# A Comparative Study of the Fractional-Order Nonlinear System of Physical Models via Analytical Methods

Authors: Humaira Yasmin; Naveed Iqbal
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

This article is related to the fractional-order analysis of one- and two-dimensional nonlinear systems of third-order KdV equations and coupled Burgers equations, applying modified analytical methods. The proposed problems will be solved with the Caputo–Fabrizio fractional derivative operator and the Yang transform. The results we obtained by implementing the suggested methods are compared with the exact solution. The convergence of the method is successfully presented and mathematically proved. To show the effectiveness of the proposed methods, we compared exact and analytical results with the help of graphs and tables, which are in strong agreement with each other. Also, the results obtained by implementing the suggested methods at various fractional orders are compared, confirming that solution gets closer to exact solution as the value tends from fractional order towards integer order. Moreover, the proposed methods are attractive, easy, and highly accurate, which confirms that these methods are suitable methods for solving partial differential equations or systems of partial differential equations.


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## Body

## 1. Introduction

Fractional calculus has overtaken ordinary calculus as a popular subject in recent years. Ordinary calculus has reached its peak of achievement in terms of discovery. As a result, fractional calculus is needed by mathematicians and researchers. This allows us to more properly describe real-world phenomena than by classic “integer” order. Many mathematicians, including Fourier, Laplace, Riesz, and others, were active and made significant contributions to the topic. The Atangana–Baleanu fractional integral [1], the Caputo fractional derivative [2], and the Caputo–Fabrizio fractional derivative [3] are modern examples of modern definitions of fractional-order derivatives and integrals that have brought in a new era in the history of fractional derivatives. Models based on fractional calculus can accurately represent many processes in engineering, physics, chemistry, and other fields [4]. Furthermore, fractional calculus is used to simulate the frequency-dependent damping behavior of several viscoelastic materials [5], economics [6], and dynamics of interfaces between nanoparticles and substrates [7].Many physical and engineering phenomena are mathematically expressed using fractional differential equations. They have obtained a lot of attention in the natural and social sciences because they can accurately model phenomena dominated by memory effects. Because FDEs represent values on each point and differentiate the gaps between the two integers, they are a generalization of ordinary differential equations (ODEs). This is why, with the discovery of fractional calculus, it was discovered that FDEs have more real-world applications than ODEs [8, 9]. In many mathematical and scientific fields, FDEs in fractional calculus are one of the most popular subjects, such as biophysics, blood flow phenomena, aerodynamics, viscoelasticity, electrical circuits, electro-analytical chemistry, biology, control theory, finance, hydrology, and control systems [10–15].In all of these fields of study, finding the actual or approximate solutions of FDEs is essential, but as we do not have a technique for finding the exact solution of these types of FDEs, we must focus on approximation to exact solution [16–19]. In mathematics, determining the exact solution of such FDEs and other scientific applications is a challenging task. In comparison to the approximate solution [20], the exact solution helps us understand the mechanism and sophistication of the problem. Dealing with the difficulties of computations in these equations makes obtaining exact analytic solutions of FDEs extremely difficult, if not impossible. As a result, it is necessary to seek out some useful approximations and numerical techniques, such as the homotopy perturbation method [21], variation iteration method [22], residual power series method [23], approximate-analytical method [24], Elzaki transform decomposition method [25], iterative laplace transform method [26], Adomian decomposition method [27], reduced differential transform method [28], and others.In this paper, we proposed two analytical techniques with the aid of Laplace transform and the Atangana–Baleanu fractional derivative operator for solving fractional-order systems. The first technique is the mixture of Laplace transform (LT) and the variational iteration method known as the variational iteration transform method (VITM), which was first introduced by He [24] and is an effective approach for a wide range of problems in applied sciences [29–31]. The second important technique is the combination of Laplace transform and the Adomian decomposition method, first introduced by George Adomian (1923–1996) in the 1980s for solving nonlinear functional equations. The method is based on the decomposition of nonlinear functional equation solution into a series of functions. A polynomial created by a power series expansion of analytic function yields each series term. This method for solving various nonlinear fractional-order differential equations is exciting, simple, and accurate.Harry Bateman initially introduced Burgers equation in 1915 [32], and it was later termed the Burgers equation. The Burgers equation has a wide range of applications in science and engineering, particularly for situations involving nonlinear equations. The importance and interest of Burgers equation applications by mathematical scientists and researchers have grown. This equation is recognized to represent a variety of phenomena, including dynamics modeling, heat conduction, acoustic waves, turbulence, and many others [33–35]. Korteweg and Vries derived the Korteweg-De Vries (KDV) equation for the first time in 1895. Long waves, tides, solitary waves, and wave propagation in a shallow canal are all modeled using KDV equation. Fluid mechanics, signal processing, hydrology, viscoelasticity, and fractional kinetics are just a few fields where KDV equation is used.


## 2. Preliminary Concepts

We provide fundamental definitions that will be used throughout the article. For the purpose of simplification, we write the exponential decay kernel as,Kτ,ϱ=e−℘τ−ϱ/1−℘.Definition 1.
The Caputo–Fabrizio derivative is given as follows [36]:(1)DτδCFℙτ=Nδ1−δ∫0τℙ′ϱKτ,ϱdϱ,n−1<δ≤n.
Nδ is the normalization function with N0=N1=1.(2)DτδCFℙτ=Nδ1−δ∫0τℙτ−ℙϱKτ,ϱdϱ.Definition 2.
The fractional integral Caputo–Fabrizio is given as [36](3)DτδCFℙτ=1−δNδℙτ+δNδ∫0τℙϱdϱ,τ≥0,δ∈0,1.Definition 3.
ForNδ=1, the following result shows the Caputo–Fabrizio derivative of Laplace transformation [36]:(4)LDτδCFℙτ=sLℙτ−ℙ0s+δ1−s.Definition 4.
The Yang transformation ofℙτ is expressed as [37](5)Yℙτ=χs=∫0∞ℙτe−τ/sdτ.τ>0.Remark 1.
Yang transformation of few useful functions is defined as follows:(6)Y1=s,Yτ=s2,Yτi=Γi+1si+1.Lemma 1 (Laplace–Yang duality).
Let the Laplace transformation ofℙτ be Fs; then, χs=F1/s [38].Proof.
From equation (6), we can achieve another type of Yang transformation by putting τ/s=ζ as(7)Lℙτ=χs=s∫0∞ℙsζeζdζ.ζ>0.
SinceLℙτ=Fs, this implies that(8)Fs=Lℙτ=∫0∞ℙτe−sτdτ.
Puttingτ=ζ/s in (17), we have(9)Fs=1s∫0∞ℙζseζdζ.
Thus, from equation (17), we achieve(10)Fs=χ1s.
Also, from equations (6) and (17), we achieve(11)F1s=χs.
Connections (17) and (17) represent the duality link between the Laplace and Yang transformation.Lemma 2.
Letℙτ be continuous function, then the Caputo–Fabrizio derivative Yang transformation of ℙτ is defined by [38](12)Yℙτ=Yℙτ−sℙ01+δs−1.Proof.
Caputo–Fabrizio fractional Laplace transformation is given by(13)Lℙτ=Lsℙτ−ℙ0s+δ1−s.
Also, we have that connection among Laplace and Yang property, i.e.,χs=F1/s. To achieve the necessary result, we substitute s by 1/s in equation (17), and we get(14)Yℙτ=1/sYℙτ−ℙ01/s+δ1−1/s,Yℙτ=Yℙτ−sℙ01+δs−1.
The proof is completed.


## 3. Idea of YDM

Here, we discuss the methodology of YDM for solving fractional-order partial differential equations.(15)Dℑ℘CFμξ,ℑ+G¯1ξ,ℑ+N1ξ,ℑ=ℱξ,ℑ,0<℘≤1,having initial terms(16)μξ,0=ξξ,∂∂ℑμξ,0=ζξ.Here, the fractional-order CF operator is indicated fromDℑ℘=∂℘/∂ℑ℘ having orders ℘, G¯1,andN1 are linear and nonlinear operators and ℱξ,ℑ represents the source term.On taking the Yang transform of (15), we get(17)YDℑ℘μξ,ℑ+G¯1ξ,ℑ+N1ξ,ℑ=Yℱξ,ℑ.By the property of YT differentiation, we obtain(18)Yμξ,ℑ=Θξ,ω−1+℘ω−1YG¯1ξ,ℑ+N1ξ,ℑ,where Θξ,ω=1/ω℘−1ω℘g0ξ+ω℘−1g1ξ+⋯+g1ξ+1+℘ω−1ℱξ,ℑ.Now, using the inverse Yang transform, we get(19)μξ,ℑ=Θξ,ω−Y−11+℘ω−1YG¯1ξ,ℑ+N1ξ,ℑ,where Θξ,ω shows the term that comes from the source factor. YDM generates the result of the infinite series of μξ,ℑ:(20)μξ,ℑ=∑m=0∞μmξ,ℑ,and we decompose the nonlinear operator N1 as(21)N1ξ,ℑ=∑m=0∞Am,where Am are Adomian polynomials given as(22)Am=1m!∂m∂ℓmN1∑k=0∞ℓkξk,∑k=0∞ℓkℑkℓ=0.Putting equations (20) and (22) into (19) gives(23)∑m=0∞μmξ,ℑ=Θξ,ω−Y−11+℘ω−1YG¯1∑m=0∞ξm,∑m=0∞ℑm+∑m=0∞Am.The terms listed are defined as follows:(24)μ0ξ,ℑ=Θξ,ω,μ1ξ,ℑ=Y−11+℘ω−1YG¯1ξ0,ℑ0+A0.As a result, all of the components form≥1 are calculated as(25)μm+1ξ,ℑ=Y−11+℘ω−1YG¯1ξm,ℑm+Am.


## 4. VITM Formulation

Here, we discuss the methodology of VITM for solving fractional-order partial differential equations.(26)Dℑ℘CFμξ,ℑ+ℳμξ,ℑ+Nμξ,ℑ−Pξ,ℑ=0,m−1<℘≤m,having initial term(27)μξ,0=g1ξ.Here, the fractional-order CF operator is indicated fromDℑ℘=∂℘/∂ℑ℘. ℳ and N are the linear and nonlinear operator, and P represents the source term.On taking the Yang transform of (26), we get(28)YDℑ℘CFμξ,ℑ+Yℳμξ,ℑ+Nμξ,ℑ−Pξ,ℑ=0.By the property of YT differentiation, we obtain(29)Yμξ,ℑ=11+℘ω−1Yℳμξ,ℑ+Nμξ,ℑ−Pξ,ℑ.The method of iteration for equation (29).(30)μm+1ξ,ℑ=μmξ,ℑ+℘s11+℘ω−1Yℳμξ,ℑ+Nμξ,ℑ−Pξ,ℑ.℘s is the Lagrange multiplier, and(31)℘s=−1+℘ω−1.Equation (30) series form solution is obtained by using the inverse Yang transform.(32)μ0ξ,ℑ=μ0+Y−1℘sY−Pξ,ℑ,μ1ξ,ℑ=Y−1℘sYℳμξ,ℑ+Nμξ,ℑ,⋮μn+1ξ,ℑ=Y−1℘sYℳμ0ξ,ℑ+μ1ξ,ℑ+⋯,μnξ,ℑ+Nμ0ξ,ℑ+μ1ξ,ℑ,…,μnξ,ℑ.


## 5. Applications

To show the validity and capability of suggested techniques, we implemented proposed methods for solving four nonlinear systems.Problem 1.
Consider the system of homogeneous KdV equation having order three.(33)∂℘CFμ∂ℑ℘=∂3μ∂ξ3+μ∂μ∂ξ+ν∂ν∂ξ,∂℘CFν∂ℑ℘=−2∂3ν∂ξ3+μ∂ν∂ξ,0<℘≤1,with an initial source(34)μξ,0=3−6tanh2ξ/2νξ,0=−3l2tanh2ξ/2.
On taking the Yang transform of (34), we get(35)11+℘ω−1Yμξ,ℑ−ωμξ,0=Y∂3μ∂ξ3+μ∂μ∂ξ+ν∂ν∂ξ,11+℘ω−1Yνξ,ℑ−ωνξ,0=Y−2∂3ν∂ξ3+μ∂ν∂ξ.
When we use the Yang inverse transform, we obtain(36)μξ,ℑ=3−6tanh2ξ/2+Y−11+℘ω−1Y∂3μ∂ξ3+μ∂μ∂ξ+ν∂ν∂ξ,νξ,ℑ=−3l2tanh2ξ/2+Y−11+℘ω−1Y−2∂3ν∂ξ3+μ∂ν∂ξ.
Assume that the solutionsμξ,ℑ and νξ,ℑ in the form of infinite series are given by(37)μξ,ℑ=∑m=0∞μmξ,ℑ,νξ,ℑ=∑m=0∞νmξ,ℑ.Here, μμξ=∑m=0∞Am, ννξ=∑m=0∞ℬm and μνξ=∑m=0∞Cm are the so-called Adomian polynomials that represent nonlinear terms, and so equation (36) is rewritten as follows:(38)∑m=0∞μmξ,ℑ=3−6tanh2ξ/2+Y−11+℘ω−1Y∂3μ∂ξ3+∑m=0∞Am+∑m=0∞ℬm,∑m=0∞νmξ,ℑ=−3l2tanh2ξ/2+Y−11+℘ω−1Y−2∂3ν∂ξ3+∑m=0∞Cm.
The decomposition of nonlinear terms by Adomian polynomials is defined as in equation (22).(39)A0=μ0μ0ξ,A1=μ1μ0ξ+μ0μ1ξ,A2=μ2μ0ξ+μ1μ1ξ+μ0μ2ξ,ℬ0=ν0ν0ξ,ℬ1=ν1ν0ξ+ν0ν1ξ,ℬ2=ν2ν0ξ+ν1ν1ξ+ν0ν2ξ,C0=μ0ν0ξ,C1=μ1ν0ξ+μ0ν1ξ,C2=μ2ν0ξ+μ1ν1ξ+μ0ν2ξ.
As a result, when comparing the two sides of equation (38), we get(40)μ0ξ,ℑ=3−6tanh2ξ/2,ν0ξ,ℑ=−3l2tanh2ξ/2.
Form=0,(41)μ1ξ,ℑ=−6sech2ξ2tanhξ21+℘ℑ−℘,ν1ξ,ℑ=3l2sech2ξ2tanhξ21+℘ℑ−℘.
Form=1,(42)μ2ξ,ℑ=322sech2ξ/2+7sech4ξ/2−15sech6ξ/21−℘2℘ℑ+1−℘2+℘2ℑ22,ν2ξ,ℑ=3l242sech2ξ/2+21sech4ξ/2−24sech6ξ/21−℘2℘ℑ+1−℘2+℘2ℑ22.
The approximate solution to the series is written as follows:(43)μξ,ℑ=∑m=0∞μmξ,ℑ=μ0ξ,ℑ+μ1ξ,ℑ+μ2ξ,ℑ+⋯,νξ,ℑ=∑m=0∞νmξ,ℑ=ν0ξ,ℑ+ν1ξ,ℑ+ν2ξ,ℑ+⋯,μξ,ℑ=3−6tanh2ξ/2−6sech2ξ/2tanhξ/21+℘ℑ−℘+322sech2ξ/2+7sech4ξ/2−15sech6ξ/21−℘2℘ℑ+1−℘2+℘2ℑ22+⋯,νξ,ℑ=−3l2tanh2ξ/2+3l2sech2ξ2tanhξ21+℘ℑ−℘+3l242sech2ξ/2+21sech4ξ/2−24sech6ξ/21−℘2℘ℑ+1−℘2+℘2ℑ22+⋯.
We achieve the exact solution by putting℘=1.(44)μξ,ℑ=3−6tanh2ℑ+ξ/2,νξ,ℑ=−3l2tanh2ℑ+ξ/2.
VITM analytical results are as follows.
For equation (33), we have the iteration formula as(45)μm+1ξ,ℑ=μmξ,ℑ−Y−11+℘ω−1Y∂3μm∂ξ3+μm∂μm∂ξ+νm∂νm∂ξ,νm+1ξ,ℑ=νmξ,ℑ−Y−11+℘ω−1Y−2∂3νm∂ξ3μm∂νm∂ξ,where(46)μ0ξ,ℑ=3−6tanh2ξ/2,ν0ξ,ℑ=−3l2tanh2ξ/2.
Form=0,1,2,⋯,(47)μ1ξ,ℑ=μ0ξ,ℑ−Y−11+℘ω−1Y∂3μ0∂ξ3+μ0∂μ0∂ξ+ν0∂ν0∂ξ,μ1ξ,ℑ=−6sech2ξ2tanhξ21+℘ℑ−℘,ν1ξ,ℑ=ν0ξ,ℑ−Y−11+℘ω−1Y−2∂3ν0∂ξ3μ0∂ν0∂ξ,ν1ξ,ℑ=3l2sech2ξ2tanhξ21+℘ℑ−℘,μ2ξ,ℑ=μ1ξ,ℑ−Y−11+℘ω−1Y∂3μ1∂ξ3+μ1∂μ1∂ξ+ν1∂ν1∂ξ,μ2ξ,ℑ=322sech2ξ/2+7sech4ξ/2−15sech6ξ/21−℘2℘ℑ+1−℘2+℘2ℑ22,ν2ξ,ℑ=ν1ξ,ℑ−Y−11+℘ω−1Y−2∂3ν1∂ξ3μ1∂ν1∂ξ,ν1ξ,ℑ=3l242sech2ξ/2+21sech4ξ/2−24sech6ξ/21−℘2℘ℑ+1−℘2+℘2ℑ22,μξ,ℑ=∑m=0∞μmξ,ℑ=3−6tanh2ξ/2−6sech2ξ2tanhξ21+℘ℑ−℘+322sech2ξ/2+7sech4ξ/2−15sech6ξ/21−℘2℘ℑ+1−℘2+℘2ℑ22+⋯,νξ,ℑ=∑m=0∞νmξ,ℑ=−3l2tanh2ξ/2+3l2sech2ξ2tanhξ21+℘ℑ−℘+3l242sech2ξ/2+21sech4ξ/2−24sech6ξ/21−℘2℘ℑ+1−℘2+℘2ℑ22+⋯.
We achieve the exact solution by putting℘=1.(48)μξ,ℑ=3−6tanh2ℑ+ξ/2,νξ,ℑ=−3l2tanh2ℑ+ξ/2.
Analytical solution and exact solution are shown in Figures1(a) and 1(b) at ℘=1 and −5≤ξ≥5. Figure 1(c) shows the absolute error and 1d gives the solution in various fractional-order graphs for μξ,ℑ. The behavior of exact solution and analytical solution for νξ,ℑ are seen in Figures 2(a) and 2(b). From the figures, it is clear that the presented solution methodology is in good agreement with exact solution.Figure 1
Exact solution, analytical solution, absolute error, and various fractional-order solutions forμξ,ℑ of Problem 1.
(a)(b)(c)(d)Figure 2
Exact solution and analytical solution forνξ,ℑ Problem 1.
(a)(b)Problem 2.
Consider the generalized coupled Hirota–Satsuma KdV system.(49)∂℘CFμ∂ℑ℘=12∂3μ∂ξ3−3μ∂μ∂ξ+3∂∂ξνℓ,∂℘CFν∂ℑ℘=3μ∂ν∂ξ−∂3ν∂ξ3,∂℘CFℓ∂ℑ℘=3μ∂ℓ∂ξ−∂3ℓ∂ξ3,0<℘≤1,with an initial source(50)μξ,0=−13+2tanh3ξ,νξ,0=tanhξ,ℓξ,0=83tanhξ.
On taking the Yang transform of (49), we get(51)11+℘ω−1Yμξ,ℑ−ωμξ,0=Y12∂3μ∂ξ3−3μ∂μ∂ξ+3∂∂ξνℓ,11+℘ω−1Yνξ,ℑ−ωνξ,0=Y3μ∂ν∂ξ−∂3ν∂ξ3,11+℘ω−1Yℓξ,ℑ−ωℓξ,0=Y3μ∂ℓ∂ξ−∂3ℓ∂ξ3.
When we use the Yang transform, we obtain(52)μξ,ℑ=−13+2tanh3ξ+Y−11+℘ω−1Y12∂3μ∂ξ3−3μ∂μ∂ξ+3∂∂ξνℓ,νξ,ℑ=tanhξ+Y−11+℘ω−1Y3μ∂ν∂ξ−∂3ν∂ξ3,ℓξ,ℑ=83tanhξ+Y−11+℘ω−1Y3μ∂ℓ∂ξ−∂3ℓ∂ξ3.
Assume that the solutionsμξ,ℑ, νξ,ℑ, and ℓξ,ℑ in the form of infinite series are given by(53)μξ,ℑ=∑m=0∞μmξ,ℑ,νξ,ℑ=∑m=0∞νmξ,ℑ,ℓξ,ℑ=∑m=0∞ℓmξ,ℑ,where μμξ=∑m=0∞Am, νℓξ=∑m=0∞ℬm, μνξ=∑m=0∞Cm, and μℓξ=∑m=0∞Dm are the so-called Adomian polynomials that represent the nonlinear terms, and so equation (52) is rewritten as follows:(54)∑m=0∞μmξ,ℑ=−13+2tanh3ξ+Y−11+℘ω−1Y∂3μ∂ξ3−3∑m=0∞Am+3∑m=0∞ℬm,∑m=0∞νmξ,ℑ=tanhξ+Y−11+℘ω−1Y3∑m=0∞Cm−∂3ν∂ξ3,∑m=0∞ℓmξ,ℑ=83tanhξ+Y−11+℘ω−1Y3∑m=0∞Dm−∂3ℓ∂ξ3.
The decomposition of nonlinear terms by Adomian polynomials is defined as in equation (22),(55)A0=μ0μ0ξ,A1=μ1μ0ξ+μ0μ1ξ,A2=μ2μ0ξ+μ1μ1ξ+μ0μ2ξ,ℬ0=ν0ℓ0ξ+ℓ0ν0ξ,ℬ1=ν0ℓ0ξ+ν1ℓ0ξ+ℓ1ν0ξ+ℓ0ν1ξ,ℬ2=ν2ℓ0ξ+ν1ℓ1ξ+ν0ℓ2ξ+ℓ2ν0ξ+ℓ1ν1ξ+ℓ0ν2ξ,C0=μ0ν0ξ,C1=μ1ν0ξ+μ0ν1ξ,C2=μ2ν0ξ+μ1ν1ξ+μ0ν2ξ,D0=μ0ℓ0ξ,D1=μ1ℓ0ξ+μ0ℓ1ξ,D2=μ2ℓ0ξ+μ1ℓ1ξ+μ0ℓ2ξ.
As a result, when comparing the two sides of equation (54), we get(56)μ0ξ,ℑ=−13+2tanh2ξ,ν0ξ,ℑ=tanhξ,ℓ0ξ,ℑ=83tanhξ.
Form=0,(57)μ1ξ,ℑ=4sech2ξtanhξ1+℘ℑ−℘,ν1ξ,ℑ=sech2ξ1+℘ℑ−℘,ℓ1ξ,ℑ=83sech2ξ1+℘ℑ−℘.
Form=1,(58)μ2ξ,ℑ=4sech2ξ1−3tanh2ξ1−℘2℘ℑ+1−℘2+℘2ℑ22,ν2ξ,ℑ=−sech2ξtanhξ1−℘2℘ℑ+1−℘2+℘2ℑ22,ℓ2ξ,ℑ=−83sech2ξtanhξ1−℘2℘ℑ+1−℘2+℘2ℑ22.
The approximate solution to the series is written as follows:(59)μξ,ℑ=∑m=0∞μmξ,ℑ=μ0ξ,ℑ+μ1ξ,ℑ+μ2ξ,ℑ+⋯,νξ,ℑ=∑m=0∞νmξ,ℑ=ν0ξ,ℑ+ν1ξ,ℑ+ν2ξ,ℑ+⋯,ℓξ,ℑ=∑m=0∞ℓmξ,ℑ=ℓ0ξ,ℑ+ℓ1ξ,ℑ+ℓ2ξ,ℑ+⋯,μξ,ℑ=−13+2tanh2ξ+4sech2ξtanhξ1+℘ℑ−℘+4sech2ξ1−3tanh2ξ1−℘2℘ℑ+1−℘2+℘2ℑ22+⋯,νξ,ℑ=tanhξ+sech2ξ1+℘ℑ−℘−sech2ξtanhξ1−℘2℘ℑ+1−℘2+℘2ℑ22+⋯,ℓξ,ℑ=83tanhξ+83sech2ξ1+℘ℑ−℘−83sech2ξtanhξ1−℘2℘ℑ+1−℘2+℘2ℑ22+⋯.
We achieve the exact solution by putting℘=1.(60)μξ,ℑ=−13+2tanh2ℑ+ξ,νξ,ℑ=tanhℑ+ξ,ℓξ,ℑ=83tanhℑ+ξ.
VITM analytical results are as follows.
For equation (49), we have the iteration formula as(61)μm+1ξ,ℑ=μmξ,ℑ−Y−11+℘ω−1Y12∂3μm∂ξ3−3μm∂μm∂ξ+3∂∂ξνmℓm,νm+1ξ,ℑ=νmξ,ℑ−Y−11+℘ω−1Y3μm∂νm∂ξ−∂3νm∂ξ3,ℓm+1ξ,ℑ=ℓmξ,ℑ−Y−11+℘ω−1Y3μm∂ℓm∂ξ−∂3ℓm∂ξ3,where(62)μ0ξ,ℑ=−13+2tanh2ξ,ν0ξ,ℑ=tanhξ,ℓ0ξ,ℑ=83tanhξ.
Form=0,1,2,⋯,(63)μ1ξ,ℑ==μ0ξ,ℑ−Y−11+℘ω−1Y12∂3μ0∂ξ3−3μ0∂μ0∂ξ+3∂∂ξν0ℓ0,μ1ξ,ℑ=4sech2ξtanhξ1+℘ℑ−℘,ν1ξ,ℑ=ν0ξ,ℑ−Y−11+℘ω−1Y3μm∂ν0∂ξ−∂3ν0∂ξ3,ν1ξ,ℑ=sech2ξ1+℘ℑ−℘,ℓ1ξ,ℑ=ℓ0ξ,ℑ−Y−11+℘ω−1Y3μ0∂ν0∂ξ−∂3ν0∂ξ3,ℓ1ξ,ℑ=83sech2ξ1+℘ℑ−℘,μ2ξ,ℑ==μ1ξ,ℑ−Y−11+℘ω−1Y12∂3μ1∂ξ3−3μ1∂μ1∂ξ+3∂∂ξν1ℓ1,μ2ξ,ℑ=4sech2ξ1−3tanh2ξ1−℘2℘ℑ+1−℘2+℘2ℑ22,ν2ξ,ℑ=ν1ξ,ℑ−Y−11+℘ω−1Y3μ1∂ν1∂ξ−∂3ν1∂ξ3,ν2ξ,ℑ=−sech2ξtanhξ1−℘2℘ℑ+1−℘2+℘2ℑ22,ℓ2ξ,ℑ=ℓ1ξ,ℑ−Y−11+℘ω−1Y3μ1∂ν1∂ξ−∂3ν1∂ξ3,ℓ2ξ,ℑ=−83sech2ξtanhξ1−℘2℘ℑ+1−℘2+℘2ℑ22,μξ,ℑ=−13+2tanh2ξ+4sech2ξtanhξ1+℘ℑ−℘+4sech2ξ1−3tanh2ξ1−℘2℘ℑ+1−℘2+℘2ℑ22+⋯,νξ,ℑ=tanhξ+sech2ξ1+℘ℑ−℘−sech2ξtanhξ1−℘2℘ℑ+1−℘2+℘2ℑ22+⋯.ℓξ,ℑ=83tanhξ+83sech2ξ1+℘ℑ−℘−83sech2ξtanhξ1−℘2℘ℑ+1−℘2+℘2ℑ22+⋯.
We achieve the exact solution by putting℘=1.(64)μξ,ℑ=−13+2tanh2ℑ+ξ,νξ,ℑ=tanhℑ+ξ,ℓξ,ℑ=83tanhℑ+ξ.
The analytical solution and exact solution forμξ,ℑ of example 2 at ℘=1 and −5≤ξ≥5 are shown in Figures 3(a) and 3(b), whereas Figures 3(c) and 3(d) show the absolute error and the solution at various fractional orders. The graphical behavior of exact solution and analytical solution for νξ,ℑ is shown in Figures 4(a) and 4(b), while Figures 4(c) and 4(d) show the absolute error and the solution at different fractional orders. Figures 5(a) and 5(b) give the solution graph for ℓξ,ℑ. From the results of the figures, it is confirmed that our method solution converges quickly towards exact solution.Figure 3
Exact solution, analytical solution, absolute error, and various fractional-order solutions forμξ,ℑ of Problem 2.
(a)(b)(c)(d)Figure 4
Exact solution, analytical solution, absolute error, and various fractional-order solutions forνξ,ℑ of Problem 2.
(a)(b)(c)(d)Figure 5
Exact solution and analytical solution for1ξ,ℑ Problem 2.
(a)(b)Problem 3.
We consider the coupled Burgers equation in one dimension.(65)∂℘CFμ∂ℑ℘=∂2μ∂ξ2+2μ∂μ∂ξ−∂∂ξμν,∂℘CFν∂ℑ℘=∂2ν∂ξ2+2ν∂ν∂ξ−∂∂ξμν,0<℘≤1,with an initial source(66)μξ,0=cosξ,νξ,0=cosξ.
On taking the Yang transform of (65), we get(67)11+℘ω−1Yμξ,ℑ−ωμξ,0=Y∂2μ∂ξ2+2μ∂μ∂ξ−∂∂ξμν,11+℘ω−1Yνξ,ℑ−ωνξ,0=Y∂2ν∂ξ2+2ν∂ν∂ξ−∂∂ξμν.
When we use the Yang transform, we obtain(68)μξ,ℑ=cosξ+Y−11+℘ω−1Y∂2μ∂ξ2+2μ∂μ∂ξ−∂∂ξμν,νξ,ℑ=cosξ+Y−11+℘ω−1Y∂2ν∂ξ2+2ν∂ν∂ξ−∂∂ξμν.
Assume that the solutionsμξ,ℑ and νξ,ℑ in the form of infinite series are given by(69)μξ,ℑ=∑m=0∞μmξ,ℑ,νξ,ℑ=∑m=0∞νmξ,ℑ,where μμξ=∑m=0∞Am, μνξ=∑m=0∞ℬm, and ννξ=∑m=0∞Cm are the so-called Adomian polynomials that represent nonlinear terms, and so equation (68) is rewritten as(70)∑m=0∞μmξ,ℑ=cosξ+Y−11+℘ω−1Y∂2μ∂ξ2+2∑m=0∞Am−∑m=0∞ℬm,∑m=0∞νmξ,ℑ=cosξ+Y−11+℘ω−1Y∂2ν∂ξ2+∑m=0∞Cm−∑m=0∞ℬm.
The decomposition of nonlinear terms by Adomian polynomials is defined as in equation (22).(71)A0=μ0μ0ξ,A1=μ1μ0ξ+μ0μ1ξ,A2=μ2μ0ξ+μ1μ1ξ+μ0μ2ξ,ℬ0=μ0ν0ξ+ν0μ0ξ,ℬ1=μ0ν0ξ+μ1ν0ξ+ν1μ0ξ+ν0μ1ξ,ℬ2=μ2ν0ξ+μ1ν1ξ+μ0ν2ξ+ν2μ0ξ+ν1μ1ξ+ν0μ2ξ,C0=ν0ν0ξ,C1=ν1ν0ξ+ν0ν1ξ,C2=ν2ν0ξ+ν1ν1ξ+ν0ν2ξ.
As a result, when comparing the two sides of equation (70), we get(72)μ0ξ,ℑ=cosξ,ν0ξ,ℑ=cosξ.
Form=0,(73)μ1ξ,ℑ=−cosξ1+℘ℑ−℘,ν1ξ,ℑ=−cosξ1+℘ℑ−℘.
Form=1,(74)μ2ξ,ℑ=cosξ1−℘2℘ℑ+1−℘2+℘2ℑ22,ν2ξ,ℑ=cosξ1−℘2℘ℑ+1−℘2+℘2ℑ22.
The approximate solution to the series is written as(75)μξ,ℑ=∑m=0∞μmξ,ℑ=μ0ξ,ℑ+μ1ξ,ℑ+μ2ξ,ℑ+⋯,νξ,ℑ=∑m=0∞νmξ,ℑ=ν0ξ,ℑ+ν1ξ,ℑ+ν2ξ,ℑ+⋯,μξ,ℑ=cosξ−cosξ1+℘ℑ−℘+cosξ1−℘2℘ℑ+1−℘2+℘2ℑ22+⋯,νξ,ℑ=cosξ−cosξ1+℘ℑ−℘+cosξ1−℘2℘ℑ+1−℘2+℘2ℑ22+⋯.
We achieve the exact solution by putting℘=1.(76)μξ,ℑ=cosξ1−ℑ+ℑ22−⋯,νξ,ℑ=cosξ1−ℑ+ℑ22−⋯.
In a closed form,μξ,ℑ=cosξexp−ℑ and νξ,ℑ=cosξexp−ℑ.
VITM analytical results as follows.
For equation (65), we have the iteration formula as(77)μm+1ξ,ℑ=μmξ,ℑ−Y−11+℘ω−1Y∂2μm∂ξ2+2μm∂μm∂ξ−∂∂ξμmνm,νm+1ξ,ℑ=νmξ,ℑ−Y−11+℘ω−1Y∂2νm∂ξ2+2νm∂ν∂ξ−∂∂ξμmνm,where(78)μ0ξ,ℑ=cosξ,ν0ξ,ℑ=cosξ.
Form=0,1,2,⋯,(79)μ1ξ,ℑ=μ0ξ,ℑ−Y−11+℘ω−1Y∂2μ0∂ξ2+2μ0∂μ0∂ξ−∂∂ξμ0ν0,μ1ξ,ℑ=−cosξ1+℘ℑ−℘,ν1ξ,ℑ=ν0ξ,ℑ−Y−11+℘ω−1Y∂2ν0∂ξ2+2ν0∂ν∂ξ−∂∂ξμ0ν0,ν1ξ,ℑ=−cosξ1+℘ℑ−℘,μ2ξ,ℑ=μ1ξ,ℑ−Y−11+℘ω−1Y∂2μ1∂ξ2+2μ1∂μ1∂ξ−∂∂ξμ1ν1,μ2ξ,ℑ=cosξ1−℘2℘ℑ+1−℘2+℘2ℑ22,ν2ξ,ℑ=ν1ξ,ℑ−Y−11+℘ω−1Y∂2ν1∂ξ2+2ν1∂ν∂ξ−∂∂ξμ1ν1,ν1ξ,ℑ=cosξ1−℘2℘ℑ+1−℘2+℘2ℑ22,μξ,ℑ=cosξ−cosξ1+℘ℑ−℘+cosξ1−℘2℘ℑ+1−℘2+℘2ℑ22+⋯,νξ,ℑ=cosξ−cosξ1+℘ℑ−℘+cosξ1−℘2℘ℑ+1−℘2+℘2ℑ22+⋯.
We achieve the exact solution by putting℘=1.(80)μξ,ℑ=cosξ1−ℑ+ℑ22−⋯,νξ,ℑ=cosξ1−ℑ+ℑ22−⋯.
In a closed form,μξ,ℑ=cosξexp−ℑ and νξ,ℑ=cosξexp−ℑ.
In Figure6, we display solution graphs 6(a) and 6(b) at ℘=1 for μξ,ℑ,νξ,ℑ in the domain −5≤ξ≤5 and Figures 6(c) and 6(d) show the solution graph of the absolute error and various fractional-order solutions. It is verified from figures and tables that our solution is closely related with the exact solution.Figure 6
Exact solution, analytical solution, absolute error, and various fractional-order solutions forμξ,ℑ and νξ,ℑ of Problem 3.
(a)(b)(c)(d)Problem 4.
We consider the coupled Burgers equation in two dimensions.(81)∂℘CFμ∂ℑ℘=∂2μ∂ξ2+∂2μ∂ϖ2+2μ∂μ∂ξ−∂∂ξμν,∂℘CFν∂ℑ℘=∂2ν∂ξ2+∂2ν∂ϖ2+2ν∂ν∂ξ−∂∂ξμν,0<℘≤1,with an initial source(82)μξ,ϖ,0=cosξ+ϖ,νξ,ϖ,0=cosξ+ϖ.
On taking the Yang transform of (81), we get(83)11+℘ω−1Yμξ,ℑ−ωμξ,0=Y∂2μ∂ξ2+∂2μ∂ϖ2+2μ∂μ∂ξ−∂∂ξμν,11+℘ω−1Yνξ,ℑ−ωνξ,0=Y∂2ν∂ξ2+∂2ν∂ϖ2+2ν∂ν∂ξ−∂∂ξμν.
We obtain (when we use the Yang transform).(84)μξ,ϖ,ℑ=cosξ+ϖ+Y−11+℘ω−1Y∂2μ∂ξ2+∂2μ∂ϖ2+2μ∂μ∂ξ−∂∂ξμν,νξ,ϖ,ℑ=cosξ+ϖ+Y−11+℘ω−1Y∂2ν∂ξ2+∂2ν∂ϖ2+2ν∂ν∂ξ−∂∂ξμν.
Assume that the solutionsμξ,ϖ,ℑ and νξ,ϖ,ℑ in the form of infinite series are given by(85)μξ,ϖ,ℑ=∑m=0∞μmξ,ϖ,ℑ,νξ,ϖ,ℑ=∑m=0∞νmξ,ϖ,ℑ,where μμξ=∑m=0∞Am, μνξ=∑m=0∞ℬm, and ννξ=∑m=0∞Cm are the so-called Adomian polynomials that represent the nonlinear terms, and so equation (84) is rewritten as(86)∑m=0∞μmξ,ϖ,ℑ=cosξ+ϖ+Y−11+℘ω−1Y∂2μ∂ξ2+∂2μ∂ϖ2+2∑m=0∞Am−∑m=0∞ℬm,∑m=0∞νmξ,ϖ,ℑ=cosξ+ϖ+Y−11+℘ω−1Y∂2ν∂ξ2+∂2μ∂ϖ2+∑m=0∞Cm−∑m=0∞ℬm.
The decomposition of nonlinear terms by Adomian polynomials is defined as in equation (22).(87)A0=μ0μ0ξ,A1=μ1μ0ξ+μ0μ1ξ,A2=μ2μ0ξ+μ1μ1ξ+μ0μ2ξ,ℬ0=μ0ν0ξ+ν0μ0ξ,ℬ1=μ0ν0ξ+μ1ν0ξ+ν1μ0ξ+ν0μ1ξ,ℬ2=μ2ν0ξ+μ1ν1ξ+μ0ν2ξ+ν2μ0ξ+ν1μ1ξ+ν0μ2ξ,C0=ν0ν0ξ,C1=ν1ν0ξ+ν0ν1ξ,C2=ν2ν0ξ+ν1ν1ξ+ν0ν2ξ.
As a result, when comparing the two sides of equation (54), we get(88)μ0ξ,ϖ,ℑ=cosξ+ϖ,ν0ξ,ϖ,ℑ=cosξ+ϖ.
Form=0,(89)μ1ξ,ϖ,ℑ=−2cosξ+ϖ1+℘ℑ−℘,ν1ξ,ϖ,ℑ=−2cosξ+ϖ1+℘ℑ−℘.
Form=1,(90)μ2ξ,ϖ,ℑ=2cosξ+ϖ1−℘2℘ℑ+1−℘2+℘2ℑ22,ν2ξ,ϖ,ℑ=2cosξ+ϖ1−℘2℘ℑ+1−℘2+℘2ℑ22.
The approximate solution to the series is written as follows:(91)μξ,ϖ,ℑ=∑m=0∞μmξ,ℑ=μ0ξ,ℑ+μ1ξ,ℑ+μ2ξ,ℑ+⋯,νξ,ϖ,ℑ=∑m=0∞νmξ,ℑ=ν0ξ,ℑ+ν1ξ,ℑ+ν2ξ,ℑ+⋯,μξ,ϖ,ℑ=cosξ+ϖ−2cosξ+ϖ1+℘ℑ−℘+2cosξ+ϖ1−℘2℘ℑ+1−℘2+℘2ℑ22+⋯,νξ,ϖ,ℑ=cosξ+ϖ−2cosξ+ϖ1+℘ℑ−℘+2cosξ+ϖ1−℘2℘ℑ+1−℘2+℘2ℑ22+⋯.
We achieve the exact solution by putting℘=1.(92)μξ,ϖ,ℑ=cosξ+ϖ1−2ℑ+4ℑ22!−⋯,νξ,ϖ,ℑ=cosξ+ϖ1−2ℑ+4ℑ22!−⋯.
In a closed form,μξ,ϖ,ℑ=cosξ+ϖexp−2ℑ and νξ,ϖ,ℑ=cosξ+ϖexp−2ℑ.
VITM analytical results as follows.
For equation (81), we have the iteration formula as(93)μm+1ξ,ϖ,ℑ=μmξ,ℑ−Y−11+℘ω−1Y∂2μm∂ξ2+∂2μm∂ϖ2+2μm∂μm∂ξ−∂∂ξμmνm,νm+1ξ,ϖ,ℑ=νmξ,ℑ−Y−11+℘ω−1Y∂2νm∂ξ2+∂2νm∂ϖ2+2νm∂νm∂ξ−∂∂ξμmνm,where(94)μ0ξ,ϖ,ℑ=cosξ+ϖ,ν0ξ,ϖ,ℑ=cosξ+ϖ.
Form=0,1,2,⋯,(95)μ1ξ,ϖ,ℑ=μ0ξ,ℑ−Y−11+℘ω−1Y∂2μ0∂ξ2+∂2μ0∂ϖ2+2μ0∂μ0∂ξ−∂∂ξμ0ν0,μ1ξ,ϖ,ℑ=−2cosξ+ϖ1+℘ℑ−℘,ν1ξ,ϖ,ℑ=ν0ξ,ℑ−Y−11+℘ω−1Y∂2ν0∂ξ2+∂2ν0∂ϖ2+2ν0∂ν0∂ξ−∂∂ξμ0ν0,ν1ξ,ϖ,ℑ=−2cosξ+ϖ1+℘ℑ−℘,μ2ξ,ϖ,ℑ=μ1ξ,ℑ−Y−11+℘ω−1Y∂2μ1∂ξ2+∂2μ1∂ϖ2+2μ1∂μ1∂ξ−∂∂ξμ1ν1,μ2ξ,ϖ,ℑ=2cosξ+ϖ1−℘2℘ℑ+1−℘2+℘2ℑ22,ν2ξ,ϖ,ℑ=ν1ξ,ℑ−Y−11+℘ω−1Y∂2ν1∂ξ2+∂2ν1∂ϖ2+2ν1∂ν1∂ξ−∂∂ξμ1ν1,ν2ξ,ϖ,ℑ=2cosξ+ϖ1−℘2℘ℑ+1−℘2+℘2ℑ22,μξ,ϖ,ℑ=cosξ+ϖ−2cosξ+ϖ1+℘ℑ−℘+2cosξ+ϖ1−℘2℘ℑ+1−℘2+℘2ℑ22−⋯,νξ,ϖ,ℑ=cosξ+ϖ−2cosξ+ϖ1+℘ℑ−℘+2cosξ+ϖ1−℘2℘ℑ+1−℘2+℘2ℑ22−⋯.
We achieve the exact solution by putting℘=1.(96)μξ,ϖ,ℑ=cosξ+ϖ1−2ℑ+4ℑ22!−⋯,νξ,ϖ,ℑ=cosξ+ϖ1−2ℑ+4ℑ22!−⋯.
In a closed form,μξ,ϖ,ℑ=cosξ+ϖexp−2ℑ and νξ,ϖ,ℑ=cosξ+ϖexp−2ℑ. In Figure 7, Figures 7(a) and 7(b) demonstrate the layout of exact solution and analytical solution, whereas the graphical view for various fractional orders is demonstrated with the help of Figures 7(c) and 7(d). Finally, it is clear from the figures and tables that the proposed methods have sufficient degree of accuracy and quick convergence towards the exact solution.Figure 7
Exact solution, analytical solution, absolute error, and various fractional-order solutions forμξ,ℑ and νξ,ℑ of Problem 4.
(a)(b)(c)(d)


## 6. Conclusion

YDM and VITM were used for solving the coupled nonlinear partial differential equations. Comparing the results of these methods with the exact solution shows that the proposed methods are extremely simple and easy to handle nonlinear terms. The obtained results converge quickly in the form of series towards the exact solution. Four nonlinear systems are solved, which shows that solutions are in strong agreement with the exact solution with suggested techniques. It is confirmed that the proposed methods need less computational work, which shows fast convergence. Furthermore, YDM and VITM are very effective and efficient for finding approximate analytic solutions for a wide range of real-world problems arising in engineering and science.



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