# Numerical Solutions of Time Fractional Zakharov-Kuznetsov Equation via Natural Transform Decomposition Method with Nonsingular Kernel Derivatives

Authors: Mei-Xiu Zhou; A. S. V. Ravi Kanth; K. Aruna; K. Raghavendar; Hadi Rezazadeh; Mustafa Inc; Ayman A. Aly
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, we have studied the time-fractional Zakharov-Kuznetsov equation (TFZKE) via natural transform decomposition method (NTDM) with nonsingular kernel derivatives. The fractional derivative considered in Caputo-Fabrizio (CF) and Atangana-Baleanu derivative in Caputo sense (ABC). We employed natural transform (NT) on TFZKE followed by inverse natural transform, to obtain the solution of the equation. To validate the method, we have considered a few examples and compared with the actual results. Numerical results are in accordance with the existing results.


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## Body

## 1. Introduction

Fractional calculus is an emerging field in various branches of engineering science. Fractional differential equations attracted researchers as they used to model a variety of diverse applications such as visco elasticity, heat conduction, biology, and dynamical systems [1–7]. Due to its importance in diverse fields, considerable methods developed to study the exact and computational solutions of fractional differential equations. Other than the modelling, divergence and convergence of the solutions are also equally important. A suitable definition is essential for a fractional generalization of a physical model. Several fractional derivative definitions developed in the last few decades. Some of the popular definitions in the literature are Riemann-Liouville (R-L), Caputo, CF, ABC, Grunwald-Letnikov, and Riesz fractional derivatives. For more details, we refer to [8, 9] and the references therein. R-L and Caputo fractional derivatives have a singular kernel. Recently, two nonsingular kernel fractional derivative definitions are developed by Atangana-Baleanu and Caputo-Fabrizio. Several methods are being investigated for the analysis of fractional differential equations for accuracy and reliable solutions. Some of the popular semi analytical and numerical methods are variational iteration method (VIM) [10], fractional differential transform method [11–14], homotopy perturbation transform method (HPTM) [15], homotopy analysis transform method [16, 17], residual power series method (RPS) [18], q-homotopy analysis transform method (q-HATM) [19–21], operational matrix method [22], tension spline method [23], parametric cubic spline [24], exponential B-spline method [25–27], fractional natural decomposition method (FNDM) [28], Adam Bashforth’s Moulton method [29], and references therein.In this paper, we have considered the TFZKE(1)Dtμux,ζ,t+auξx+buηxxx+cuδxζζ=0,with initial condition(2)ux,ζ,0=ϕx,ζ.This model illustrates the behavior of weakly nonlinear ion-acoustic waves in a plasma bearing cold ions and hot isothermal electrons in the presence of a uniform magnetic field. This problem has been solved by many techniques such as VIM [10], homotopy perturbation method [30], HPTM [15], perturbation iteration algorithm and RPS [31], and new iterative Sumudu transform method [32]. Recently, Veeresha and Prakasha [28] presented the applications of q-HATM and FNDM for solving TFZKE.The aim of this paper is to implement NTDM to solve TFZKE. Rawashdeh and Maitama [33] introduced NTDM for a class of nonlinear partial differential equations. NTDM do not require linearization; prescribe assumptions, perturbation, or discretization; and prevent any round-off errors. Recently, NTDM employed to time-fractional Fisher’s equation [34] and 2+1−dimensional time-fractional coupled Burger equations [35]. The paper is organized as follows. Basic definitions of singular and nonsingular definitions of fractional calculus and NT and its fractional derivatives are discussed briefly in Section 2. In Section 4, we presented the convergence and uniqueness of the solutions. In Section 3, we presented the NTDM for nonsingular definitions to solve TFZKE. In Section 5, few examples of TFZKE are given to validate the present methods. Section 6 presents the results and discussions. In Section 7, brief conclusion of this paper is presented.


## 2. Basic Definitions

There are various fractional derivative definitions that are available in the literature; for more details, we refer [8, 36–38]. In this section, we give the definitions of R-L, Caputo, CF, and ABC fractional derivatives for the benefit of the readers.Definition 1 (see [36]).
The R-L left-sided fractional integral operator of a functionf∈Cν, ν≥−1 is given as(3)Iμfω=1Γμ∫0ωω−ςμ−1fςdς,μ>0,ω>0,and I0fω=fω.Definition 2 (see [1]).
The Caputo sense fractional derivative offωis defined by(4)0CDωμfω=Im−μDmfω=1m−μ∫0ωω−ςm−μ−1fmςdς,for m−1<μ≤m, m∈ℕ, ω>0, f∈Cνm, and ν≥−1.Definition 3 (see [39]).
The CF fractional derivative offω is given by(5)0CFDωμfω=Bμ1−μ∫0ωexp−μω−ς1−μDfςdς,where 0<μ<1 and Bμ is a normalization function, where B0=B1=1.Definition 4 (see [40]).
The ABC fractional derivative offω is presented as(6)0ABCDωμfω=Bμ1−μ∫0ωEμ−μω−ς1−μDfςdς,where 0<μ<1. Normalization function is Bμ, and the Mittag-Leffler function is Eμz=∑l=0∞zl/Γμl+1. These definitions widely used to study the fractional differential equation solutions using numerous integral transform techniques such as Sumudu transform, Shehu transform, and Laplace transform. Recently, natural transform of these definitions applied to study various differential equations; for more details, we refer [33, 41, 42]. Computational time can be reduced in this transform than other traditional methods while preserving the efficiency. When v=1 and s=1, the NT reduced to the Laplace transform and Sumudu integral transform, respectively.Definition 5.
The natural transform ofut is defined by(7)NTut=Us,v=∫−∞∞e−stuvtdt,s,v∈−∞,∞.
Fort∈0,∞, natural transform of ut is defined by(8)NTutHt=NT+ut=U+s,v=∫0∞e−stuvtdt,s,v∈0,∞,where Ht is the Heaviside function.Definition 6.
The inverse natural transform ofUs,v is given by(9)NT−1Us,v=ut,∀t≥0.Lemma 7 (linearity property).
If natural transform ofu1t is u1s,v and u2t is u2s,v, then(10)NTc1u1t+c2u2t=c1NTu1t+c2NTu2t=c1u1s,v+c2u2s,v,where c1 and c2 are constants.Lemma 8 (inverse linearity property).
If inverse natural transform ofu1s,v and u2s,v is u1t and u2t, respectively, then(11)NT−1c1u1s,v+c2u2s,v=c1NT−1u1s,v+c2NT−1u2s,v=c1u1t+c2u2t,where c1 and c2 are constants.Definition 9 (see [43]).
Natural transform ofDtμut by means of Caputo sense is given as(12)NT0CDtμut=svμNTut−1su0.Definition 10 (see [44]).
Natural transform ofDtμut by means of CF is defined as(13)NT0CFDtμut=11−μ+μv/sNTut−1su0.With this motivation, we defined natural transform of ABC derivative as follows.Definition 11.
Natural transform ofDtμut by means of ABC derivative is defined as(14)NT0ABCDtμut=Mμ1−μ+μv/sμNTut−1su0.


## 3. Methodology

In this section, we present a novel approximate analytical procedure based on natural transform [42] to the following equation(15)Dtμuζ,t=Luζ,t+Nuζ,t+hζ,t,with the initial condition(16)uζ,0=ϕζ,where N, L, and hζ,t are nonlinear, linear, and source terms, respectively. Now we employing NT on equation (15) by considering fractional derivative by means of three fractional definitions.Case 1.
(NTDMCF). By taking natural transform of equation (15), by means of CF fractional derivative, we obtain(17)1pμ,v,sNTuζ,t−ϕζs=NTMζ,t,where(18)pμ,v,s=1−μ+μvs.
By taking inverse natural transform using (8), we rewrite (17) as(19)uζ,t=NT−1ϕζs+pμ,v,sNTMζ,t.
Nuζ,t can be decomposed into(20)Nuζ,t=∑l=0∞Al,where Al is the Adomian polynomials [45, 46]. We assume that equation (15) has the analytical expansion(21)uζ,t=∑l=0∞ulζ,t.
By substituting equations (20) and (21) into (19), we obtain(22)∑l=0∞ulζ,t=NT−1ϕζs+pμ,v,sNThζ,t+NT−1pμ,v,sNT∑l=0∞Lulζ,t+Al.
From (22), we get(23)u0CFζ,t=NT−1ϕζs+pμ,v,sNThζ,tu1CFζ,t=NT−1pμ,v,sNTLu0ζ,t+A0⋮ul+1CFζ,t=NT−1pμ,v,sNTLulζ,t+Al,l=1,2,⋯.
By substituting (23) into (21), we get the NTDMCF solution of (15) and (16) as(24)uCFζ,t=u0CFζ,t+u1CFζ,t+u2CFζ,t+⋯.Case 2.
(NTDMABC). By taking natural transform of equation (15), by means of ABC derivative, we acquire(25)1qμ,v,sNTuζ,t−ϕζs=NTMζ,t,where(26)qμ,v,s=1−μ+μv/sμBμ.
By taking inverse natural transform using (8), we rewrite (25), as(27)uζ,t=NT−1ϕζs+qμ,v,sNTMζ,t.
Nuζ,t can be decomposed into(28)Nuζ,t=∑l=0∞Al,where Al is the Adomian polynomials. We assume that equation (15) has the analytical expansion(29)uζ,t=∑l=0∞ulζ,t.
By substituting equations (28) and (29) into (27), we obtain(30)∑l=0∞ulζ,t=NT−1ϕζs+qμ,v,sNThζ,t+NT−1qμ,v,sNT∑l=0∞Lulζ,t+Al.
From (30), we get(31)u0ABCζ,t=NT−1ϕζs+NT−1qμ,v,sNThζ,tu1ABCζ,t=NT−1qμ,v,sNTLu0ζ,t+A0⋮ul+1ABCζ,t=NT−1qμ,v,sNTLulζ,t+Al,l=1,2,⋯.
By substituting (31) into (29), we get the NTDMABC solution of (15)–(16) as(32)uABCζ,t=u0ABCζ,t+u1ABCζ,t+u2ABCζ,t+⋯


## 4. Convergence Analysis

We have presented uniqueness and convergence of theNTDMCF and NTDMABC in this section.Theorem 12.
TheNTDMCF solution of (15) is unique when 0<δ1+δ21−μ+μt<1.Proof.
LetF=CJ,. be the Banach space with the norm ϕt=maxt∈Jϕt, ∀ continuous functions on J. Let G:F⟶F is a nonlinear mapping, where(33)ul+1Cζ,t=u0C+NT−1pμ,v,sNTLulζ,t+Nulζ,t,l≥0.
Suppose thatLu−Lu∗<δ1u−u∗ and Nu−Nu∗<δ2u−u∗, where δ1 and δ2 are Lipschitz constants and u≔uζ,t and u∗≔u∗ζ,t are two different function values.(34)Gu−Gu∗≤maxt∈JNT−1pμ,v,sNTLu−Lu∗+pμ,v,sNTNu−Nu∗≤maxt∈Jδ1NT−1pμ,v,sNTu−u∗+δ2NT−1pμ,v,sNTu−u∗≤maxt∈Jδ1+δ2NT−1pμ,v,sNTu−u∗≤δ1+δ2NT−1pμ,v,sNTu−u∗=δ1+δ21−μ+μtu−u∗.
G is contraction as 0<δ1+δ21−μ+μt<1. The solution of (15) is unique from Banach fixed point theorem.☐Theorem 13.
TheNTDMABC solution of (15) is unique when 0<δ1+δ21−μ+μtμ/Γμ+1<1.Proof.
LetF=CJ,. be the Banach space with the norm ϕt=maxt∈Jϕt, ∀ continuous functions on J. Let G:F⟶F is a nonlinear mapping, where(35)ul+1Cζ,t=u0C+NT−1qμ,v,sNTLulζ,t+Nulζ,t,l≥0.
Suppose thatLu−Lu∗<δ1u−u∗ and Nu−Nu∗<δ2u−u∗, where δ1 and δ2 are Lipschitz constants and u≔uζ,t and u∗≔u∗ζ,t are two different function values.(36)Gu−Gu∗≤maxt∈JNT−1qμ,v,sNTLu−Lu∗+qμ,v,sNTNu−Nu∗≤maxt∈Jδ1NT−1qμ,v,sNTu−u∗+δ2NT−1qμ,v,sNTu−u∗≤maxt∈Jδ1+δ2NT−1qμ,v,sNTu−u∗≤δ1+δ2NT−1qμ,v,sNTu−u∗=δ1+δ21−μ+μtμΓμ+1u−u∗.
G is contraction as 0<δ1+δ21−μ+μtμ/Γμ+1<1. The solution of (15) is unique from Banach fixed point theorem.☐Theorem 14.
NTDMCF solution of (15) is convergent.Proof.
Letum=∑r=0murζ,t. To prove that um is a Cauchy sequence in F. Consider(37)um−un=maxt∈Jum−un=maxt∈J∑r=n+1mur,n=1,2,3,⋯.≤maxt∈JNT−1pμ,v,sNT∑r=n+1mLur−1+Nur−1=maxt∈JNT−1pμ,v,sNT∑r=nm−1Lur+Nur≤maxt∈JNT−1pμ,v,sNTLum−1−Lun−1+Num−1−Nun−1≤δ1maxt∈JNT−1pμ,v,sNTLum−1−Lun−1+δ2maxt∈JNT−1pμ,v,sNTNum−1−Nun−1=δ1+δ21−μ+μtum−1−un−1.
Letm=n+1, then(38)un+1−un≤δun−un−1≤δ2un−1−un−2≤⋯≤δnu1−u0,where δ=δ1+δ21−μ+μt. Similarly, we have(39)um−un≤un+1−un+un+2−un+1+⋯+um−um−1,≤δn+δn+1+⋯+δm−1u1−u0≤δn1−δm−n1−δu1,
As0<δ<1, we get 1−δm−n<1. Therefore(40)um−un≤δn1−δmaxt∈Ju1.
Sinceu1<∞. um−un⟶0 when n⟶∞. Hence, um is a Cauchy sequence in F; therefore, the series um is convergent.☐Theorem 15.
NTDMABC solution of (15) is convergent.Proof.
Letum=∑r=0murζ,t. To prove that um is a Cauchy sequence in F. Consider(41)um−un=maxt∈Jum−un=maxt∈J∑r=n+1mur,n=1,2,3,⋯.≤maxt∈JNT−1qμ,v,sNT∑r=n+1mLur−1+Nur−1=maxt∈JNT−1qμ,v,sNT∑r=nm−1Lur+Nur≤maxt∈JNT−1qμ,v,sNTLum−1−Lun−1+Num−1−Nun−1≤δ1maxt∈JNT−1qμ,v,sNTLum−1−Lun−1+δ2maxt∈JNT−1qμ,v,sNTNum−1−Nun−1=δ1+δ21−μ+μtμΓμ+1∥um−1−un−1∥.
Letm=n+1, then(42)un+1−un≤δun−un−1≤δ2un−1−un−2≤⋯≤δnu1−u0,where δ=δ1+δ21−μ+μtμ/Γμ+1. Similarly, we have(43)um−un≤un+1−un+un+2−un+1+⋯+um−um−1,≤δn+δn+1+⋯+δm−1u1−u0≤δn1−δm−n1−δu1,
As0<δ<1, we get 1−δm−n<1. Therefore(44)um−un≤δn1−δmaxt∈Ju1.
Sinceu1<∞. um−un⟶0 when n⟶∞. Hence, um is a Cauchy sequence in F; therefore, the series um is convergent.☐


## 5. Numerical Examples

This section includes the approximate analytical solutions for a few examples of TFZKE. We have chosen these equations as the closed form solutions are available and also well-known methods employed to study the solutions in the literature.Example 1.
TFZKE (1) is considered with the following parameters. Let ξ=η=δ=2, a=1,b=c=1/8, and ux,ζ,0=4/3λsinh2x+ζ [47, 48]. When μ=1, exact solution [49] is ux,ζ,t=4/3λsinh2x+ζ−λt.NTDMCF: By employing NTDMCF, we get(45)u0CFx,ζ,t=43λsinh2x+ζ,u1CFx,ζ,t=−89λ2μt−1+15sinh4x+ζ−4sinh2x+ζ,u2CFx,ζ,t=3227λ32+2μ2−4μ+4μ−4μ2t+μ2t2×13cosh2x+ζ+75cosh6x+ζ−70cosh4x+ζ,⋮Substitutingu0CFx,ζ,t, u1CFx,ζ,t, in (24), we obtain the NTDMCF solution as(46)uCFx,ζ,t≈43λsinh2x+ζ−89λ2μt−1+15sinh4x+ζ−4sinh2x+ζ+3227λ32+2μ2−4μ+4μ−4μ2t+μ2t2×13cosh2x+ζ+75cosh6x+ζ−70cosh4x+ζ.NTDMABC: By employing the NTDMABC, we get(47)u0ABCx,ζ,t=43λsinh2x+ζ,u1ABCx,ζ,t=−8λ2−μΓμ+Γμ+tμ5sinh4x+ζ9Γμ−4sinh2x+ζ,u2ABCx,ζ,t=64λ313cosh2x+y−70cosh4x+y+75cosh6x+y27ΓμΓ2μ+1×μΓμ+1t2μ+μ−1Γ2μ+1μ−1Γμ−2tμ,⋮Substitutingu0ABCx,ζ,t, u1ABCx,ζ,t, in (28), we obtain the NTDMABC solution as(48)uABCx,ζ,t≈43λsinh2x+ζ−8λ2−μΓμ+Γμ+tμ5sinh4x+ζ−4sinh2x+ζ9Γμ,64λ313cosh2x+y−70cosh4x+y+75cosh6x+y27ΓμΓ2μ+1×μΓμ+1t2μ+μ−1Γ2μ+1μ−1Γμ−2tμ.Example 2.
TFZKE (1) is considered with the following parameters. Let ξ=η=δ=3, a=1,b=c=2, and ux,ζ,0=3/2λsinhx+ζ/6 [47, 48]. When μ=1, exact solution [49] is given by ux,ζ,t=3/2λsinhx+ζ−λt/6.NTDMCF: By employing the NTDMCF, we get(49)u0CFx,ζ,t=32λsinhx+ζ6,u1CFx,ζ,t=332λ3μt−1+15coshx+ζ6−9coshx+ζ2,u2CFx,ζ,t=3λ51024−621sinhx+ζ2+70sinhx+ζ6+765sinh5x+ζ6×μ2t−4t+2+4μt−1+2,⋮Substitutingu0CFx,ζ,t, u1CFx,ζ,t, in (24), we obtain the NTDMCF solution as(50)uCFx,ζ,t≈32λsinhx+ζ6+332λ3μt−1+15coshx+ζ6−9coshx+ζ2+3λ51024−621sinhx+ζ2+70sinhx+ζ6+765sinh5x+ζ6×μ2t−4t+2+4μt−1+2+.NTDMABC: By employing the NTDMABC, we get(51)u0ABCx,ζ,t=32λsinhx+ζ6,u1ABCx,ζ,t=−3λ3−μΓμ+Γμ+tμcoshx+ζ/69coshx+ζ/3−716Γμ,u2ABCx,ζ,t=3λ5μΓμ+1t2μ+μ−1Γ2μ+1μ−1Γμ−2tμ512ΓμΓ2μ+1×−621sinhx+ζ2+70sinhx+ζ6+765sinh5x+ζ6,⋮Substitutingu0ABCx,ζ,tu1ABCx,ζ,t, in (32), we obtain the NTDMABC solution as(52)uABCx,ζ,t≈32λsinhx+ζ6−3λ3−μΓμ+Γμ+tμcoshx+ζ/69coshx+ζ/3−716Γμ+3λ5μΓμ+1t2μ+μ−1Γ2μ+1μ−1Γμ−2tμ512ΓμΓ2μ+1×−621sinhx+ζ2+70sinhx+ζ6+765sinh5x+ζ6+.


## 6. Numerical Results and Discussion

Tables1 and 2 demonstrates the comparison of absolute errors with the existing methods and approximate solutions for different fractional orders with different fractional derivatives, respectively, of Example 1. Absolute errors of Example 1 graphically represented in Figure 1 for fixed t when μ=1. In Figure 2, we plotted approximate solutions for different values of μ for fixed t of Example 1. Figure 3 presents the comparison of NTDMCF and NTDMABC solutions of Example 1 with exact solution for different values of fractional order μ for fixed x and ζ. In Table 3, we presented absolute errors of two fractional derivative solutions and existing results of Example 2. We have tabulated the approximate solution of Example 2 for noninteger fractional values in Table 4. For Example 2, NTDMCF and NTDMABC absolute errors graphically represent in Figure 4 for fixed t when μ=1, and in Figure 5, we plotted approximate solutions for different values of μ for specific value of t. Figure 6 presents the comparison of NTDMCF and NTDMABC with exact solution for noninteger values of μ for fixed t and ζ of Example 2. It is observed from tables and figures that the two-term approximate solution is having good accordance with the existing results and exact solution. For noninteger values, NTDMCF and NTDMABC are showing same behavior.Table 1
Absolute errors ofNTDMCF and NTDMABC with PIA [31] and RPSM [31] of Example 1 for μ=1 and λ=0.001.
xζtNTDMCFNTDMABCPIA [31]RPSM [31]0.10.10.23.8519E-073.8519E-073.8521E-073.8521E-070.35.7584E-075.7584E-075.7591E-075.7591E-070.47.6517E-077.6517E-077.6535E-077.6535E-070.60.60.24.6474E-054.6474E-054.6634E-054.6639E-050.36.8157E-056.8157E-056.8606E-056.8631E-050.48.8582E-058.8582E-058.9824E-058.9905E-050.90.90.24.9249E-044.9249E-045.1213E-045.1424E-040.36.7503E-046.7503E-047.3819E-047.4845E-040.48.1510E-048.1510E-049.5794E-049.8914E-04Table 2
Approximate solutions ofNTDMCF and NTDMABC for μ=0.67 and μ=0.75 of Example 1 for t=1 and λ=0.001.
xζμ=0.67μ=0.75NTDMCFNTDMABCNTDMCFNTDMABC0.020.021.7406E-061.7189E-061.7380E-061.7176E-060.044.1951E-064.1584E-064.1923E-064.1581E-060.067.7192E-067.6674E-067.7161E-067.6679E-060.081.2317E-051.2250E-051.2313E-051.2251E-050.101.7994E-051.7911E-051.7990E-051.7913E-050.040.024.1951E-064.1584E-064.1923E-064.1581E-060.047.7192E-067.6674E-067.7161E-067.6679E-060.061.2317E-051.2250E-051.2313E-051.2251E-050.081.7994E-051.7911E-051.7990E-051.7913E-050.102.4757E-052.4658E-052.4752E-052.4660E-050.060.027.7192E-067.6674E-067.7161E-067.6679E-060.041.2317E-051.2250E-051.2313E-051.2251E-050.061.7994E-051.7911E-051.7990E-051.7913E-050.082.4757E-052.4658E-052.4752E-052.4660E-050.103.2616E-053.2500E-053.2610E-053.2503E-050.080.021.2317E-051.2250E-051.2313E-051.2251E-050.041.7994E-051.7911E-051.7990E-051.7913E-050.062.4757E-052.4658E-052.4752E-052.4660E-050.083.2616E-053.2500E-053.2610E-053.2503E-050.104.1581E-054.1448E-054.1575E-054.1451E-050.100.021.7994E-051.7911E-051.7990E-051.7913E-050.042.4757E-052.4658E-052.4752E-052.4660E-050.063.2616E-053.2500E-053.2610E-053.2503E-050.084.1581E-054.1448E-054.1575E-054.1451E-050.105.1665E-055.1513E-055.1657E-055.1516E-05Figure 1
(a) Exact solution, (b) absolute error of NTDMCF, and (c) absolute error of NTDMABC of Example 1 for μ=1, t=0.5, and λ=0.001.
(a)(b)(c)Figure 2
(a) NTDMCF for μ=0.60, (b) NTDMCF for μ=0.75, (c) NTDMABC for μ=0.60, and (d) NTDMABC for μ=0.75, when t=0.5 and λ=0.001 of Example 1.
(a)(b)(c)(d)Figure 3
(a) NTDMCF and (b) NTDMABC of Example 1, for x=0.5, ζ=0.5, and λ=0.001 for different values of μ.
(a)(b)Table 3
Absolute errors of NTDMCF and NTDMABC with existing methods of Example 2 when λ=0.001.
x,ζtNTDMCFNTDMABCFNDM [28]q−HATM [28]0.020.024.9926E-094.9926E-094.9926E-094.9926E-090.049.9852E-099.9852E-099.9852E-099.9852E-090.061.4979E-081.4979E-081.4979E-081.4979E-080.081.9970E-081.9970E-081.9970E-081.9970E-080.102.4963E-082.4963E-082.4963E-082.4963E-080.040.024.9929E-094.9929E-094.9929E-094.9929E-090.049.9859E-099.9859E-099.9859E-099.9859E-090.061.4979E-081.4979E-081.4979E-081.4979E-080.081.9972E-081.9972E-081.9972E-081.9972E-080.102.4965E-082.4965E-082.4965E-082.4965E-080.060.024.9934E-094.9934E-094.9934E-094.9934E-090.049.9869E-099.9869E-099.9869E-099.9869E-090.061.4980E-081.4980E-081.4980E-081.4980E-080.081.9974E-081.9974E-081.9974E-081.9974E-080.102.4967E-082.4967E-082.4967E-082.4967E-080.080.024.9942E-094.9942E-094.9942E-094.9942E-090.049.9884E-099.9884E-099.9884E-099.9884E-090.061.4983E-081.4983E-081.4983E-081.4983E-080.081.9977E-081.9977E-081.9977E-081.9977E-080.102.4971E-082.4971E-082.4971E-082.4971E-080.100.024.9952E-094.9952E-094.9952E-094.9952E-090.049.9904E-099.9904E-099.9904E-099.9904E-090.061.4986E-081.4986E-081.4986E-081.4986E-080.081.9981E-081.9981E-081.9981E-081.9981E-080.102.4976E-082.4976E-082.4976E-082.4976E-08Table 4
Approximate solutions of NTDMCF and NTDMABC for μ=0.67 and μ=0.75 of Example 2 when t=1 and λ=0.001.
xζμ=0.67μ=0.75NTDMCFNTDMABCNTDMCFNTDMABC0.020.029.9997E-069.9997E-069.9997E-069.9997E-060.041.5000E-051.5000E-051.5000E-051.5000E-050.062.0000E-052.0000E-052.0000E-052.0000E-050.082.5001E-052.5001E-052.5001E-052.5001E-050.103.0002E-053.0002E-053.0002E-053.0002E-050.040.021.5000E-051.5000E-051.5000E-051.5000E-050.042.0000E-052.0000E-052.0000E-052.0000E-050.062.5001E-052.5001E-052.5001E-052.5001E-050.083.0002E-053.0002E-053.0002E-053.0002E-050.103.5003E-053.5003E-053.5003E-053.5003E-050.060.022.0000E-052.0000E-052.0000E-052.0000E-050.042.5001E-052.5001E-052.5001E-052.5001E-050.063.0002E-053.0002E-053.0002E-053.0002E-050.083.5003E-053.5003E-053.5003E-053.5003E-050.104.0004E-054.0004E-054.0004E-054.0004E-050.080.022.5001E-052.5001E-052.5001E-052.5001E-050.043.0002E-053.0002E-053.0002E-053.0002E-050.063.5003E-053.5003E-053.5003E-053.5003E-050.084.0004E-054.0004E-054.0004E-054.0004E-050.104.5006E-054.5006E-054.5006E-054.5006E-05100.023.0002E-053.0002E-053.0002E-053.0002E-050.043.5003E-053.5003E-053.5003E-053.5003E-050.064.0004E-054.0004E-054.0004E-054.0004E-050.084.5006E-054.5006E-054.5006E-054.5006E-050.105.0009E-055.0009E-055.0009E-055.0009E-05Figure 4
(a) Exact solution, (b) absolute error of NTDMCF, and (c) absolute error of NTDMABC, for μ=1, t=0.5, and λ=0.001 of Example 2.
(a)(b)(c)Figure 5
Example2. (a) NTDMCF for μ=0.60. (b) NTDMCF for μ=0.75. (c) NTDMABC for μ=0.60. (d) NTDMABC for μ=0.75 when t=0.5 and λ=0.001.
(a)(b)(c)(d)Figure 6
(a) NTDMCF and (b) NTDMABC, for x=0.02, ζ=0.02, and λ=0.001 for Example 2 with various values of μ.
(a)(b)


## 7. Conclusions

In this paper, we have studied the TFZKE through natural transformation by means of CF and ABC derivatives. We compared numerical results with the existing results. It is observed that the present method results are in accordance with existing methods. The NTDM is simple in its principles; also, NTDM is effective in solving nonlinear fractional differential equations, and promising method for a large varieties of such equations arises in mathematical physics.



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