# Conformable Double Laplace–Sumudu Transform Decomposition Method for Fractional Partial Differential Equations

Authors: Jia Honggang; Zhao Yanmin
Journal: Complexity
 (2022)

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## Abstract

In this work, we proposed a new method called conformable fractional double Laplace–Sumudu transform decomposition method (CFDLSTDM) to solve fractional partial differential equations (FPDEs).This method is a combination of the Laplace–Sumudu transform method and the Adomian decomposition method. Besides, we presented some excellent properties and results of conformable double Laplace–Sumudu transform (CDLST). Illustrative examples results are given to show that the CFDLSTDM is an effective and accurate approach for fractional partial differential equations.


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## Body

## 1. Introduction

Recently, fractional calculus has gotten the attention of more and more scientists. It is widely employed in mechanics, mathematics, biomedical engineering, chemical engineering and other scientific and engineering fields. Research studies show that fractional calculus can more accurately reflect some nonlinear memory and effects. A large number of researches on fractional order theory and its applications have emerged. There have been many definitions of fractional derivatives: Riemann Liouville, Gr¨unwald-Letnikov, Riesz, Caputo, and conformable derivative [1–4], and most of the above definitions lack some basic properties of integer order case, but conformable derivative satisfies the classic properties of integer derivative; thus, it is simpler to use it to solve nonlinear fractional partial differential equations, and many applications using this new derivative combining with other technique are reported [5–9] because of its easy applicability. Bhanotar and Kaabar [5] presented the conformable triple Laplace transform decomposition method for obtaining the analytical solutions of the nonlinear fractional partial differential equations. Fractional double Laplace transform technique was proposed by Ozkan and Kurt [6] for obtaining the solution of conformable fractional partial integro-differential equations. The conformable Laplace transform method (CLT) method was [7] employed to solve fractional differential equation with the conformable derivative. Bouaouid et al. [8] solved nonlocal telegraph equation in sense of the conformable time-fractional derivative. Zhour et al. [9] presented some new views about the conformable fractional Sumudu transform and solved some fractional differential equations.Some new integral transform techniques have been reported in Refs. [10–13], in which are presented some definitions and results, and some classes of partial differential equations are solved. Ahmed et al. [10] employed double Laplace–Sumudu transform to solve the integer order integral differential equations. Reference [11] obtained the solution of integer order partial differential equations by new Laplace–Sumudu transform. Khan et al. [12] analyzed some properties and presented some examples about conformable fractional Laplace transform. Khan et al. [13] extended the triple Laplace transform to solve fractional differential equations.The double transform technique has some advantages over other methods, such as it has faster convergence and has no restrictive assumption of the solution [14]; however, this method fails to solve nonlinear partial differential equations; therefore, in this work, we employed the conformal fractional double Laplace–Sumudu transform decomposition method to solve the linear and nonlinear fractional differential equations.We organize the rest of this article as follows. In Section2, the fundamental theories about conformal fractional double Laplace–Sumudu transform are given. In Section 3, the analytical method of general fractional partial differential equations by using conformable fractional double Laplace–Sumudu transform decomposition method (CFDLSTDM) is presented. In Section 4, illustrative examples are provided using the proposed method to validate the obtained results in Section 3. Finally, conclusions are presented in Section 5.


## 2. Basic Definitions and Theorems of Conformal Fractional Double Laplace–Sumudu Transform

In this section, we generalize the double Laplace–Sumudu transform [10, 11] in the sense of integer order derivative to the conformable fractional double Laplace–Sumudu transform case, and some preliminary knowledge and related definitions given in this section will be used later.Definition 1.
Forfx,t,x,t>0 To be a real value function, then the conformable fractional partial derivative of fx,t order 0<α,β≤1 with respect to x,t are denoted as(1)Dxαfx,t=limε⟶0fx+εx1−α,t−fx,tε,x,t>0,Dtβfx,t=limε⟶0fx,t+εt1−β−fx,tε,x,t>0.Definition 2.
Forfx,t,x,t>0 to be a real value function, then the conformable single Laplace transform of fx,t with respect to x is defined as(2)Lxαfx,t=Fαs,t=∫0∞e−sxα/αfx,txα−1dx,0<α≤1.Definition 3.
Within the following set of functions(3)Aβ=fx,t:∃M,τ1,τ2>0,fx,t<Me−tβ/τjβ,iftβ∈−1j×0,∞,j=1,2,then the conformable single fractional Sumudu transform of fx,t with respect to t can be defined as(4)Sβtfx,t=Fβx,u=1u∫0∞e−tβ/uβfx,ttβ−1dt,t>0,provided that the integral exists.Definition 4.
Letfx,t,x,t>0 be a real value function, then the conformable fractional double Laplace–Sumudu transform of the function fx,t can be defined as(5)LxαStβfx,t=Us,u=1u∫0∞∫0∞e−sxα/α−tβ/βufx,txα−1tβ−1dxdt,0<α,β≤1.Definition 5.
Under the condition of Definition 4, then the conformable fractional inverse double Laplace–Sumudu transform can be defined as(6)Lx−1St−1Us,u=fx,t=12πi∫γ−i∞γ+i∞esxα/αds12πi∫ω−i∞ω+i∞1uetβ/βuUs,udu,0<α,β≤1.Theorem 1 (see [12]).
Letfx,t be a real value function that fx,t,Lxαfx,t are continuous with respect to variable x, then the conformable single Laplace transform of ∂mα/∂xmαfx,t with respect to x can be given as(7)Lx′α∂mα∂xmαfx,t=smFαs,t−∑i=0m−1sm−1−iDαixFα0,t.Theorem 2 (see [9]).
Forfx,t to be a n times differentiable real value function with respect to variable t, the conformable single fractional Sumudu transform of ∂nβ/∂tnβfx,t with respect to t can be given as(8)StβDtnβfx,t=Stβfx,tun−fx,0un=Ltβfx,tβ1/β|s=1/uun+1−fx,0un,0<β≤1,n∈N.Theorem 3.
Iffx,t is a given real value function, 0<β≤1, then based on the idea in Ref. [9], we get(9)Stβfx,t=1uLtβfx,tβ1/β|s=1/u,Stβfx,t=1uFβx,1u.Theorem 4.
Let0<α,β≤1, then we have the conformable double Laplace–Sumudu transform for some special functions:(a)
LxαStβC=C/s,Cis a constant(b)
LxαStβxα/αmtβ/βn=m!n!/sm+1un,m,n∈N(c)
LxαStβecxα/α+dtβ/β=1/s−c1−du,s>c,u>1/d(d)
LxαStβsindcxα/αcosdtβ/β=c/s2+c21+d2u2,s>0,u>1/d(e)
LxαStβsinhcxα/αcoshdtβ/β=c/s2−c21−d2u2,s>c,u>1/d(f)
LxαStβxptq=αp/αβq/βuq/β/s1+p/αΓ1+p/αΓ1+q/β,p/α,q/β>−1Proof.
(a)
Using the Proposition 1 in Ref. [5] and Theorem 3 in Ref. [9], we have(10)LxαStβc=Lxα1Stβc=1sc=cs.(b)
Using the Proposition 1 in Ref. [5] and Theorem 3 in Ref [9], we get(11)LxαStβxααmtββn=LxαxααmStβtββn=m!sm+1n!un.
Similarly, we can prove (c)–(f).Theorem 5.
IfLxαStβf1x,t=U1s,u and LxαStβf2x,t=U2s,u exist, then we have the following convolution theorem of conformable fractional double Laplace–Sumudu transform:(12)LxαStβf1x,t∗∗f2x,t=uU1s,uU2s,u.Proof.
By using the definition of conformable fractional double Laplace–Sumudu transform and Theorem 6 [10], we get(13)LxαStβf1x,t∗∗f2x,t=1u∫0∞∫0∞e−sxα/α−tβ/βuf1x,t∗∗f2x,txα−1tβ−1dxdt=1u∫0∞∫0∞e−sxα/α−tβ/βu∫0x∫0tf1x−δ,t−εf2δ,εxα−1tβ−1dxdt=∫0∞∫0∞f2δ,εdδdε1u∫0∞∫0∞e−sxα/α−tβ/βuf1x−δ,t−εHx−δ,t−εxα−1tβ−1dxdt=∫0∞∫0∞f2δ,ε1u∫0∞∫0∞e−sδ−ε/uU1s,udδdε=U1s,u1u∫0∞∫0∞e−sδ−ε/uf2δ,εdδdε=uU1s,uU2s,u.Theorem 6.
If0<α,β≤1, then the conformable fractional double Laplace–Sumudu transform of fractional partial derivative can be given as(14)LxαStβ∂α∂xαfx,t=sUs,u−Stβf0,t,(15)LxαStβ∂β∂tβfx,t=1uUs,u−1uLxαfx,0.Proof.
(14)
Using the Definition4, we have(16)LxαStβ∂α∂xαfx,t=1u∫0∞∫0∞e−sxα/α∂∂xfx,tx1−αxα−1e−tβ/βutβ−1dxdt,=1u∫0∞∫0∞e−sxα/α∂∂xfx,te−tβ/βutβ−1dxdt.
Then by integrating by part, we have(17)=1u∫0∞∫0∞e−sxα/α∂∂xfx,te−tβ/βutβ−1dxdt,=−1u∫0∞f0,te−xβ/βutβ−1dt+su∫0∞∫0∞e−sxα/αe−tβ/βufx,txα−1tβ−1dt,=sUs,u−Stβf0,t.
Similarly, the proof of (15) can be completed.Theorem 7.
If0<α,β≤1, then the conformable fractional double Laplace–Sumudu transform for higher order fractional partial derivatives can be given as(18)LxαStβ∂mα∂xmαfx,t=smUs,u−∑i=0m−1sm−1−iStβ∂iα∂xiαf0,t,LxαStβ∂nβ∂tnβfx,t=1unUs,u−∑j=0n−11un−jLxα∂jβ∂tjβfx,0.Proof.
Follows by using Theorem 6 and then using the induction process onn.


## 3. Solving Nonlinear Partial Fractional Differential Equation Using the Conformable Double Laplace–Sumudu Transform Decomposition Method

Now, the following general nonhomogeneous conformable fractional partial fractional differential equations are to be solved:(19)∂mα∂xmαfx,t+∂nβ∂tnβfx,t+Rfx,t+Nfx,t=gx,t,where 0<α,β≤1,m,n∈N. R,N denote linear and nonlinear fractional differential operators, respectively, and gx,t is the source term.Subjecting to the following initial and boundary conditions, we get(20)∂jβ∂tjβfx,0=hjx,0,j=0,1,2,…,n−1,∂iα∂xiαf0,t=Hix,0,i=0,1,2,…,m−1.Now, we need the following steps to solve equation (19).Step 1.
Operating the conformable fractional double Laplace–Sumudu transform from equation (19) on both sides, we get(21)LxαLtβ∂mα∂xmαfx,t+LxαLtβ∂nβ∂tnβfx,t+LxαLtβRfx,t+LxαLtβNfx,t=LxαLtβgx,t.
Applying Theorems6 and 7 and initial and boundary conditions in equation (20) yields:(22)smUs,u−∑i=0m−1sm−1−iStβ∂iα∂xiαf0,t+1unUs,u−∑j=0n−11un−jLxα∂jβ∂tjβfx,0+LxαStβRfx,t+LxαStβNfx,t=LxαStβgx,t.Step 2.
Simplify it and then operating the conformable fractional inverse double Laplace–Sumudu transform, we get(23)fx,t=Lx−1St−11sm+u−n∑i=0m−1sm−1−i∂iα∂xiαf0,t+1sm+u−n∑j=0n−11un−jLxα∂jβ∂tjβfx,0+1sm+u−nLxαStβgx,t−Lx−1St−11sm+u−nLxαStβRfx,t+1sm+u−nLxαStβNfx,t=Gx,t−Lx−1St−11sm+u−nLxαStβRfx,t+1sm+u−nLxαStβNfx,t,where Gx,t is the term originating from the source term and initial or boundary conditions terms.Step 3.
For the conformal fractional double Laplace–Sumudu transform decomposition method, we suppose the solution of equation (19) be a form of infinite series:(24)fx,t=∑l=0∞flx,t.
Then the nonlinear term can be decomposed as the following:(25)Nfx,t=∑l=0∞Al,where Al is the Adomian polynomials of f1x,t,f2x,t,…,flx,t, and it can be expressed by(26)Al=1l!∂l∂σlN∑i=1∞σiuiσ=0,l=0,1,2,….
Substituting (25) and (26) in (24), we get(27)∑l=0∞flx,t=Gx,t−Lx−1St−11sm+u−nLxαStβR∑l=0∞flx,t+1sm+u−nLxαStβN∑l=0∞Al,Step 4.
Comparing on both sides of equation (27), we get(28)f0x,t=Gx,t,f1x,t=−Lx−1St−11sm+u−nLxαStβRf0x,t+1sm+u−nLxαStβNA0,f2x,t=−Lx−1St−11sm+u−nLxαStβRf1x,t+1sm+u−nLxαStβNA1,⋮fl+1x,t=−Lx−1St−11sm+u−nLxαStβRflx,t+1sm+u−nLxαStβNAl,where m,n=1,2,3,…,l=0,1,2,3,….
At the end, the approximate analytical solution of equation (19) can be(29)fx,t=∑l=0∞flx,t.


## 4. Illustrative Examples

Here, two examples called homogeneous and nonhomogeneous fractional partial differential equations are solved by employing the proposed conformable fractional double Laplace–Sumudu decomposition method.Example 1.
Consider the following nonhomogeneous space-time conformable fractional partial differential equation [15]:(30)∂2α∂x2αfx,t−∂β∂tβfx,t=−cosxαα,
With the initial and boundary conditions, we get(31)fx,0=0,f0,t=1−e−tβ/β,∂α∂xαf0,t=0.
Operating on the conformable fractional double Laplace–Sumudu from (25), and then using the property of partial derivative of the conformable double Laplace–Sumudu transform to (25), and then applying the initial and boundary conditions (26), we get(32)s2Us,u−∑i=01s1−iStβ∂iα∂xiαf0,t+1uUs,u−1uLxαfx,0+LxαStβ0=LxαStβ−cosxαα.
Simplifying (32) and taking the inverse of the conformable fractional double Laplace–Sumudu on both sides of equation (27) and using Theorem 4, we get(33)f0x,t=Gx,t=Lx−1St−1us1+u1+s2=cosxαα1−e−tβ/β,fl+1x,t=−Lx−1St−11sm+u−nLxαStβRflx,t+1sm+u−nLxαStβNAl=0,l=0,1,….
Therefore, the approximate analytic solution of (30) is(34)fx,t=cosxαα1−e−tβ/β.
This solution is exactly the same as the solution obtained in [15] by using the conformable double Laplace transform method.Example 2.
We consider the following nonlinear fractional equation in the sense of conformable fractional derivative:(35)fx,t∂α∂xαfx,t+∂β∂tβfx,t=0,0<α,β≤1,x≥0,t>0,subject to the following initial and boundary conditions:(36)fx,0=sinxαα,f0,t=tββ.
First, applying the conformable fractional double Laplace–Sumudu on both sides of (35), and using the property of partial derivative of the conformable double Laplace–Sumudu transform, and lastly, applying initial and boundary conditions (36), we get(37)fx,t=Lx−1St−11u−11uLxαsinxαα−Lx−1St−11u−1LxαStβfx,t∂α∂xαfx,t=sinxαα−Lx−1St−11u−1LxαStβfx,t∂α∂xαfx,t,f0x,t=sinxαα,∑l=0∞flx,t=Lx−1St−11u−11uLxαsinxαα−Lx−1St−11u−1LxαStβ∑l=0∞flx,t∂α∂xα∑l=0∞flx,t=f0x,t−Lx−1St−11u−1LxαStβ∑l=0∞flx,t∂α∂xα∑l=0∞flx,t,and the recursive relation is given by(38)fl+1x,t=−Lx−1St−1uLxαStβAl,l=0,1,2,…,where Al is the Adomian polynomial, and it is expressed by(39)Al=1l!∂l∂σlN∑i=1∞σiuiσ=0,l=0,1,2,….
Suppose the nonlinear term is given as(40)Ffl=fl∂αfl∂xα.
Substituting (40) in equations (38) and (39) can be rewritten as follows:
Forn=0, we get(41)A0=Ff0=f0∂αf0∂xασ=0=sinxααcosxαα.
Thus, substituting (41) into (38), we get(42)f1x,t=−Lx−1St−11u−1LxαStβA0=−12tββsin2xαα.
Forn=1, we get(43)A1=11!∂∂σFf0+σf1σ=0=∂∂σf0+σf1∂α∂xαf0+σf1σ=0=f1∂αf0∂xα+f0∂αf1∂xα=−tββ12sin2xααcosxαα+sinxααcos2xαα.
Thus, substituting (43) into equation (38), we get(44)f2x,t=−Lx−1St−1uLxαStβA1=t2β2β212sin2xααcosxαα+sinxααcos2xαα.
Forn=2, we get(45)A2=12!∂2∂σ2Ff0+σf1+σ2f2σ=0=12!∂2∂σ2f0+σf1+σ2f2∂α∂xαf0+σf1+σ2f2σ=0=f0∂αf2∂xα+f1∂αf1∂xα+f2∂αf0∂xα=t2β4β2sin4xαα−5sin2xααsin2xαα.
Therefore, substituting (45) into (38) yields(46)f3x,t=−Lx−1St−1uLxαStβA2=−t3β12β3sin4xαα−5sin2xααsin2xαα,and so on….
Thus, the third term approximate analytic solution of (35) is given by(47)fx,t=sinxαα−tβ2βsin2xαα+t2β2β212sin2xααcosxαα+sinxααcos2xαα−t3β12β3sin4xαα−5sin2xααsin2xαα.


## 5. Conclusion

In this work, the conformable fractional double Laplace–Sumudu transform decomposition method for solving linear and nonlinear conformable fractional partial differential equations is used. We presented the related theorems and excellent properties of the new proposed conformable fractional double integral transform, and some examples are given. The examples show that the new Laplace–Sumudu transform technique is effective in solving the linear and nonlinear fractional partial differential equations.



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