# Analysis of the Fuzzy Fractional-Order Solitary Wave Solutions for the KdV Equation in the Sense of Caputo-Fabrizio Derivative

Authors: Muhammad Naeem; Hadi Rezazadeh; Ahmed A. Khammash; Rasool Shah; Shamsullah Zaland
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this paper, we construct a system for analysis of an analytic solution of fractional fuzzy solitary wave solutions for the Korteweg–De Vries (KdV) equation. We apply the iterative method and the Laplace transform under the fractional Caputo-Fabrizio operator. The obtained series form the solution was calculated and approached the estimate values of the proposed problems. The upper and lower portions of the fuzzy result in all three problems were simulation applying two different fractional order among zero and one. The fractional operator is nonsingular and global since the exponential function is present. It provides all types of fuzzy results occurring among zero and one at any fractional order because its dynamic behaviour is globalised of the suggested problems. Because the fuzzy number provides the result in a fuzzy form, with lower and upper branches, fuzziness is also incorporated in the unknown quantity.


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## Body

## 1. Introduction

Fractional calculus has a wide range of implementations in areas where data are imprecise, such as natural, biological, sciences, and engineering [1, 2]. Let us first learn about the fuzzy set before delving into such problems. In 1965 [3], Zadeh developed the concept of a fuzzy set, which describes how to quantify uncertainty in some phenomena. The fuzzy set theory is, therefore, extended to a few other branches such as algebra, topology, fuzzy logic, analysis, automata, and several more branches of mathematics and science. They expand on the fundamental idea by describing fuzzy function and control. Based on these findings, the scholars expanded on the idea by introducing basic fuzzy calculus. Fuzzy fractional integral and differential equations, denoted as FFIEs and FFDEs, have gained popularity in recent times due to their validity in constructing real-world phenomena. In [4], Esmail et al. investigated some basic mathematical analytical solutions. Fractional fuzzy integral equations (FFIEs) and fractional fuzzy differential equations (FFDEs) can be utilised to model these forms of problems. Many more scholars and scientists believe that this method can be used to analyse both fractional and integer fuzzy differential equations. Because of the numerous uses of the fuzzy differential equation to model unknown phenomena in a variety of areas, including business, physical sciences and biological sciences, refer to [5–9].A wide range of applied sciences and engineering fields, such as porous media, signal processing, electrical circuits, thermal systems, acoustics, robotics, and optimal control, has turned to fuzzy differential equations in recent decades to describe the physical processes and uncertain parameters [10–16]. Fuzzy equations can be used to model physical phenomena more realistically and to achieve a better explanation of their underlying causes by applying fractional operators to them [17]. Consequently, the uniqueness and existence of fuzzy fractional differential equation solution have been shown in [18] built on the idea suggested in [19]. Agarwal et al. [19] introduced the idea of fuzzy fractional differential equations in this regard. Laplace fuzzy transformation has been used to analysis fuzzy fractional differential equations under the Riemann–Liouville H-differentiability [20]. Nonlaopon et al. [21] provided explicit solutions to fuzzy fractional differential equations in the way of Riemann–Liouville generalised H-differentiability utilising Mittag–Leffler functions. In [22], the fractional derivative generalised Atangana–Baleanu differentiability has been implemented to solve fuzzy fractional differential equations. Additionally, Allahviranloo and Ghanbari [17] investigated the ABC fractional derivative in order to handle with fuzzy fractional differential equations expressed in parametric interval type. Hoa et al. [23] proposed a new technique for obtaining analytical solutions to fuzzy fractional differential equations using the fractional derivative Caputo Katugampola.As a consequence, different researchers evaluated the solution of fractional fuzzy differential equations to use these problems [24]. The investigators [25, 26] applied an accurate computational method to analyse analytic results to nonlinear Lane Emden equations and an effective numerical technique for the fractional vibration and advection dispersion problems that occur in the porous media. Numerous fractional partial differential equations using ion-acoustic waves in plasma, hydromagnetic waves in a cold plasma, and magnetoacoustic waves in analytical and numerical methods are applied [27, 28]. As far as new ideas are introduced, we propose an initial value solitary wave solution for the KdV equation when the fuzzy Caputo fractional operator is used to solve it. It is because of the fuzzy number that we use an unknown quantity’s fuzziness and start-up conditions to figure out its fuzzy type with two branch solutions. A lot of people have been working on integer and fractional-order diffusion equations and fuzzy heat equations, as well as a lot of other things. For the analysis of the suggested problem with different external source components, we investigated both fractional order and fuzzy [10, 29–32].Daftardar-Gejji and Jafari developed an innovative iterative technique of analysis for solving nonlinear equations in 2006 [35, 36]. In the first implementation of Laplace transform in an iterative method by Jafari et al., the iterative Laplace transform method [37] was presented as a straightforward technique for evaluating approximate implications of the fractional partial differential equation system. The iterative Laplace transform method was used to solve linear and nonlinear partial differential equations such as fractional-order Fornberg Whitham equations (38), time-fractional Zakharov–Kuznetsov equation [39], and fractional-order Fokker Planck equations (40).


## 2. Basic Definitions

In this section, we represent some definitions, theorems, and lemmas of fractional calculus and well-known operators.Definition 1.
The fractional integral fuzzy Caputo-Fabrizio operator with respect toη, then the fuzzy continuous function U˜η on a 0,b subset of ℛ, is given as [14, 33, 34](1)CFIνΦ˜η=1−νMνΦ˜ν+νMν∫0ηΦ˜τdτ,ν,τ∈0,∞,where M0=M1=1. Next, if Φ˜η∈LF0,b∩CF0,b, as CF0,b is the space of the fuzzy continuous function while LF0,b is the space of fuzzy Lebesgue integral functions, respectively, then the Caputo-Fabrizio noninteger order fuzzy integral is defined as(2)CFIνΦ˜ηr=IνΦ¯rη,IνΦ¯rη,0≤r≤1,CFIνΦ¯rη=1−νMνΦ¯ν+νMν∫0ηΦ˜τdτ,ν,τ∈0,∞,CFIνΦ¯rη=1−νMνΦ¯ν+νMν∫0ηΦ˜τdτ,ν,τ∈0,∞.Definition 2.
Similarly, for an operatorΦ˜η∈LF0,b∩CF0,b, as Φ˜η=Φ¯η,Φ¯η, 0≤r≤1, and 0<η0<b, the Caputo-Fabrizio (CF) fractional operator in the sense of fuzzy is define as [14, 33, 34](3)CFDνΦ˜ηr=DνΦ¯η0,DνΦ¯η0,0<ν≤1,CFDνΦ¯η0=Mν1−ν∫0ηΦ¯τ′exp−νη−τ1−νdτ,CFDνΦ¯η0=Mν1−ν∫0ηΦ¯τ′exp−νη−τ1−νdτ,where the integral converges or exists and m=⌈ν⌉+1. As ν lies in the intervals 0,1, m=1.Definition 3.
The Laplace transformation of fuzzy functionℱx is expressed as [14, 33, 34](4)ℱx=Lfx=∫0∞e−xηfηdη,η>0.Definition 4.
The Laplace transformation for CF is(5)LCFDν+nΦ˜η=ωn+1Φ˜ω−ωnΦ˜0−ωn−1Φ˜′0−⋯−Φ˜n0ω+ν1−ω.Definition 5.
The “Mittag–Leffler” operatorEβη is [14, 33, 34](6)Eβη=∑n=0∞ηnΓ1+nβ,where 0<β.Definition 6.
A mappingk:Re⟶0,1 is named as a number of fuzzy if the following criteria hold [14, 33, 34]:1
k is continuous until it reaches a maximum value2
kμy1+μy2≥minky1,ky23
∃y0∈ℜ;ky0=1, i.e., k is normal4
cly∈R,ky>0 is continuous and bounded, where cl shows closure for the supports of y
We denote the set of fuzzy numbers collectively asε.


## 3. General Implementation of Fuzzy

In this portion, we examine our conceptual model for an approximate result. The fractional CF derivative is applied along with an iterative transformation technique as(7)ℒCFDηνΦ˜ε,ζ,η=ℒDε2Φ˜ε,ζ,η+Dζ2Φ˜ε,ζ,η+k˜rℱε,ζ,η,where ν∈0,1; therefore, the Laplace transform of (7) is(8)ωℒΦ˜ε,ζ,η−Φ˜ε,ζ,0ω+ν1−ω=ℒDε2Φ˜ε,ζ,η+Dζ2Φ˜ε,ζ,η+k˜rℱε,ζ,η.On using the initial condition, we obtain(9)ωℒΦ˜ε,ζ,η=gε,ζ+ω+ν1−ωℒDε2Φ˜ε,ζ,η+Dζ2Φ˜ε,ζ,η+k˜rℱε,ζ,η,ℒΦ˜ε,ζ,η=gε,ζω+ω+ν1−ωωℒDε2Φ˜ε,ζ,η+Dζ2Φ˜ε,ζ,η+k˜rℱε,ζ,η.We write the solution ofΦ˜ε,ζ,η=∑n=0∞Φ˜nε,ζ,η; then, (7) is expressed as,(10)ℒ∑n=0∞Φ˜nε,ζ,η=gε,ζω+ω+ν1−ωωℒDε2∑n=0∞Φ˜nε,ζ,η+Dζ2∑n=0∞Φ˜nε,ζ,η+k˜rℱε,ζ,η.We can write the following results:(11)ℒΦ˜0ε,ζ,η=gε,ζω+ω+ν1−ωωℒk˜rℱε,ζ,η,ℒΦ˜1ε,ζ,η=ω+ν1−ωωℒDε2Φ˜0ε,ζ,η+Dζ2Φ˜0ε,ζ,η,ℒΦ˜2ε,ζ,η=ω+ν1−ωωℒDε2Φ˜1ε,ζ,η+Dζ2Φ˜1ε,ζ,η,⋮ℒΦ˜n+1ε,ζ,η=ω+ν1−ωωℒDε2Φ˜nε,ζ,η+Dζ2Φ˜nε,ζ,η.Taking the inverse Laplace transform, we obtain(12)Φ¯0ε,ζ,η=gε,ζ+ℒ−1ω+ν1−ωωℒk¯rℱε,ζ,η,Φ¯0ε,ζ,η=gε,ζ+ℒ−1ω+ν1−ωωℒk¯rℱε,ζ,η,Φ¯1ε,ζ,η=ℒ−1ω+ν1−ωωℒDε2Φ¯0ε,ζ,η+Dζ2Φ¯0ε,ζ,η,Φ¯1ε,ζ,η=ℒ−1ω+ν1−ωωℒDε2Φ¯0ε,ζ,η+Dζ2Φ¯0ε,ζ,η.(13)Φ¯2ε,ζ,η=ℒ−1ω+ν1−ωωℒDε2Φ¯1ε,ζ,η+Dζ2Φ¯1ε,ζ,η,Φ¯2ε,ζ,η=ℒ−1ω+ν1−ωωℒDε2Φ¯1ε,ζ,η+Dζ2Φ¯1ε,ζ,η,⋮Φ¯n+1ε,ζ,η=ℒ−1ω+ν1−ωωℒDε2Φ¯nε,ζ,η+Dζ2Φ¯nε,ζ,η,Φ¯n+1ε,ζ,η=ℒ−1ω+ν1−ωωℒDε2Φ¯nε,ζ,η+Dζ2Φ¯nε,ζ,η.Thus, the solution becomes(14)Φ¯ε,ζ,η=Φ¯0ε,ζ,η+Φ¯1ε,ζ,η+Φ¯2ε,ζ,η+⋯,Φ¯ε,ζ,η=Φ¯0ε,ζ,η+Φ¯1ε,ζ,η+Φ¯2ε,ζ,η+⋯.This equation is the series form result.


## 4. Examples

Example 1.
We consider the fuzzy fractional nonlinear dispersive equation as(15)CFDηνΦ˜ε,η+Φ˜2ε+Φ˜2εεε=0,0<ν≤1,0<ε<1,with the initial condition(16)Φ˜ε,0=k˜4α3cos2ε4.
Using the scheme of (12), we obtain(17)Φ¯0ε,η=k¯r4α3cos2ε4,Φ¯0ε,η=k¯r4α3cos2ε4,Φ¯1ε,η=k¯rα3sinε21+νη−ν,Φ¯1ε,η=k¯rα3sinε21+νη−ν,Φ¯2ε,η=−k¯rα312cosε21−ν2νη+1−ν2+ν2η22,Φ¯2ε,η=−k¯rα312cosε21−ν2νη+1−ν2+ν2η22,Φ¯3ε,η=−k¯rα472sinε21−ν23νη+1−ν3+3ν21−νη22+ν3η33!,Φ¯3ε,η=−k¯rα472sinε21−ν23νη+1−ν3+3ν21−νη22+ν3η33!.
The series form result is achieved by applying equation (10); we get(18)Φ˜ε,η=Φ˜0ε,η+Φ˜1ε,η+Φ˜2ε,η+Φ˜3ε,η+Φ˜4ε,η+⋯.
The upper and lower forms of solution are given as(19)Φ¯ε,η=Φ¯0ε,η+Φ¯1ε,η+Φ¯2ε,η+Φ¯3ε,η+Φ¯4ε,η+⋯,Φ¯ε,η=Φ¯0ε,η+Φ¯1ε,η+Φ¯2ε,η+Φ¯3ε,η+Φ¯4ε,η+⋯.Φ¯ε,η=k¯r4α3cos2ε4+k¯rα3sinε21+νη−ν−k¯rα312cosε21−ν2νη+1−ν2+ν2η22−k¯rα472sinε21−ν23νη+1−ν3+3ν21−νη22+ν3η33!+⋯,Φ¯ε,η=k¯r4α3cos2ε4+k¯rα3sinε21+νη−ν−k¯rα312cosε21−ν2νη+1−ν2+ν2η22−k¯rα472sinε21−ν23νη+1−ν3+3ν21−νη22+ν3η33!+⋯.
The exact result is(20)Φ˜ε,η=4α3cos2ε−αη4,ε−αη|≤2π0,otherwise.
Figure1 shows that this method’s accuracy by lower and upper branches of fuzzy solution of example 1 link with the fuzzy Laplace transformation and Caputo-Fabrizio operator is shown in this article. Figure 2 shows the three-dimensional upper and lower branch graphs. The behaviour defines the variance in the mappings Φ˜ε,η on the space coordinate ε with the consideration of η and the unpredictability parameters rε0,1. The graph shows that, with time, the mappings Φ˜ε,η will become much more intricate.Figure 1
Upper and lower fuzzy figure of analytical solution of example1.Figure 2
Various factional order of the upper and lower fuzzy figure of analytical solution of example1.Example 2.
We consider the fuzzy fractional nonlinear dispersive equation as(21)CFDηνΦ˜ε,η+Φ˜ε,ηΦ˜εε,η+Φ˜εεεε,η=0,0<ν≤1,0<ε<1,with the initial condition(22)Φ˜ε,0=−2k˜w2ewε1+ewε2.
Using the scheme of (21), we obtain(23)Φ¯0ε,η=−2k¯rw2ewε1+ewε2,Φ¯0ε,η=−2k¯rw2ewε1+ewε2,Φ¯1ε,η=−2k¯rw5ewεewεε−11+ewε31+νη−ν,Φ¯1ε,η=−2k¯rw5ewεewεε−11+ewε31+νη−ν,Φ¯2ε,η=−2k¯rw5ewεe2wε−4ewε+11+ewε41−ν2νη+1−ν2+ν2η22,Φ¯2ε,η=−2k¯rw5ewεe2wε−4ewε+11+ewε41−ν2νη+1−ν2+ν2η22,Φ¯3ε,η=−k¯rw11ewεe3wε−11e2wε+11ewε−131+ewε51−ν23νη+1−ν3+3ν21−νη22+ν3η33!,Φ¯3ε,η=−k¯rw11ewεe3wε−11e2wε+11ewε−131+ewε51−ν23νη+1−ν3+3ν21−νη22+ν3η33!,
The series form result is achieves applying equation (21); we get(24)Φ˜ε,η=Φ˜0ε,η+Φ˜1ε,η+Φ˜2ε,η+Φ˜3ε,η+Φ˜4ε,η+⋯.
The upper and lower forms of solution are given as(25)Φ¯ε,η=Φ¯0ε,η+Φ¯1ε,η+Φ¯2ε,η+Φ¯3ε,η+Φ¯4ε,η+⋯,Φ¯ε,η=Φ¯0ε,η+Φ¯1ε,η+Φ¯2ε,η+Φ¯3ε,η+Φ¯4ε,η+⋯.Φ¯ε,η=−2k¯rw2ewε1+ewε2−2k¯rw5ewεewεε−11+ewε31+νη−ν−2k¯rw5ewεe2wε−4ewε+11+ewε41−ν2νη+1−ν2+ν2η22−k¯rw11ewεe3wε−11e2wε+11ewε−131+ewε51−ν23νη+1−ν3+3ν21−νη22+ν3η33!+⋯,Φ¯ε,η=−2k¯rw2ewε1+ewε2−2k¯rw5ewεewεε−11+ewε31+νη−ν−2k¯rw5ewεe2wε−4ewε+11+ewε41−ν2νη+1−ν2+ν2η22−k¯rw11ewεe3wε−11e2wε+11ewε−131+ewε51−ν23νη+1−ν3+3ν21−νη22+ν3η33!+⋯.
The exact result is(26)Φ˜ε,η=−2w2ewε−w2η1+ewε−w2η2.
Figure3 shows that this method’s accuracy by lower and upper branches of fuzzy solution of example 2 link with the fuzzy Laplace transformation and Caputo-Fabrizio operator is shown in this article. Figure 4 shows the three-dimensional upper and lower branch graphs. The behaviour defines the variance in the mappings Φ˜ε,η on the space coordinate ε with the consider of η and the unpredictability parameters rε0,1. The graph shows that, with time, the mappings Φ˜ε,η will become much more intricate.Figure 3
Upper and lower fuzzy figure of analytical solution of example2.Figure 4
Various factional orders of the upper and lower fuzzy figure of analytical solution of example2.Example 3.
We consider the fuzzy fractional nonlinear dispersive equation as(27)CFDηνΦ˜ε,η−6Φ˜ε,ηΦ˜εε,η+Φ˜εεεε,η=0,0<ν≤1,0<ε<1,with the initial condition(28)Φ˜ε,0=−24k˜e2ε1+e2ε2.
Using the scheme of (27), we obtain(29)Φ¯0ε,η=−24k¯re2ε1+e2ε2,Φ¯0ε,η=−24k¯re2ε1+e2ε2,Φ¯1ε,η=192k¯re2ε1+25e2ε−25e4ε−e6ε1+e2ε51+νη−ν,Φ¯1ε,η=192k¯re2ε1+25e2ε−25e4ε−e6ε1+e2ε51+νη−ν,Φ¯2ε,η=−768k¯re2ε1+240e2ε+183e4ε−2128e6ε+183e2ε+240e10ε+e12ε1+e2ε51−ν2νη+1−ν2+ν2η22,Φ¯2ε,η=−768k¯re2ε1+240e2ε+183e4ε−2128e6ε+183e2ε+240e10ε+e12ε1+e2ε51−ν2νη+1−ν2+ν2η22.
The series form result is achieved by applying equation (28); we get(30)Φ˜ε,η=Φ˜0ε,η+Φ˜1ε,η+Φ˜2ε,η+Φ˜3ε,η+Φ˜4ε,η+⋯.
The upper and lower forms of solution are given as(31)Φ¯ε,η=Φ¯0ε,η+Φ¯1ε,η+Φ¯2ε,η+Φ¯3ε,η+Φ¯4ε,η+⋯,Φ¯ε,η=Φ¯0ε,η+Φ¯1ε,η+Φ¯2ε,η+Φ¯3ε,η+Φ¯4ε,η+⋯,Φ¯ε,η=−24k¯re2ε1+e2ε2+192k¯re2ε1+25e2ε−25e4ε−e6ε1+e2ε51+νη−ν−768k¯re2ε1+240e2ε+183e4ε−2128e6ε+183e2ε+240e10ε+e12ε1+e2ε51−ν2νη+1−ν2+ν2η22+⋯,Φ¯ε,η=−24k¯re2ε1+e2ε2+192k¯re2ε1+25e2ε−25e4ε−e6ε1+e2ε51+νη−ν−768k¯re2ε1+240e2ε+183e4ε−2128e6ε+183e2ε+240e10ε+e12ε1+e2ε51−ν2νη+1−ν2+ν2η22+⋯.
The exact result is(32)Φ˜ε,η=12∂∂εe8η−2ε+2e64η−4ε+e72η−6ε1+3e8η−2ε+3e64η−4ε+e72η−6ε.Figure 5 shows that this method’s accuracy by lower and upper branches of fuzzy solution of example 3 link with the fuzzy Laplace transformation and Caputo-Fabrizio operator are shown in this article. Figure 6 shows the three-dimensional upper and lower branch graphs. The behaviour defines the variance in the mappings Φ˜ε,η on the space coordinate ε with the consider of η and the unpredictability parameters rε0,1. The graph shows that, with time, the mappings Φ˜ε,η will become much more intricate.Figure 5
Upper and lower fuzzy figure of analytical solution of example1.Figure 6
Various factional orders of the upper and lower fuzzy figure of analytical solution of example1.


## 5. Conclusion

This investigation aimed to provide a semianalytical result to the fuzzy fractional solitary wave solutions for the KdV equation by considering the Caputo-Fabrizio fractional derivatives. In contrast, the crisp operators are unable to simulate any physical mechanism in an unpredictable setting. Therefore, fuzzy operators are a preferable means to describe the physical phenomenon in such a scenario. We explored the the KdV equation in a fuzzy approach, taking into account the uncertainty in the initial condition. We have generalised the fractional KdV equation to the fuzzy fractional KdV model in this research. To obtain the approximate expression of the suggested problem in its parametric form, we then used a new iterative transform method. We identified 26 numerous illustrations to support the intended methodology and achieved a parametric solution for each case. In the end, it is not simple to find analytical solutions for many types of fuzzy fractional partial differential equations. In the future, it is important to look at and solve fractional fuzzy partial differential, dynamical, and integral differential equations based on the Caputo-Fabrizio operator of the fractional orderν>1.



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