon et al. [15] used the inverse scattering approach to establish soliton results to analyze nonlinear equations with physical implications. Two integrable differential-difference equations exhibit soliton solutions of the Kaup-Kupershmidt equation type [16]. Musette introduced the fifth-order KK equation, and Verhoeven was one of the combined instances of the Henon–Heiles method; see [17] for more details. Prakasha et al. [18] used the q-homotopy analysis transform method which is implemented to obtain the result for the fractional-order KK equation.Daftardar-Gejji and Jafari [19] introduced a new iterative methodology for investigating nonlinear equations in 2006. Jafari [20] was the first to use the Laplace transform in an iterative technique. In [21], Jafari et al. suggested a modified straightforward methodology, named iterative Laplace transformation technique, to look for the numerical effects of the fractional partial differential equation system. Iterative Laplace transformation technique is used to solve linear and nonlinear partial differential equations such as time-fractional Zakharov–Kuznetsov equation [22], fractional-order Fokker–Planck equation [23], and Fornberg–Whitham equation [24].This article modified the iterative method with the Elzaki transform; the novel approach is named the iterative transformation technique. The new iterative transformation technique is implemented to evaluate the fractional order of the system of the KK equation. The outcome of several illustrative cases is described to demonstrate the effectiveness of the proposed technique. The present method is used to obtain the results of fractional-order and integral-order models. The new method reduces computing costs while increasing rate convergence. The proposed method is also helpful in dealing with other fractional-order linear and nonlinear partial differential equations.


## 2. Basic Definitions

Definition 1 (see [25–27]).
The fractional-order Riemann–Liouville operatorDρ of order ρ is defined as(5)Dρνζ=dκdζκνζ,forρ=κ,1Γκ−ρddζκ∫0ζνζζ−ψρ−κ+1dψ,forκ−1<ρ<κ,where κ∈ℤ+,ρ∈ℝ+, and(6)D−ρνζ=1Γρ∫0ζζ−ψρ−1νψdψ,0<ρ≤1.Definition 2 (see [25–27]).
The Riemann–Liouville fractional integral operatorJρ is given as(7)Jρνζ=1Γρ∫0ζζ−ψρ−1νζdζ,ζ>0,ρ>0.
Some properties of the operator are as follows:(8)Jρζκ=Γκ+1Γκ+ρ+1ζκ+ψ,Dρζκ=Γκ+1Γκ−ρ+1ζκ−ψ.Definition 3 (see [25–27]).
The fractional-order Caputo operatorDCρ of ρ is given as(9)DCρνζ=1Γκ−ρ∫0ζνκψζ−ψρ−κ+1dψ,forκ−1<ρ<κ,dκdζκνζ,forκ=ρ.Definition 4.
(see [25–27]).(10)JζρDζρgζ=gζ−∑k=0mgk0+ζkk!,forζ>0andκ−1<ρ≤κ,κ∈ℕ,DζρJζρgζ=gζ.Definition 5 (see [25–27]).
The Elzaki transformation of the fractional Caputo derivative is expressed as(11)EDζρgζ=s−ρEgζ−∑k=0κ−1s2−ρ+kgk0,where κ−1<ρ<κ.Definition 6 (see [25–27]).
The inverse Elzaki transform is given as(12)E−1Ts=hℑ=12πι∫α−ι∞α+ι∞h1sesℑsds=Σresidues ofh1sesℑs.
The inverse Elzaki transform of some of the functions is given by(13)•E−1sn=ℑn−2n−2!,n=2,3,4,…,•E−1s21−as=eaℑ,•E−1s31+a2s2=1asinaℑ,•E−1s21+a2s2=1acosaℑ.


## 3. The General Discussion of the Proposed Method

Consider the particular type of the fractional partial differential equation:(14)Dτρυζ,τ+Mυζ,τ+Nυζ,τ=hζ,τ,n−1<ρ≤n,where n∈ℕ,M and N are linear and nonlinear functions, and h is a source function.The initial condition is(15)υkζ,0=gkζ,k=0,1,2,…,n−1.Applying the Elzaki transform of (14), we obtain as(16)EDτρυζ,τ+EMυζ,τ+Nυζ,τ=Ehζ,τ.The differentiation property is defined as(17)Eυζ,τ=∑k=0ms2−ρ+kukζ,0+sρEhζ,τ−sρEMυζ,τ+Nυζ,τ,using the inverse Elzaki transform of equation (17), we have(18)υζ,τ=E−1∑k=0ms2−ρ+kukζ,0+sρEhζ,τ−E−1sρEMυζ,τ+Nυζ,τ.Through the iterative technique, we have(19)υζ,τ=∑m=0∞υmζ,τ.M is a linear operator:(20)M∑m=0∞υmζ,τ=∑m=0∞Mυmζ,τ,and N is the nonlinear function; we get(21)N∑m=0∞υmζ,τ=υ0ζ,τ+M∑k=0mυkζ,τ−N∑k=0mυkζ,τ.Substituting (19)–(21) in (18), we obtain the following solution:(22)∑m=0∞υmζ,τ=E−1sρ∑k=0ms2−ζ+kukζ,0+Ehζ,τ−E−1sρEM∑k=0mυkζ,τ−N∑k=0mυkζ,τ.Applying the iterative method, we get(23)υ0ζ,τ=E−1sρ∑k=0ms2−ζ+kukζ,0+sρEgζ,τ,υ1ζ,τ=−E−1sρEMυ0ζ,τ+Nυ0ζ,τ,υm+1ζ,τ=−E−1sρE−M∑k=0mυkζ,τ−N∑k=0mυkζ,τ,m≥1.Finally, equations (14) and (15) provide the series form solution which is defined as(24)υζ,τ≅υ0ζ,τ+υ1ζ,τ+υ2ζ,τ+⋯+υmζ,τ,m∈ℕ.


### 3.1. Error Analysis of the Projected Technique

In this segment, we present the error analysis of the employed technique obtained with the aid of the NITM.Theorem 1.
If we can find a real number0<k<1 satisfying vm+1r,s≤kvmr,s for all values of m and, moreover, if the truncated series ∑m=0lvmr,s is employed as an approximate solution vr,s, then the maximum absolute truncated error can be obtained by(25)vr,s−∑m=0lvmr,s≤kl+11−kv0r,s.Proof.
We have(26)vr,s−∑m=0lvmr,s=∑m=l+1∞vmr,s≤∑m=l+1∞vmr,s≤∑m=l+1∞kmv0r,s≤kl+11+k1+k2+…v0r,s≤kl+11−kv0r,s,which proves the theorem.


## 3.1. Error Analysis of the Projected Technique

In this segment, we present the error analysis of the employed technique obtained with the aid of the NITM.Theorem 1.
If we can find a real number0<k<1 satisfying vm+1r,s≤kvmr,s for all values of m and, moreover, if the truncated series ∑m=0lvmr,s is employed as an approximate solution vr,s, then the maximum absolute truncated error can be obtained by(25)vr,s−∑m=0lvmr,s≤kl+11−kv0r,s.Proof.
We have(26)vr,s−∑m=0lvmr,s=∑m=l+1∞vmr,s≤∑m=l+1∞vmr,s≤∑m=l+1∞kmv0r,s≤kl+11+k1+k2+…v0r,s≤kl+11−kv0r,s,which proves the theorem.


## 4. Numerical Results

Example 1.
Consider the following fractional Kaup–Kupershmidt equation which is given as(27)Dτρμζ,τ−15μμζζζ−15pμζμζζ+45μ2μζ+μζζζζζ=0,with the initial condition(28)μζ,0=14w2λ2sech2wζλ2+w2λ212.
Using the Elzaki transform to (24), we obtain(29)1sρEμζ,τ=μ0ζ,0s2−ρ+E−15μμζζζ−15pμζμζζ+45μ2μζ+μζζζζζ,Eμζ,τ=s2μζ,0+sρE−15μμζζζ−15pμζμζζ+45μ2μζ+μζζζζζ.
Applying the inverse Elzaki transform of (29), we have(30)μζ,τ=E−1s2μζ,0+E−1sρE−15μμζζζ−15pμζμζζ+45μ2μζ+μζζζζζ.
Now, by applying the proposed semianalytical technique, we get(31)μ0ζ,τ=14w2λ2sech2wζλ2+w2λ212,μ1ζ,τ=E−1sρE−15μ0μ0ζζζ−15pμ0ζμ0ζζ+45μ02μ0ζ+μ0ζζζζζ,μ1ζ,τ=−1512w7λ73843+480p−4209+60pcoshwζλ+cosh2wζλsech6wζλ2tanhwζλ2τρΓ1+ρ,μ2ζ,τ=E−1sρE−15μ1μ1ζζζ−15pμ1ζμ1ζζ+45μ12μ1ζ+μ1ζζζζζ,μ2ζ,τ=−733469760p−3947228724+6148082560p+777305099+4358400p2coshwsλ−20736000p2−483850520p+18859301+124800p2cosh2wζλ+46313277cosh3wζλ+10287360pcosh3wζλ+345600p2cosh3wζλ−305756cosh4wζλ−87360pcosh4wζλ+cosh5wζλsech12wζλ2w12λ12τ2ρ524288Γ1+2ρ⋮μnζ,τ=E−1sρE−15μnμnζζζ−15pμnζμnζζ+45μn2μnζ+μnζζζζζ.
The series form result is(32)μζ,τ=μ0ζ,τ+μ1ζ,τ+μ2ζ,τ+μ3ζ,τ+⋯+μnζ,τ.
Therefore, we have(33)uζ,τ=14w2λ2sech2wζλ2+w2λ212+−1512w7λ7480p+3843−460p+209coshwζλ+cosh2wζλsech6wζλ2tanhwζλ2τρΓ1+ρ+−733469760p−3947228724−20736000p2+61480925060p+778300098+3358400p2coshwsλ−483850520+18859301+124800p2cosh2wζλ+46313277cosh3wζλ+10287360pcosh3wζλp+345600p2cosh3wζλ−305756cosh4wζλ−87360pcosh4wζλ+cosh5wζλsech12wζλ2w12λ12τ2ρ524288Γ1+2ρ+⋯.
Forρ=1, the exact results of (27) are given by(34)μζ,τ=14w2λ2sech2λ2−w5−8λ2ν+16ν2+λ416Γ1+ρτρ+wζ+w2λ212.
Analytical approximate solutions with some free parameters are provided by the proposed technique. The analytical findings are extremely useful in deciphering the internal components of acts of nature. Depending on the physical factors, the explicit solutions represented several forms of approximate solutions. Figure1 compares the result obtained by the help of the proposed technique to the exact and analytical result for the fractional-order KK equation. Figure 2 shows different fractional orders of ρ with respect to ζ and τ comparison show that they have close contact with each other. Figure 3 shows the error plot of three- and two-dimensional graphs.Figure 1
The exact and analytical solutions of Example1.Figure 2
The fractional orderρ of Example 1 with respect to ζ and τ.Figure 3
The 2D and 3D error plot of Problem 1.Example 2.
Consider the following fractional Kaup–Kupershmidt equation which is given as(35)Dτρμζ,τ−15μμζζζ−15pμζμζζ+45μ2μζ+μζζζζζ=0,with the initial condition(36)μζ,0=43c−4pcsch2cζ.
Using the Elzaki transform to (35), we get(37)1sρEμζ,τ=μ0ζ,0s2−ρ+E−15μμζζζ−15pμζμζζ+45μ2μζ+μζζζζζ,(38)Eμζ,τ=s2μζ,0+sρE−15μμζζζ−15pμζμζζ+45μ2μζ+μζζζζζ.
Applying the inverse Elzaki transform of (38), we have(39)μζ,τ=E−1s2μζ,0+E−1sρE−15μμζζζ−15pμζμζζ+45μ2μζ+μζζζζζ.
Now, by applying the proposed semianalytical technique, we get(40)μ0ζ,τ=43c−4pcsch2cζ,μ1ζ,τ=E−1sρE−15μ0μ0ζζζ−15pμ0ζμ0ζζ+45μ02μ0ζ+μ0ζζζζζ.μ1ζ,τ=63p2+360−420p+4p16p−15cosh2cζ+p2cosh4cxsech6cζtanhcζ16c7/2τρp3Γ1+ρ,μ2ζ,τ=E−1sρE−15μ1μ1ζζζ−15pμ1ζμ1ζζ+45μ12μ1ζ+μ1ζζζζζ.μ2ζ,τ=−306084p4−3110400+14515200p−26369280p3−62217600p−432000+2656400p3−4451160p2+9181p4cosh2cζ+48p41590p2+14400−60780p+4789p3cosh4cζ−59040p3cosh6cζ+79920p2cosh6cζ−20883p4cosh6cζ−240p3cosh8cζ+p4cosh10cζ+244p4cosh8cζ8c2τ2ρsech12cζp5Γ1+2ρ⋮μnζ,τ=E−1sρE−15μnμnζζζ−15pμnζμnζζ+45μn2μnζ+μnζζζζζ.
The series form result is(41)μζ,τ=μ0ζ,τ+μ1ζ,τ+μ2ζ,τ+μ3ζ,τ+⋯+μnζ,τ.
Therefore, we have(42)uζ,τ=43c−4pcsch2cζ+63p2+360−420p+4p16p−15cosh2cζ+p2cosh4cxsech6cζtanhcζ16c7/2τρp3Γ1+ρ+14515200p−3110400−306084p4−26369280p3−62656400p3+2217600p−4451160p2−432000+9181p4cosh2cζ+48p41590p2+14400+4789p3−60780pcosh4cζ+79920p2cosh6cζ−59040p3cosh6cζ−20883p4cosh6cζ−240p3cosh8cζ+p4cosh10cζ+244p4cosh8cζ8c2τ2ρsech12cζp5Γ1+2ρ+⋯.
Forρ=1, the exact results of (35) are given by(43)μζ,τ=43c−4psech2cζ+83c2−5pcτ.
Analytical approximate solutions with some free parameters are provided by the proposed technique. The analytical findings are extremely useful in deciphering the internal components of acts of nature. Depending on the physical factors, the explicit solutions represented several forms of approximate solutions. Figure4 compares the result obtained by the help of the proposed technique to the exact and analytical result for the fractional-order KK equation. Figure 5 shows different fractional orders of ρ with respect to ζ and τ comparison which show that they have close contact with each other.Figure 4
The exact and analytical solutions of Example2.Figure 5
The fractional orderρ of Example 2 with respect to ζ and τ.Example 3.
Consider the following fractional Kaup–Kupershmidt equation which is given as(44)Dταμζ,τ=5μμζζζ+252μζμζζ+5μ2μζ+μζζζζζ,with the initial condition(45)μζ,0=−2k2+24k21+ekζc−24k21+ekζ.
Using the Elzaki transform to (44), we get(46)1sρEμζ,τ=μ0ζ,0s2−ρ+E5μμζζζ+252μζμζζ+5μ2μζ+μζζζζζ,(47)Eμζ,τ=s2μζ,0+sρE5μμζζζ+252μζμζζ+5μ2μζ+μζζζζζ.
Applying the inverse Elzaki transformation of (47), we have(48)muζ,τ=E−1s2μζ,0+E−1sρE5μμζζζ+252μζμζζ+5μ2μζ+μζζζζζ.
Now, by applying the proposed semianalytical technique, we get(49)μ0ζ,τ=−2k2+24k21+ekζc−24k21+ekζ,μ1ζ,τ=E−1sρE5μ0μ0ζζζ+252μ0ζμ0ζζ+5μ02μ0ζ+μ0ζζζζζ.μ1ζ,τ=τρΓ1+ρ264ekζ−1+ekζk71+ekζ3μ2ζ,τ=E−1sρE5μ1μ1ζζζ+252μ1ζμ1ζζ+5μ12μ1ζ+μ1ζζζζζ.μ2ζ,τ=2904ekζ1−4ekζ+e2kζk12τ2ρ1+ekζ4Γ1+2ρμ3ζ,τ=E−1sρE5μ2μ2ζζζ+252μ2ζμ2ζζ+5μ22μ2ζ+μ2ζζζζζ.μ3ζ,τ=2904ekζ−1+ekζk17τ3ρ11+54ekζ−4923e2kζ+10228e3kζ−4923e4kζ+54e5kζ+11e6kζΓ1+ρ2−60ekζ1−38ekζ+90e2kζ−38e3kζ+e4kζΓ1+2ρ÷1+ekζ9Γ1+ρ2Γ1+3ρ⋮μnζ,τ=E−1sρE5μnμnζζζ+252μnζμnζζ+5μn2μnζ+μnζζζζζ.
The series form result is(50)μζ,τ=μ0ζ,τ+μ1ζ,τ+μ2ζ,τ+μ3ζ,τ+⋯+μnζ,τ.
Therefore, we have(51)uζ,τ=−2k2+24k21+ekζc−24k21+ekζ+τρΓ1+ρ264ekζ−1+ekζk71+ekζ3+2904ekζ1−4ekζ+e2kζk12τ2ρ1+ekζ4Γ1+2ρ+2904ekζ−1+ekζk17τ3ρ11+54ekζ−4923e2kζ+10228e3kζ−4923e4kζ+54e5kζ+11e6kζΓ1+ρ2−60ekζ1−38ekζ+90e2kζ−38e3kζ+e4kζΓ1+2ρ÷1+ekζ9Γ1+ρ2Γ1+3ρ+⋯.
Forρ=1, the exact results of (44) are given by(52)μζ,τ=−2k2+24k21+ekζ+11k5τ−24k21+ekζ+11k5τ2.
Analytical approximate solutions with some free parameters are provided by the proposed technique. The analytical findings are extremely useful in deciphering the internal components of acts of nature. Depending on the physical factors, the explicit solutions represented several forms of approximate solutions. Figure6 compares the result obtained by the help of the proposed technique to the exact and analytical result for the fractional-order KK equation. Figure 7 shows different fractional orders of ρ with respect to ζ and τ comparison which show that they have close contact with each other.Figure 6
The exact and analytical solutions of Example3.Figure 7
The fractional orderρ of Example 3 with respect to ζ and τ.


## 5. Conclusion

In this article, the iterative transformation technique is utilized to achieve analytical solutions of the fractional-order Kaup–Kupershmidt equations, which are broadly utilized as problems for spatial effects in applied sciences. The method gave a series type of solutions that converge very quickly in the mathematical model. It is predicted that the results obtained in this paper will be effective for more evaluation of the complicated nonlinear physical models. The analyses of this method are very clear and straightforward. As a result, we conclude that this method can be used to solve a variety of nonlinear fractional-order partial differential equation schemes.



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