# Application of Conformable Sumudu Decomposition Method for Solving Conformable Fractional Coupled Burger’s Equation

Authors: Hassan Eltayeb; Said Mesloub
Journal: Journal of Function Spaces
 (2021)

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## Abstract

The conformable double Sumudu decomposition method (CDSDM) is a combination of decomposition method (DM) and a conformable double Sumudu transform. It is an approximate analytical method, which can be used to solve linear and nonlinear partial differential equations. In this work, one-dimensional conformable functional Burger’s equation has been solved by applying conformable double Sumudu decomposition. Two examples are used to illustrate the method.


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## Body

## 1. Introduction

Burgers’ equation is represented by nonlinear wave motion with linear diffusion. These equations have appeared in the area of applied sciences such as fluid mechanics and mathematical modeling. The importance of getting the exact or approximate solutions of Burgers’ equations in mathematics and physics is still a significant problem that needs new methods to discover exact or approximate solutions. Theoretical solution of Burger equations, based on Fourier series analysis, using the appropriate initial and boundary conditions was discussed in [1]. The authors in [2] studied different exact solutions of Burger’s like equations and their classifications. The author in [3] discussed (2+1)-dimensional time-fractional Bogoyavlensky-Konopelchenko (BK) equation by using a subequation method that is predicated on the Riccati equation. The new wave solutions of the fractional Camassa-Holm equation are studied by applying a new extended direct algebraic method (see [4]). The wave solutions of the time-fractional Benjamin-Ono equation are established by using the Jacobi elliptic function expansion method for more details (see [5]). The solitary and traveling wave solutions for the time-fractional pKP equation have been obtained by applying the new direct algebraic method by means of the conformable derivative see [6]. Many powerful methods were used to solve nonlinear partial differential equations, such as the fractional Sumudu transform [7, 8]. The exact solutions for time-fractional Burgers’ equations were studied by the first integral method [9]. The authors in [10] employed the generalized two-dimensional differential transform method (DTM) and obtained the solution for the coupled Burgers’ equations with space- and time-fractional derivatives. In recent time, the new definition of the conformable derivative has been introduced for more specifics; we refer to [11–13]. Furthermore, the authors in [11, 14] are solved the fractional differential equations by conformable Laplace transform technique. In [15], the researchers applied the conformable double Laplace transform method and obtained the solution for the fractional partial differential equations. The conformable double Laplace decomposition method [16] has been proposed to obtain exact and approximate solutions of regular and singular one-dimensional conformable fractional coupled burgers’ equations. The exact solutions of the time-fractional Burgers’ equations were established by using the first integral method (see [9]). We introduce a new method called the conformable double Sumudu decomposition method (CDSDM) for solving the nonlinear equations in the present paper. The proposed method is an elegant combination of the conformable double Sumudu transform method and the decomposition method. This paper considers the effectiveness of the conformable double Sumudu decomposition method (CDSDM) in solving regular and singular one-dimensional conformable fractional coupled burgers’ equations, and the advantage of this method is easy to apply for solving fractional nonlinear partial differential equations. In the following, we list some definitions from the conformable derivatives which are used further in this paper.Definition 1 (see [13, 17–19]).
Given a functionf:0,∞⟶R, then the conformable fractional derivative of f of order α is defined by
(1)dαdtαft=limε⟶0ft+εt1−α−ftε,t>0,0<α≤1.Definition 2 (see [20]).
Given a functionfxα/α,tβ/β:R×0,∞⟶R. Then, the conformable space fractional partial derivative of order α a function fxα/α,tβ/β is defined as follows:
(2)∂α∂xαfxαα,tββ=limε⟶0fxα/α,+εx1−α,t−fxα/α,tβ/βε,xαα,tββ>0,0<α,β≤1,Definition 3 (see [20]).
Given a functionfxα/α,tβ/β:R×0,∞⟶R. Then, the conformable time fractional partial derivative of order β a function fxα/α,tβ/β is defined as follows:
(3)∂β∂tβfxαα,tββ=limσ⟶0fxα/α,tβ/β+σt1−β−fxα/α,tβ/βσ,xαα,tββ>0,0<α,β≤1,The following are the conformable fractional derivative of certain functions:Example 4 (see [16]).
(4)∂α∂xαxααtββ=tββ,∂α∂xαxααntββ=nxααn−1tββ,∂β∂tβxααtββ=xαα,∂β∂tβxααntββm=mxααntββm−1,∂β∂tβsinxααsintββ=sinxααcostββ,∂α∂xαsinaxααsintββ=acosxααsintββ,∂α∂xαeλxα/α+τtβ/β=λeλxα/α+τtβ/β,∂β∂tβeλxα/α+τtβ/β=τeλxα/α+τtβ/β.
We introduce the conformable double Sumudu transform and solve the conformable fractional coupled burgers’ equation by using this new definition.Definition 5.
Over the set of function(5)A=ftββ:∃M,τ1,τ2>0,ftββ<Metβ/βτj,ift∈−1j×0,∞,the conformable Sumudu transform is given by
(6)Stβftββ=Fβu=∫0∞ftββe−1/utβ/βtβ−1dt,u∈−τ1,τ2.Definition 6.
Letfxα/α,tβ/β be function we defined conformable double Sumudu transform of function fxα/α,tβ/β, xα/α,tβ/β∈ℝ+ is given by
(7)Fα,βu,v=SxαStβfxαα,tββ=1uv∫0∞∫0∞e−1/uxα/α+1/vtβ/βfxαα,tββxα−1tβ−1dxdtor
(8)Fα,βu,v=SxαStβfxαα,tββ=∫0∞∫0∞e−xα/α+tβ/βfuxαα,vtββxα−1tβ−1dxdt.where SxαStβ represent the conformable double Sumudu transform. The relation between the usual and the conformable double Sumudu transforms is given below.Example 7.
The double conformable Sumudu transform for certain functions are given by(9)SxαStβxααntββm=SxStxntm=n!m!unvm,SxαStβeλxαα+τtββ=SxSteλx+τt=11−λu1−τv,SxαStβsinλxααsinτtββ=SxStsinλxsinλt=λu1+λ2u2λv1+τ2v2,SxαStβcosxααcostββ=SxStcosλxcosλt=11+λ2u211+λ2v2.
Now, we present the conditions for the existence of the conformable double Sumudu transform:
Iffxα/α,tβ/β is an exponential order a and b as xα/α⟶∞, tβ/β⟶∞ and if ∃K>0, then for all xα/α>X and tβ/β>T,
(10)fxαα,tββ≤Keaxα/α+btβ/β,and we write
(11)fxαα,tββ=Oeaxα/α+btβ/βasxαα⟶∞,tββ⟶∞,equivalently,
(12)limxαα⟶∞tββ⟶∞e−1/uxα/α−1/ηtβ/βfxαα,tββ=Klimxαα⟶∞tββ⟶∞e−1/u−axα/α−1/η−btβ/β=0,where 1/u>a and 1/η>b. The function fxα/α,tβ/β does not grow faster than Keaxα/α+btβ/β as xα/α⟶∞, tβ/β⟶∞.Theorem 8.
The functionfxα/α,tβ/β is defined on 0,X and 0,T and of exponential order eaxα/α+btβ/β; then the conformable Sumudu transform of fxα/α,tβ/β exists for all Re1/u>1/μ,Re1/v>1/η.Proof.
From equation (29), we obtain
(13)Fα,βu,v=1uv∫0∞∫0∞e−1/uxα/α+1/vtβ/βfxαα,tββxα−1tβ−1dxdt≤K1u∫0∞e−1/u−axα/α−1/v−btβ/βxα−1tβ−1dxdt=k1−au1−bv.
ForRe1/u>1/μ,Re1/v>1/η, from equation (13), we get
(14)limxαα⟶∞tββ⟶∞Fα,βu,v=or
(15)limxαα⟶∞tββ⟶∞Fα,βu,v=0.
Now the next result deals with the multiplication of the functionfxα/α,tβ/β by xα/α,tβ/β.Theorem 9.
If conformable double Sumudu transform of the functionfxα/α,tβ/β is given by Fα,βu,v=SxαStβfxα/α,tβ/β, then the conformable double Sumudu transforms of the functions
(16)xααfxαα,tββ,tββfxαα,tββ,xαα2fxαα,tββ,tββ2fxαα,tββ,xααtββfxαα,tββare given by
(17)SxαStβxααfxαα,tββ=u2∂Fα,βu,v∂u+uFα,βu,v,(18)SxαStβtββfxαα,tββ=v2∂Fα,βu,v∂v+vFα,βu,v,(19)SxαStβxαα2fxαα,tββ=u4∂2Fα,βu,v∂u2+4u3∂Fα,βu,v∂u+2u2Fα,βu,v,(20)SxαStβtββ2fxαα,tββ=v4∂2Fα,βu,v∂v2+4v3∂Fα,βu,v∂v+2v2Fα,βu,v,(21)Stβxααtββfxαα,tββ=uv∂2Fα,βu,v∂u∂v+uv2∂Fα,βu,v∂v+u2v∂Fα,βu,v∂u+uvFα,βu,v,respectively.Proof.
By using the definition of conformable double Sumudu transform for equation (17), we get
(22)∂Fα,βu,v∂u=∂∂u∫0∞∫0∞1uve−1/uxα/α+1/vtβ/βfxαα,tββxα−1tβ−1dxdt,=∫0∞1ve−1vtββ∫0∞∂∂u1ue−1/uxα/αfxαα,tββxα−1dxtβ−1dt,by calculating the partial derivative inside brackets, we can easily get
(23)∫0∞∂∂u1ue−1/uxα/αfxαα,tββxα−1dx=∫0∞1u3xαα−1u2e−1/uxα/αfxαα,tββxα−1dx=∫0∞1u3xααe−1/uxα/αfxαα,tββxα−1dx−∫0∞1u2e−1/uxα/αfxαα,tββxα−1dx,substituting equation (23) into equation (22), we obtain
(24)∂Fα,βu,v∂u=∫0∞1ve−1/vtβ/β∫0∞1u3xααe−1/uxα/αfxαα,tββxα−1dxtβ−1dt−∫0∞1ve−1/vtβ/β∫0∞1u2e−1/uxα/αfxαα,tββxα−1dxtβ−1dt,by applying equation (7), we get equation (17)
(25)SxαStβxααfxαα,tββ=u2∂Fα,βu,v∂u+uFα,βu,v.
For equation (19), by using second partial derivative for Fα,βu,v with respect to u and definition of conformable double Sumudu transform, we have
(26)∂2Fα,βu,v∂u2=∂2∂u2∫0∞∫0∞1uve−1/uxα/α+1/vtβ/βfxαα,tββxα−1tβ−1dxdt,=0∞1ve−1/vtβ/β0∞∂2∂u21ue−1/uxα/αfxαα,tββxα−1dxtβ−1dt,=∫0∞1ve−1/vtβ/β∫0∞1u5xαα2−3u4xααe−1/uxα/αfxαα,tββxα−1dxtβ−1dt−∫0∞1ve−1/vtβ/β∫0∞1u4xαα−2u3e−1/uxα/αfxαα,tββxα−1dxtβ−1dt.
Applying equations (7) and (19), we obtain equation (19)
(27)SxαStβxαα2fxαα,tββ=u4∂2Fα,βu,v∂u2+4u3∂Fα,βu,v∂u+2u2Fα,βu,v.
Similarly, in the same way, one can prove equations (18), (20), and (21).
The conformable double Sumudu transform of the first partial derivative with respect tox and t is given by
(28)SxαStβ∂αψ∂xα=1uΨα,βu,v−1uΨβ0,v,SxαStβ∂βψ∂tβ=1vΨα,βu,v−1vΨαu,0,and the conformable double Sumudu transform of second partial derivative concerning x and t is denoted by
(29)SxαStβ∂2αψ∂x2α=1u2Ψα,βu,v−1u2Ψβ0,v−1uStβ∂αψ0,tβ/β∂xαSxαStβ∂2βψ∂t2β=1v2Ψα,βu,v−1v2Ψαu,0−1vSxα∂βψxα/α,0∂tβ.
The following theorem gives the conformable double Sumudu transform of the fractional partial derivativesxα/α∂βψ/∂tβ and xα/α∂2βψ/∂t2β.Theorem 10.
If conformable double Sumudu transform of the fractional partial derivativesxα/α∂βψ/∂tβ and xα/α∂2βψ/∂t2β are given by
(30)SxαStβxαα∂βψ∂tβ=uvdduuΨα,βu,v−uvdduuΨαu,0,(31)SxαStβxαα∂2βψ∂t2β=uv2dduuΨα,βu,v−uv2dduuΨαu,0−uvdduuSxα∂βψxα/α,0∂tβ,respectively.Proof.
By using the definition of double Sumudu transform, we have(32)dduSxαStβ∂βψ∂tβ=ddu∫0∞∫0∞1uve−1/uxα/α+1/vtβ/β∂βψ∂tβxα−1tβ−1dxdt,=∫0∞1ve−1/vtβ/β∂βψ∂tβ∫0∞ddu1ue−1/uxα/αxα−1dxtβ−1dt,we calculate the partial derivative inside brackets as follows:
(33)∫0∞ddu1ue−1/uxα/αxα−1dx=∫0∞1u3xαα−1u2e−1/uxα/αxα−1dx=∫0∞1u3xααe−1/uxα/αxα−1dx−∫0∞1u2e−1/uxα/αxα−1dx,substituting equation (33) into equation (32), we get
(34)dduSxαStβ∂βψ∂tβ=∫0∞1ve−1/vtβ/β∫0∞1u3xααe−1/uxα/α∂βψ∂tβxα−1dxtβ−1dt−∫0∞1ve−1/vtβ/β∫0∞1u2e−1/uxα/α∂βψ∂tβxα−1dxtβ−1dt,applying equation (7), we obtain
(35)SxαStβxαα∂βψ∂tβ=u2dduSxαStβ∂βψ∂tβ+uSxαStβ∂βψ∂tβ,and on using equation (28), we have
(36)SxαStβxαα∂βψ∂tβ=uvdduuΨα,βu,v−uvdduuΨαu,0.
In a similar manner, one can prove equation (31).


## 2. One-Dimensional Fractional Coupled Burgers’ Equation

In this section, the solution of regular and irregular one-dimensional conformable fractional coupled burgers’ equations is discussed by using conformable double Sumudu decomposition methods (CDSDM). By takingα=1 and β=1 in the following problems, one can get the problems which were discussed in [21]:The first problem is as follows: we consider the following one-dimensional conformable fractional coupled burgers’ equations:(37)∂βψ∂tβ−∂2αψ∂x2α+ηψ∂αψ∂xα+ζ∂α∂xαψφ=fxαα,tββ,∂βφ∂tβ−∂2αφ∂x2α+ηv∂αφ∂xα+μ∂α∂xαψφ=gxαα,tββ,appropriate initial conditions
(38)ψxαα,0=f1xαα,φxαα,0=g1xαα,where fxα/α,tβ/β, gxα/α,tβ/β, f1xα/α, and g1xα/α are known functions and η,ζ, and μ are arbitrary constants depending on the system parameters. Employing the conformable double Sumudu transform for both sides of equation (37) and conformable single Sumudu transform for equation (38), we obtain
(39)Ψα,βu,v=Ψαu,0+vSxαStβfxαα,tββ+∂2αψ∂x2α−ηψ∂αψ∂xα−ζ∂α∂xαψφ,(40)Φα,βu,v=Φαu,0+vSxαStβgxαα,tββ+∂2αφ∂x2α−ηφ∂αφ∂xα−ζ∂α∂xαψφ.The solution of one-dimensional conformable fractional coupled burgers’ equations is defined by the infinite series as follows:(41)ψxαα,tββ=∑n=0vφnxαα,tββ,φxαα,tββ=∑n=0vφnxαα,tββ.The nonlinear operatorsAn, Bn, and Cn defined by Adomian polynomials are given as follows:
(42)An=n=0∞ψnψxn,Bn=∑n=0vφnφxn,Cn=∑n=0vψnφn.The few components of the Adomian polynomials are given as follows:(43)A0=ψ0ψ0x,A1=ψ0ψ1x+ψ1ψ0x,A2=ψ0ψ2x+ψ1ψ1x+ψ2ψ0x,A3=ψ0ψ3x+ψ1ψ1x+ψ2ψ1x+ψ3ψ0x,A4=ψ0ψ4x+ψ1ψ3x+ψ2ψ2x+ψ3ψ1x+ψ4ψ0x,(44)B0=φ0φ0x,B1=φ0φ1x+φ1φ0x,B2=φ0φ2x+φ1φ1x+φ2φ0x,B3=φ0φ3x+φ1φ2x+φ2φ1x+φ3φ0x,B4=φ0φ4x+φ1φ3x+φ2φ2x+φ3φ1x+φ4φ0x,(45)C0=ψ0φ0,C1=ψ0φ0+ψ1φ0C2=ψ0φ2+ψ1φ1+ψ2φ0.C3=ψ0φ3+ψ1φ2+ψ2φ1+ψ3φ0,C3=ψ0φ4+ψ1φ3+ψ2φ2+ψ3φ1+ψ4φ0.Applying the inverse double Sumudu transform to equation (39), equation (40) and using equation (42), we get
(46)∑n=0∞ψnxαα,tββ=f1xαα+Su−1Sv−1vSxαStβ∂2αψn∂x2α−Su−1Sv−1vSxαStβηAn−Su−1Sv−1vSxαStβζCn,∑n=0∞φnxαα,tββ=g1xαα+Su−1Sv−1vGu,v+Su−1Sv−1vSxαStβ∂2αφn∂x2α−Su−1Sv−1vSxαStβηBn−Su−1Sv−1vSxαStβμCn.For the casen=0, we set
(47)ψ0=f1xαα+Su−1Sv−1vFu,v,φ0=g1xαα+Su−1Sv−1vGu,v.Now, we can obtain the following general form:(48)ψn+1=Su−1Sv−1vSxαStβ∂2αψn∂x2α−Su−1Sv−1vSxαStβηAn−Su−1Sv−1vSxαStβζCn,(49)φn+1=Su−1Sv−1vSxαStβ∂2αφn∂x2α−Su−1Sv−1vSxαStβηBn−Su−1Sv−1vSxαStβμCn.Here, we provide double inverse Sumudu transform with respect tou and v existing for each term in the right-hand side of the above equations. To show the method of coupled one-dimensional conformable fractional coupled burgers’ equations, the following example is considered.Example 11.
Consider the following homogeneous form of a one-dimensional conformable fractional coupled burgers’ equations:(50)∂βψ∂tβ−∂2αψ∂x2α−2ψ∂αψ∂xα+∂α∂xαψφ=0,∂βφ∂tβ−∂2αφ∂x2α−2φ∂αφ∂xα+∂α∂xαψφ=0,with initial condition
(51)ψxαα,0=sinxαα,φxαα,0=sinxαα.
According to equations (47), (48), and (49), the first three terms of the Sumudu decomposition series are derived as follows:
(52)ψ0=sinxαα,φ0=sinxααred,ψ1=Su−1Sv−1vSxαStβ∂2αψ0∂x2α+2ψ0∂αψ0∂xα−∂α∂xαψ0φ0=Su−1Sv−1vSxαStβ−sinxαα=Su−1Sv−1uv1+u2=−tββsinxαα,φ1=Su−1Sv−1vSxαStβ∂2αφ0∂x2α+2φ0∂αφ0∂xα−∂α∂xαψ0φ0=Su−1Sv−1vSxαStβ−sinxαα=Su−1Sv−1uvu2+1=−tββsinxααred,ψ2=Su−1Sv−1vSxαStβ∂2αψ1∂x2α+2ψ0∂α∂xαψ1+ψ1∂α∂xαψ0−∂α∂xαψ0φ1+ψ1φ0=Su−1Sv−1vSxαStβtββsinxαα=Su−1Sv−1uv2u2+1=tβ/β22sinxαα,φ2=Su−1Sv−1vSxαStβ∂2αφ1∂x2α+2φ0∂α∂xαφ1+φ1∂α∂xαφ0−∂α∂xαψ0φ1+ψ1φ0=Su−1Sv−1vSxαStβtββsinxαα=Su−1Sv−1uv2u2+1=tβ/β22sinxαα,ψ3=Su−1Sv−1vSxαStβ∂2αψ2∂x2α+2ψ0∂α∂xαψ2+ψ1∂α∂xαψ1+ψ2∂α∂xαψ0−Su−1Sv−1vSxαStβ∂α∂xαψ0φ2+ψ1φ1+ψ2φ0=−tβ/β36sinxαα,φ3=Su−1Sv−1vSxαStβ∂2αφ2∂x2α+2φ0∂α∂xαφ2+φ1∂α∂xαφ1+φ2∂α∂xαφ0−Su−1Sv−1vSxαStβ∂α∂xαψ0φ2+ψ1φ1+ψ2φ0=−tβ/β36sinxαα,and so on for other components. Using equation (41), the series solutions are therefore given by
(53)ψxαα,tββ=ψ0+ψ2+ψ3+⋯=1−tββ+tβ/β22!−tβ/β33!+⋯sinxαα,φxαα,tββ=φ0+φ2+φ3+⋯=1−tββ+tβ/β22!−tβ/β33!+⋯sinxαα.
Therefore, the exact solution takes the form of(54)ψxαα,tββ=e−tβ/βsinxαα,φxαα,tββ=e−tβ/βsinxαα.
By takingα=1 and β=1, the fractional solution become
(55)ψx,t=e−tsinx,φx,t=e−tsinx.
The exact solution of equation (50) for α=β=1 is shown in Figure 1 (see also Figures 2(a)–2(c)) for α≠β.
The second problem is as follows: we consider the singular one-dimensional conformable fractional coupled burgers’ equations of the form:(56)∂βψ∂tβ−αxα∂α∂xαxαα∂αψ∂xα+ηψ∂αψ∂xα+ζ∂α∂xαψφ=fxαα,tββ,∂βφ∂tβ−αxα∂α∂xαxαα∂αφ∂xα+ηψ∂αφ∂xα+μ∂α∂xαψφ=gxαα,tββ,with initial conditions
(57)ψxαα,0=f1xαα,φxαα,0=g1xαα,where α/xα∂α/∂xαxα/α∂α/∂xα is a conformable Bessel operator and ζ, μ, and η are real constants. In order to obtain the solution of equation (56). First, multiplying equation (56) by xα/α to obtain the following:
(58)xαα∂βψ∂tβ−∂α∂xαxαα∂αψ∂xα+ηxααψ∂αψ∂xα+ζxαα∂α∂xαψφ=xααfxαα,tββ,xαα∂βφ∂tβ−∂α∂xαxαα∂αφ∂xα+ηxααφ∂αφ∂xα+μxαα∂α∂xαψφ=xααgxαα,tββ.
Second, applying the conformable double Sumudu transform on both sides of equation (58) and after using the differentiation property of conformable Sumudu transform, we get
(59)SxαStβxαα∂βψ∂tβ=SxαStβ∂α∂xαxαα∂αψ∂xα−ηxααψ∂αψ∂xα−ζxαα∂α∂xαψφ+xααfxαα,tββ,SxαStβxαα∂βφ∂tβ=SxαStβ∂α∂xαxαα∂αφ∂xα−ηxααφ∂αφ∂xα−μxαα∂α∂xαψφ+xααgxαα,tββ.
On using Theorems9 and 10, we have
(60)uvdduuΨα,βu,v−uvdduuΨαu,0=u2∂Fα,βu,v∂u+uFα,βu,v++SxαStβ∂α∂xαxαα∂αψ∂xα−ηxααψ∂αψ∂xα−ζxαα∂α∂xαψφ,uvdduuΦα,βu,v−uvdduuΦαu,0=u2∂Gα,βu,v∂u+uGα,βu,v+SxαStβ∂α∂xαxαα∂αφ∂xα−ηxααφ∂αφ∂xα−μxαα∂α∂xαψφ,and arranging equation (60), we obtain
(61)dduuΨα,βu,v=uv∂Fα,βu,v∂u+vFα,βu,v+dduuΨαu,0+vuSxαStβ∂α∂xαxαα∂αψ∂xα−ηxααψ∂αψ∂xα−ζxαα∂α∂xαψφ,dduuΦα,βu,v=uv∂Gα,βu,v∂u+vGα,βu,v+dduuΦαu,0+vuSxαStβ∂α∂xαxαα∂αφ∂xα−ηxααφ∂αφ∂xα−μxαα∂α∂xαψφ.
Third, by applying the integral for both sides of equation (61) from 0 to u with respect to u, we have
(62)Ψα,βu,v=1u∫0uuv∂Fα,βu,v∂u+vFα,βu,v+dduuΨαu,0du+1u∫0uvuSxαStβ∂α∂xαxαα∂αψ∂xα−ηxααψ∂αψ∂xα−ζxαα∂α∂xαψφdu,Φα,βu,v=1u∫0uuv∂Gα,βu,v∂u+vGα,βu,v+dduuΦαu,0du+1u∫0uvuSxαStβ∂α∂xαxαα∂αφ∂xα−ηxααψ∂αφ∂xα−ζxαα∂α∂xαψφdu.
Applying the inverse double Sumudu transform, to both sides of equation (62) yields the following:
(63)ψxαα,tββ=Su−1Sv−11u∫0uuv∂Fα,βu,v∂u+vFα,βu,v+dduuΨαu,0du+Su−1Sv−11u∫0uvuSxαStβ∂α∂xαxαα∂αψ∂xα−ηxααψ∂αψ∂xα−ζxαα∂α∂xαψφdu,φxαα,tββ=Su−1Sv−11u∫0uuv∂Gα,βu,v∂u+vGα,βu,v+dduuΦαu,0du+Su−1Sv−11u∫0uvuSxαStβ∂α∂xαxαα∂αφ∂xα−ηxααψ∂αφ∂xα−ζxαα∂α∂xαψφdu.
The conformable double Sumudu decomposition method proposes a series solution of the functionsψxα/α,tβ/β and φxα/α,tβ/β by the infinite series
(64)ψxαα,tββ=∑n=0∞ψnxαα,tββ,φxαα,tββ=∑n=0∞φnxαα,tββ,∑n=0∞ψnxαα,tββ=Su−1Sv−11u∫0uuv∂Fα,βu,v∂u+vFα,βu,v+dduuΨαu,0du+Su−1Sv−11u∫0uvuSxαStβ∂α∂xαxαα∂α∂xα∑n=0∞φndu−Su−1Sv−11u∫0uvuSxαStβηxαα∑n=0∞Andu−Su−1Sv−11u∫0uvuSxαStβζxαα∑n=0∞Cndu,∑n=0∞φnxαα,tββ=Su−1Sv−11u∫0uuv∂Gα,βu,v∂u+vGα,βu,v+dduuΦαu,0du+Su−1Sv−11u∫0uvuSxαStβ∂α∂xαxαα∂α∂xα∑n=0∞φndu−Su−1Sv−11u∫0uvuSxαStβηxαα∑n=0∞Bndu−Su−1Sv−11u∫0uvuSxαStβμxαα∑n=0∞Cndu.where An, Bn, and Cn are given by equation (42). Consequently,
(65)ψ0=Su−1Sv−11u∫0uuv∂Fα,βu,v∂u+vFα,βu,v+dduuΨαu,0du,φ0=Su−1Sv−11u∫0uuv∂Gα,βu,v∂u+vGα,βu,v+dduuΦαu,0du,(66)ψn+1xαα,tββ=Su−1Sv−11u∫0uvuSxαStβ∂α∂xαxαα∂αψn∂xαdu−Su−1Sv−11u∫0uvuSxαStβηxααAndu−Su−1Sv−11u∫0uvuSxαStβζxααCndu,(67)φn+1xαα,tββ=Su−1Sv−11u∫0uvuSxαStβ∂α∂xαxαα∂αφn∂xαdu−Su−1Sv−11u∫0uvuSxαStβηxααBndu−Su−1Sv−11u∫0uvuSxαStβμxααCndu.
Assume that double inverse Sumudu transform with respectu and v exists for each term in the right-hand side of equations (65), (66), and (67).Example 12.
Consider singular one-dimensional conformable fractional coupled burgers’ equations(68)∂βψ∂tβ−αxα∂α∂xαxαα∂αψ∂xα−2ψ∂αψ∂xα+∂α∂xαψφ=xαα2etβ/β−4etβ/β,∂βφ∂tβ−αxα∂α∂xαxαα∂αφ∂xα−2φ∂αφ∂xα+∂α∂xαψφ=xαα2etβ/β−4etβ/β,subject to
(69)ψx,0=xαα2,φx,0=xαα2.
Using the above steps, we get(70)∑n=0∞ψnxαα,tββ=xαα2etβ/β−4etβ/β+4−Su−1Sv−11v∫0uSxαStβ∂α∂xαxαα∂α∂xα∑n=0∞φndu−Su−1Sv−11v∫0uSxαStβ2xαα∑n=0∞Andu+Su−1Sv−11v∫0uSxαStβxαα∂α∂xα∑n=0∞Cndu,∑n=0∞φnxαα,tββ=xαα2etβ/β−4etβ/β+4−Su−1Sv−11v∫0uSxαStβ∂α∂xαxαα∂α∂xα∑n=0∞φndu−Su−1Sv−11v∫0uSxαStβ2xαα∑n=0∞Bndu+Su−1Sv−11v∫0uSxαStβxαα∑n=0∞Cndu,where the nonlinear terms An, Bn, and Cn are denoted in equations (43), (44), and (45), respectively. In view of the recursive relations (65), (66), and (67), we obtained other components as follows:
(71)ψ0=xαα2etβ/β−4etβ/β+4,φ0=xαα2etβ/β−4etβ/β+4,ψ1=−Su−1Sv−11v∫0uSxαStβ∂α∂xαxαα∂αψ0∂xα+2xααψ0∂αψ0∂xα−xαα∂α∂xαψ0φ0du,ψ1=−Su−1Sv−11v∫0uSxαStβ4xααetββdu=4etβ/β−4,φ1=−Su−1Sv−11v∫0uSxαStβ∂α∂xαxαα∂αφ0∂xα+2xααφ0∂αφ0∂xα−xαα∂α∂xαψ0φ0du,φ1=−Su−1Sv−11v∫0uSxαStβ4xααetβ/βdu=4etβ/β−4,ψ2=−Su−1Sv−11v∫0uSxαStβ∂α∂xαxαα∂αψ0∂xαdu−Su−1Sv−11v∫0uSxαStβ2xααψ0∂αψ1∂xα+ψ1∂αψ0∂xαdu+Su−1Sv−11v∫0uSxαStβxαα∂α∂xαψ0φ1+ψ1φ0duψ2=0,φ2=0.
So the solution of singular one-dimensional conformable fractional coupled burgers’ equations in the series form is given by(72)ψxαα,tββ=ψ0+ψ1+ψ2+⋯,φxαα,tββ=φ0+φ1+φ2+⋯
Therefore, the exact solution is obtained as follows:(73)ψxαα,tββ=xαα2etβ/βandφxαα,tββ=xαα2etβ/β.
By usingα=1 and β=1, the fractional solution becomes
(74)ψx,t=x2et,φx,t=x2et.
The exact solution of equation (68) for α=β=1 is shown in Figure 3 (see also Figures 4(a)–4(c)) for α≠β.Figure 1
The exact solutionsψx,t=φx,t for equation (50) when α=β=1.Figure 2
The solutionsψx,t=φx,t for equation (50) at different values of α and βt=π/2.
(a)
The solutionsψx,t=φx,t for equation (50) at α=β=1,α=β=0.85,α=β=0.90, and α=β=0.95(b)
The solutionsψx,t=φx,t for equation (50) when α=0.99 and different values of β(c)
The solutionsψx,t=φx,t for equation (50) for different values of α at β=0.99Figure 3
The solutionsψx,t=φx,t for equation (68) when α=β=1.Figure 4
The solutionsψx,t=φx,t for equation (68) at different values of α and β.
(a)
The solutionψx,t=φx,t for (68) at α=β(b)
The solutionψx,t=φx,t for (68) when α=0.99 and different values of β(c)
The solutionψx,t=φx,t for (68) for different values of α at β=0.99


## 3. Conclusion

In this paper, we introduce the conformable double Sumudu decomposition method and give the conformable double Sumudu transform of the fractional partial derivatives. Then we discuss the solution of regular and irregular one-dimensional conformable fractional coupled burgers’ equations by using a conformable double Sumudu decomposition method. Further properties and problems can be considered by using conformable double Sumudu decomposition. We sketched all the graphs by using Matlab. In the future, we are going to generalize conformable Laplace transform to solve more general nonlinear partial differential equations.



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