# Three-Dimensional Laplace Adomian Decomposition Method and Singular Pseudoparabolic Equations

Authors: Hassan Eltayeb; Diaa Eldin Elgezouli; Adem Kilicman; Imed Bachar
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this work, the solution of the linear, nonlinear, and coupled system fractional singular two-dimensional pseudoparabolic equation is examined by using a three-dimensional Laplace Adomian decomposition method (3-DLADM). Analysis of the method is discussed, and some demonstrative examples are mentioned to confirm the power and accuracy of the recommended method, and numerical analysis is applied to sketch the exact and approximate solution.


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## Body

## 1. Introduction

The fractional derivative has been attracting much attention in physical and engineering problems, for instance, acoustics, viscoelasticity, and control. The sufficient condition for commutators of a fractional integral operator is discussed in [1]. The authors in [2] addressed the boundedness of commutators of the multidimensional Hardy-type operators with bounded mean oscillation coefficients. The fractional derivative of the Riemann zeta function was computed, using the Caputo derivative in the Ortigueira sense (for more details, see [3]). The author in [4] analyzed the fractional derivative of the Riemann function and discussed the functional equation with the distribution of prime numbers.The parabolic equation occurred in several fields of applied mathematics, for example, the heat diffusion equation and fluid mechanics (for a model, see [5–8]). The solution fractional diffusion equation problems have been obtained via the Adomian decomposition method and series expansion method by the authors [9, 10]. Several articles have been found in previous studies, which are associated with qualities and applications of a fractional derivative [11–13]. The pseudoparabolic equation represents a diversity of physical operation. The author in [14] discussed the existence, uniqueness, and continuous dependence of powerful solutions of the one-dimensional pseudoparabolic equation. Overall, certainty of the nonlinear equations of real life is so far very hard to solve either theoretically or numerically. Currently, many researchers have suggested an exact solution to a one-dimensional coupled parabolic equation (for more details, see [15, 16]). The convergence of the Adomian method was discussed by many researchers (we refer the readers to see [17–20]). The author has modified the 2-D Laplace decomposition method to solve coupled pseudoparabolic equations in order to accelerate the convergence of the series solution [21].Recently, the three-dimensional Laplace Adomian decomposition (3-DLADM) has been successfully applied to solve regular and singular coupled Burgers’ equations (see [22]). The objective of this research is to find the solution of singular 2-D fractional pseudoparabolic equations by applying a more successful technique, which is called 3-DLADM.


## 2. Some Basic Idea of the Double and Triple Laplace Transform and Caputo Fractional Derivative

In this unit, we offer basic definitions, properties of fractional calculus, Mittag-Leffler function, and Laplace transform theory which should be used in this work.Definition 1 (see [23]).
The left-sided Caputo fractional derivative ofg,g∈C−1m,m∈N∪0, is given by
(1)D∗μgt=∂μgt∂tμ=Im−μ∂mgt∂tm,m−1<μ<m,m∈N,∂mgt∂tm,μ=m.Definition 2 (see [24]).
The Caputo fractional partial derivative of functionfx1,t with respect to t is given by
(2)∂ζfx1,t∂tζ=1Γn−ζ0tt−νn−ζ−1∂nfx1,ν∂νndν,n−1<ζ<n.Definition 3 (see [25]).
Letgx1,x2 be a continuous function of two variables x1,x2; then, the two-dimensional Laplace transform (2-DLT) of gx1,x2 is given
(3)L2gx1,x2=Gp1,p2=∫0∞∫0∞e−p1x1−p2x2gx1,x2dx1dx2,where x1,x2>0, L2 indicate 2-DLT, and p1,p2 are complex variables.Definition 4 (see [26]).
Letgx1,x2,t be a piecewise continuous function on the interval 0,∞×0,∞×0,∞ of exponential order. Consider for some a,b,c∈ℝsupx1,x2,t>0,ψx1,x2,t/eax1+bx2+ct. Under these conditions, 3-DLT is defined by
(4)L3gx1,x2,t=Gp1,p2,s=∫0∞∫0∞∫0∞e−p1x1−p2x2−stgx1,x2,tdtdx2dx1,where the symbol L3 indicates 3-DLT and p1,p2,s∈ℂ.Definition 5.
The inverse 3-DLT of the functionFp1,p2,s is determined by
(5)L3−1Gp1,p2,s=gx1,x2,t=12πi∫a−i∞a+i∞ep1x1dp112πi∫c−i∞c+i∞ep2x2dp212πi∫d−i∞d+i∞estGp1,p2,sds,where L3−1 indicate inverse 3-DLT with respect to p1,p2 and s.Furthermore 3-DLT of the derivativesψx1x1,x2,t and ψtx1,x2,t are presented by
(6)L3ψx1x1,x2,t=p1Ψp1,p2,s−Ψ0,p2,s,(7)L3ψtx1,x2,t=sψp1,p2,s−Ψp1,p2,0.The 2-DLT formulas for the partial fractional Caputo derivatives are denoted by(8)L2∂βfx1,t∂xζ=p1βFp1,s−∑i=0n−1p1β−1−iLt∂i∂x1iψ0,t,(9)L2∂ζfx1,t∂tζ=sζFp1,s−∑i=0n−1sζ−1−iLx1∂i∂tiψx1,0,where ζ,β>0,m−1<ζ≤m,r−1<β≤r,r,m∈N (for more details, see [27]).In the following, we provided one and two parameters; the classical Mittag-Leffler function is useful in this work.


### 2.1. Mittag-Leffler Function

The Mittag-Leffler function of one parameter is established by(10)Ξητ=∑i=0∞τiΓηi+1,τ∈ℂ,Reη>0.The Mittag-Leffler function with two parameters is determined by(11)Ξη,γτ=∑i=0∞τiΓηi+γ,τ∈ℂ,Reη>0(see [28, 29]). If we set η=1 in Equation (11), we get Equation (10). It appears from Equation (11) that
(12)Ξ1,1τ=∑i=0∞τiΓi+1=∑i=0∞τii!=eτ,(13)Ξ1,2τ=∑k=0∞τiΓi+2=∑i=0∞τii+1!=1τ∑k=0∞τi+1i+1=eτ−1τ,(14)Ξ1,3τ=∑i=0∞τiΓi+3=∑i=0∞τii+2!=1τ2∑i=0∞τi+2i+2=eτ−1−1τ2;hence, in general,
(15)Ξ1,mτ=1τm−1eτ−∑i=0m−2τii!.Differentiation of the Mittag-Leffler function is represented by(16)dndtntη−1Ξη,ζtζ=tη−n−1Ξη,ζ−ntη(for more details, see [30]).Next, we provide the Laplace transform (LT) of Mittag-Leffler functions helpful in this research:(17)L3tζΞ1,ζ+1t=1sζs−1,L3tζΞ1,ζ+1t=1sζs−1,L3t2ζΞ1,2ζ+1t=1s2ζs−1,L3tζ−1Ξ1,ζt=1sζ−1s−1,L3t2ζ−1Ξ1,2ζλt=1s2ζ−1s−λ.In the same way, the single Laplace transform (LT) of two-parameter Mittag-Leffler functions(18)L3tη−1Ξζ,η±λtζ=sζ−ηsζ−λ.


## 2.1. Mittag-Leffler Function

The Mittag-Leffler function of one parameter is established by(10)Ξητ=∑i=0∞τiΓηi+1,τ∈ℂ,Reη>0.The Mittag-Leffler function with two parameters is determined by(11)Ξη,γτ=∑i=0∞τiΓηi+γ,τ∈ℂ,Reη>0(see [28, 29]). If we set η=1 in Equation (11), we get Equation (10). It appears from Equation (11) that
(12)Ξ1,1τ=∑i=0∞τiΓi+1=∑i=0∞τii!=eτ,(13)Ξ1,2τ=∑k=0∞τiΓi+2=∑i=0∞τii+1!=1τ∑k=0∞τi+1i+1=eτ−1τ,(14)Ξ1,3τ=∑i=0∞τiΓi+3=∑i=0∞τii+2!=1τ2∑i=0∞τi+2i+2=eτ−1−1τ2;hence, in general,
(15)Ξ1,mτ=1τm−1eτ−∑i=0m−2τii!.Differentiation of the Mittag-Leffler function is represented by(16)dndtntη−1Ξη,ζtζ=tη−n−1Ξη,ζ−ntη(for more details, see [30]).Next, we provide the Laplace transform (LT) of Mittag-Leffler functions helpful in this research:(17)L3tζΞ1,ζ+1t=1sζs−1,L3tζΞ1,ζ+1t=1sζs−1,L3t2ζΞ1,2ζ+1t=1s2ζs−1,L3tζ−1Ξ1,ζt=1sζ−1s−1,L3t2ζ−1Ξ1,2ζλt=1s2ζ−1s−λ.In the same way, the single Laplace transform (LT) of two-parameter Mittag-Leffler functions(18)L3tη−1Ξζ,η±λtζ=sζ−ηsζ−λ.


## 3. Multidimensional Laplace Transforms (n+1-DLT)

Here, we deal with the multidimensional Laplace transform (n+1-DLT) which is very useful to this work.Definition 6.
Letgx1,x2,x3,⋯,xn,t be a piecewise continuous function on the interval 0,∞×0,∞×0,∞×⋯×0,∞×0,∞ of exponential order. Consider for some a1,a2,a3,⋯,an,b∈ℝsupx1,x2,x3,⋯,xn,t>0,ψx1,x2,x3,⋯,xn,t/ea1x1+a2x2+⋯+anxn+bt. Under these conditions, n+1-DLT is defined by
(19)LnLtξ=Gp1,p2,p3,⋯,pn,s=∫0∞⋯∫0∞∫0∞e−p1x1−p2x2⋯−pnxn−stξdx1dx2⋯dxndt,where ξ=gx1,x2,x3,⋯,xn,t, the symbol LnLt indicate n+1-DLT, and p1,p2,p3,⋯,pn ands are complex variables.Definition 7.
The inversen+1-DLT of the function Gp1,p2,p3,⋯,pn,s is determined by
(20)Ln−1Ls−1F=gx1,x2,x3,⋯,xn,t∣=12πi∫c1−i∞c1+i∞ep1x1dp1⋯12πi∫cn−i∞cn+i∞epnxndpn12πi∫d−i∞d+i∞estFds,where F=Gp1,p2,p3,⋯,pn,s and Ln−1Ls−1 indicate inverse n+1-DLT with respect to p1,p2,p3,⋯,pn and s.


### 3.1. Existence Condition for the Multi-Laplace Transform

Iffx1,x2,x3,⋯,xn,t is said to be of exponential order a1,a2,a3,⋯,an (>0) and b (>0) on 0≤x1,x2,x3,⋯,xn<∞, 0≤t<∞, if there exists a positive constant K such that for all x1>X1,x2>X2,X3>x3,⋯,xn>Xn and t>T:
(21)fx1,x2,x3,⋯,xn,t≤Kea1x1+a2x2+a3x3+⋯anxn+bt,where fx1,x2,x3,⋯,xn,t can be written as
(22)fx1,x2,x3,⋯,xn,t=Oea1x1+a2x2+a3x3+⋯anxn+bt,at x1,x2,x3,⋯,xn⟶∞ and t⟶∞. Or, equivalently,
(23)limx1,x2,x3,⋯,xn→∞t→∞e−ζ1x1−ζ2x2−ζ3x3−⋯−ζnxn−βtfx11,x12,x13,⋯,x1n,t=klimx1,x2,x3,⋯,xn→∞t→∞e−ζ1−a1x1−ζ2−a2x2−ζ3−a3x3−⋯−ζn−anxn−β−bt=0,ζ1>a1,ζ2>a2,ζ3>a3⋯ζn>an,β>b,where the functionfx1,x2,x3,⋯,xn,t is simply called an exponential order as x1,x2,x3,⋯,xn⟶∞, t⟶∞, and clearly, it does not grow faster than Kea1x1+a2x2+a3x3+⋯anxn+bt as x1,x2,x3,⋯,xn⟶∞, t⟶∞.Theorem 8.
If a functionfx1,x2,x3,⋯,xn,t is a continuous function in every finite intervals 0,X1,0,X2,0,X3,⋯,0,Xn and 0,T and of exponential order ea1x1+a2x2+a3x3+⋯anxn+bt, then the double Laplace transform of fx1,x2,x3,⋯,xn,t exists for all p1,p2,p3,⋯,pn and s provided Rep1>a1,Rep2>a2,Rep3>a3,⋯,Repn>an and Res>b.Proof.
We have(24)Fp1,p2,p3,⋯,pn,s=∫0∞⋯∫0∞∫0∞e−p1x1−p2x2−⋯−pnxn−stfx1,x2,x3,⋯,xn,tdx1dx2⋯dxndt≤∫0∞⋯∫0∞e−p1−a1x1−p2−a2x2−p3−a3x3−⋯−pn−anxn−s−btdx1dx2⋯dxndt=Kp1−a1p2−a2p3−a3⋯pn−ans−b.
ForRep1>a1,Rep2>a2,Rep3>a3,⋯,Repn>an and Res>b.
In the following theorem, we generalize Equations (8) and (9) to a multidimensional Laplace transform.Theorem 9.
Letζ,β>0,m−1<ζ≤m,r−1<β≤r,r,m∈N, be so that ψ∈Clℝ+×ℝ+×⋯×ℝ+, l=maxm,r,ψl∈L10,a1×0,a2×0,a3×⋯×0,b, for any a1,a2,a3,⋯,an (>0) and b (>0):
(25)ψx1,x2,x3,⋯,xn,t≤Kea1x1+a2x2+a3x3+⋯anxn+bt,x1>a1>0,x2>a2>0,⋯,t>b>0,hold for any constant k,τ1,τ2,τ3⋯τn≥0, then
(26)LnLtDtζψx1,x2,x3,⋯,xn,t=sζΨp1,p2,p3,⋯,pn,s−∑i=0m−1sζ−1−iLn∂iψx1,x2,x3,⋯,xn,0∂ti,LnLtDx1βψx1,x2,x3,⋯,xn,t=p1βΨp1,p2,p3,⋯,pn,s−∑i=0r−1p1β−1−iLn−1Lt∂iψ0,x2,x3,⋯,xr,t∂x1i,LnLtDx2βψx1,x2,x3,⋯,xn,t=p2βΨp1,p2,p3,⋯,pn,s−∑i=0r−1p2β−1−iLn−1Lt∂iψx1,0,x3,⋯,xr,t∂x2i,(27)LnLtDxβψx1,x2,x3,⋯,xn,t=p3βΨp1,p2,p3,⋯,pn,s−∑i=0r−1p3β−1−iLn−1Lt∂iψx1,x2,0,⋯,xr,t∂x2i,where Ψp1,p2,p3,⋯,pn,s is n+1-DLT of ψx1,x2,x3,⋯,xn,t.Proof.
With the use of Theorem9 and definition of multidimensional Laplace transform, we can deduce the proof easily.Theorem 10.
Letf be piecewise continuous on 0,∞×0,∞×0,∞×⋯×0,∞×0,∞,n+1-DLT of the partial derivatives of order ζth ∏i=1nxi∂ζψ/∂tζ and ∏i=1nxifx1,x2,x3,⋯,xn,t are given by
(28)LnLt∏i=1nxi∂ζ∂tζψx1,x2,x3,⋯,xn,t=−1n∂n∂p1∂p2∂p3⋯∂pnΛ,where
(29)Λ=sζΨp1,p2,p3,⋯,pn,s∑i=0n−1sζ−1−iLn∂i∂tiψx1,x2,x3,⋯,xn,0,(30)LnLt∏i=1nxifxx1,x2,x3,⋯,xn,t=−1n∂n∂p1∂p2∂p3⋯∂pnFp1,p2,p3,⋯,pn,s.Proof.
By employing definition ofn-DLT for ∂ζψ/∂tζ, we get
(31)LnLt∂ζψ∂tζ=∫0∞⋯∫0∞∫0∞e−p1x1−p2x2⋯−pnxn−st∂ζψ∂tζx1,x2,x3,⋯,xn,tdx11dx12⋯dx1ndt;by taking partial derivatives ∂n/∂p1∂p2∂p3⋯∂pn for both sides of Equation (31), we have
(32)∂n∂p1∂p2∂p3⋯∂pnLnLt∂ζψ∂tζ=∫0∞e−st∂ζψ∂tζ∫0∞⋯∫0∞∂n∂p1∂p2∂p3⋯∂pne−p1x1−p2x2⋯−pnxndx1dx2⋯dxndt.
The integral inside bracket is determined by(33)∫0∞⋯∫0∞∂n∂p1∂p2∂p3⋯∂pne−p1x1−p2x2⋯−pnxndx1dx2⋯dxn=−1n∫0∞⋯∫0∞Δdx1dx2⋯dxn,where Δ=∏i=1nx1ie−p1x11−p2x12⋯−pnx1n; hence, we find that
(34)∂n∂p1∂p2∂p3⋯∂pnLnLt∂ζψ∂tζ=−1n∫0∞⋯∫0∞∫0∞i=1nx1ie−p1x1−p2x2⋯−pnxn−st∂ζψ∂tζdx1dx2⋯dxndt=−1nLnLt∏i=1nx1i∂ζ∂tζψx1,x2,x3,⋯,xn,t;by substituting Equations (33) and (34) into Equation (34) and using Theorem 9, we achieved
(35)LnLt∏i=1nx1i∂ζ∂tζψx1,x2,x3,⋯,xn,t=−1n∂n∂p1∂p2∂p3⋯∂pnΛ.
Similarly, we can obtain Equation (30).
In particular, atn=2, we have
(36)L2Ltx1x2∂ζψ∂tζ=∂2∂p1∂p2sζΨp1,p2,s−sζ−1Ψp1,p2,0,(37)L2Ltx1x2fx1,x2,t=∂2∂p1∂p2L2Ltfx1,x2,t.


## 3.1. Existence Condition for the Multi-Laplace Transform

Iffx1,x2,x3,⋯,xn,t is said to be of exponential order a1,a2,a3,⋯,an (>0) and b (>0) on 0≤x1,x2,x3,⋯,xn<∞, 0≤t<∞, if there exists a positive constant K such that for all x1>X1,x2>X2,X3>x3,⋯,xn>Xn and t>T:
(21)fx1,x2,x3,⋯,xn,t≤Kea1x1+a2x2+a3x3+⋯anxn+bt,where fx1,x2,x3,⋯,xn,t can be written as
(22)fx1,x2,x3,⋯,xn,t=Oea1x1+a2x2+a3x3+⋯anxn+bt,at x1,x2,x3,⋯,xn⟶∞ and t⟶∞. Or, equivalently,
(23)limx1,x2,x3,⋯,xn→∞t→∞e−ζ1x1−ζ2x2−ζ3x3−⋯−ζnxn−βtfx11,x12,x13,⋯,x1n,t=klimx1,x2,x3,⋯,xn→∞t→∞e−ζ1−a1x1−ζ2−a2x2−ζ3−a3x3−⋯−ζn−anxn−β−bt=0,ζ1>a1,ζ2>a2,ζ3>a3⋯ζn>an,β>b,where the functionfx1,x2,x3,⋯,xn,t is simply called an exponential order as x1,x2,x3,⋯,xn⟶∞, t⟶∞, and clearly, it does not grow faster than Kea1x1+a2x2+a3x3+⋯anxn+bt as x1,x2,x3,⋯,xn⟶∞, t⟶∞.Theorem 8.
If a functionfx1,x2,x3,⋯,xn,t is a continuous function in every finite intervals 0,X1,0,X2,0,X3,⋯,0,Xn and 0,T and of exponential order ea1x1+a2x2+a3x3+⋯anxn+bt, then the double Laplace transform of fx1,x2,x3,⋯,xn,t exists for all p1,p2,p3,⋯,pn and s provided Rep1>a1,Rep2>a2,Rep3>a3,⋯,Repn>an and Res>b.Proof.
We have(24)Fp1,p2,p3,⋯,pn,s=∫0∞⋯∫0∞∫0∞e−p1x1−p2x2−⋯−pnxn−stfx1,x2,x3,⋯,xn,tdx1dx2⋯dxndt≤∫0∞⋯∫0∞e−p1−a1x1−p2−a2x2−p3−a3x3−⋯−pn−anxn−s−btdx1dx2⋯dxndt=Kp1−a1p2−a2p3−a3⋯pn−ans−b.
ForRep1>a1,Rep2>a2,Rep3>a3,⋯,Repn>an and Res>b.
In the following theorem, we generalize Equations (8) and (9) to a multidimensional Laplace transform.Theorem 9.
Letζ,β>0,m−1<ζ≤m,r−1<β≤r,r,m∈N, be so that ψ∈Clℝ+×ℝ+×⋯×ℝ+, l=maxm,r,ψl∈L10,a1×0,a2×0,a3×⋯×0,b, for any a1,a2,a3,⋯,an (>0) and b (>0):
(25)ψx1,x2,x3,⋯,xn,t≤Kea1x1+a2x2+a3x3+⋯anxn+bt,x1>a1>0,x2>a2>0,⋯,t>b>0,hold for any constant k,τ1,τ2,τ3⋯τn≥0, then
(26)LnLtDtζψx1,x2,x3,⋯,xn,t=sζΨp1,p2,p3,⋯,pn,s−∑i=0m−1sζ−1−iLn∂iψx1,x2,x3,⋯,xn,0∂ti,LnLtDx1βψx1,x2,x3,⋯,xn,t=p1βΨp1,p2,p3,⋯,pn,s−∑i=0r−1p1β−1−iLn−1Lt∂iψ0,x2,x3,⋯,xr,t∂x1i,LnLtDx2βψx1,x2,x3,⋯,xn,t=p2βΨp1,p2,p3,⋯,pn,s−∑i=0r−1p2β−1−iLn−1Lt∂iψx1,0,x3,⋯,xr,t∂x2i,(27)LnLtDxβψx1,x2,x3,⋯,xn,t=p3βΨp1,p2,p3,⋯,pn,s−∑i=0r−1p3β−1−iLn−1Lt∂iψx1,x2,0,⋯,xr,t∂x2i,where Ψp1,p2,p3,⋯,pn,s is n+1-DLT of ψx1,x2,x3,⋯,xn,t.Proof.
With the use of Theorem9 and definition of multidimensional Laplace transform, we can deduce the proof easily.Theorem 10.
Letf be piecewise continuous on 0,∞×0,∞×0,∞×⋯×0,∞×0,∞,n+1-DLT of the partial derivatives of order ζth ∏i=1nxi∂ζψ/∂tζ and ∏i=1nxifx1,x2,x3,⋯,xn,t are given by
(28)LnLt∏i=1nxi∂ζ∂tζψx1,x2,x3,⋯,xn,t=−1n∂n∂p1∂p2∂p3⋯∂pnΛ,where
(29)Λ=sζΨp1,p2,p3,⋯,pn,s∑i=0n−1sζ−1−iLn∂i∂tiψx1,x2,x3,⋯,xn,0,(30)LnLt∏i=1nxifxx1,x2,x3,⋯,xn,t=−1n∂n∂p1∂p2∂p3⋯∂pnFp1,p2,p3,⋯,pn,s.Proof.
By employing definition ofn-DLT for ∂ζψ/∂tζ, we get
(31)LnLt∂ζψ∂tζ=∫0∞⋯∫0∞∫0∞e−p1x1−p2x2⋯−pnxn−st∂ζψ∂tζx1,x2,x3,⋯,xn,tdx11dx12⋯dx1ndt;by taking partial derivatives ∂n/∂p1∂p2∂p3⋯∂pn for both sides of Equation (31), we have
(32)∂n∂p1∂p2∂p3⋯∂pnLnLt∂ζψ∂tζ=∫0∞e−st∂ζψ∂tζ∫0∞⋯∫0∞∂n∂p1∂p2∂p3⋯∂pne−p1x1−p2x2⋯−pnxndx1dx2⋯dxndt.
The integral inside bracket is determined by(33)∫0∞⋯∫0∞∂n∂p1∂p2∂p3⋯∂pne−p1x1−p2x2⋯−pnxndx1dx2⋯dxn=−1n∫0∞⋯∫0∞Δdx1dx2⋯dxn,where Δ=∏i=1nx1ie−p1x11−p2x12⋯−pnx1n; hence, we find that
(34)∂n∂p1∂p2∂p3⋯∂pnLnLt∂ζψ∂tζ=−1n∫0∞⋯∫0∞∫0∞i=1nx1ie−p1x1−p2x2⋯−pnxn−st∂ζψ∂tζdx1dx2⋯dxndt=−1nLnLt∏i=1nx1i∂ζ∂tζψx1,x2,x3,⋯,xn,t;by substituting Equations (33) and (34) into Equation (34) and using Theorem 9, we achieved
(35)LnLt∏i=1nx1i∂ζ∂tζψx1,x2,x3,⋯,xn,t=−1n∂n∂p1∂p2∂p3⋯∂pnΛ.
Similarly, we can obtain Equation (30).
In particular, atn=2, we have
(36)L2Ltx1x2∂ζψ∂tζ=∂2∂p1∂p2sζΨp1,p2,s−sζ−1Ψp1,p2,0,(37)L2Ltx1x2fx1,x2,t=∂2∂p1∂p2L2Ltfx1,x2,t.


## 4. Singular2+1-D Fractional Pseudoparabolic Equation and 3-DLADM

The following is the procedure demonstrating two problems that are related to the linear and nonlinear singular2+1-D pseudoparabolic equation:Problem 11.
The 3-DLADM is an effective technique for solving linear singular two-dimensional pseudoparabolic equation.
We consider0<ζ≤1; let us consider a general fractional singular 2+1-D pseudoparabolic equation:
(38)Dtζψ−1x1x1∂∂x1ψx1−1x2x2∂∂x2ψx2−1x1x1ψx1x1t=fx1,x2,t,(39)subjecttoψx1,x2,0=f1x1,x2,where the linear parts 1/x1x1∂/∂x1x1 and 1/x2x2∂/∂x2x2 are called Bessel’s operator and fx1,x2,t, f1x1,x2 are known functions. For the objective to solve Equation (38), we are implementing the following steps.Step 1.
By multiplying the two sides of Equation (38) by x1x2(40)x1x2Dtζψ−x2x1∂∂x1ψx1−x1x2∂∂x2ψx2−x2x1∂∂x1ψx1t=x1x2fx1,x2,t.Step 2.
By implementing Equations (36), (37), and (7) for the equation in the first step and 2-DLT for condition, we get
(41)∂2∂p1∂p2Ψp1,p2,s=1s∂2∂p1∂p2Lx1Lx2f1x1,x2+1sζ∂2∂p1∂p2L3fx1,x2,t+1sζL3x2x1∂∂x1ψx1+x1x2∂∂x2ψx2+x2x1∂∂x1ψx1t.Step 3.
Operating the integral form of Equation (41), from 0 to p1 and 0 to p2 with respect to p1,p2, respectively, we get
(42)Ψp1,p2,s=1s∫0p1∫0p2∂2∂p1∂p2Lx1Lx2f1x1,x2dp1dp2+1sζ∫0p1∫0p2L3x2x1∂∂x1ψx1+x1x2∂∂x2ψx2+x2x1∂∂x1ψx1tdp1dp2+1sζ∫0p1∫0p2∂2∂p1∂p2L3fx1,x2,tdp1dp2.Step 4.
The series solution of the singular2+1-D pseudoparabolic equation is therefore entirely determined by
(43)ψx1,x2,t=∑m=0∞ψmx1,x2,t.Step 5.
Working with the 3-DLT for both sides of Equation (42) and applying Equation (43), we obtain
(44)∑m=0∞ψmx1,x2,t=f1x1,x2+L3−11sζ∫0p1∫0p2∂2∂p1∂p2Lx1Lx2f1x1,x2,tdp1dp2+L3−11sζ∫0p1∫0p2L3x2x1∂∂x1∑m=0∞ψmx1dp1+L3−11sζ∫0p1∫0p2L3x1x2∂∂x2∑n=0∞ψmx2dp1dp2+L3−11sζ∫0p1∫0p2L3x2x1∂∂x1∑n=0∞ψmx1tdp1dp2;in view of the first approximation,
(45)ψ0=f1x1,x2+L3−11sζ∫0p1∫0p2∂2∂p1∂p2Lx1Lx2f1x1,x2,tdp1dp2,and the remaining components ψm+1, m≥0, are denoted by
(46)ψm+1=+L3−11sζ∫0p1∫0p2L3x2x1ψmx1x1dp1+L3−11sζ∫0p1∫0p2L3x1x2ψmx2x2dp1dp2+L3−11sζ∫0p1∫0p2L3x2x1ψmx1x1tdp1dp2.Here, we consider that the inverse 3-DLT with respect top1,p2 and s of Equations (45) and (46) exists. To display the applicability of the method explained previously, we current the following example.Example 12.
Singular2+1-D pseudoparabolic equation is given by
(47)Dtζψ−1x1x1ψx1x1−1x2x2ψx2x2−1x1x1ψx1x1t=x12−x22et−4et0≤x1,x2,t<∞,0<ζ≤1,(48)subjecttoψx1,x2,0=x12−x22.
By applying the above steps and Theorem8 for Equation (47), all terms of the sequence are computed as follows:
(49)ψ0=x12−x22+x12tζΞ1,ζ+1t−x22tζΞ1,ζ+1t−4tζΞ1,ζ+1t,(50)ψm+1=L3−11sζ∫0p1∫0p2L3x2x1ψmx1x1dp1+L3−11sζ∫0p1∫0p2L3x1x2ψmx2x2dp1dp2+L3−11sζ∫0p1∫0p2L3x2x1ψmx1x1tdp1dp2;according to 3-DLADM, we get the following components:
(51)ψ1=L3−11sζ∫0p1∫0p2L3x2x1ψ0x1x1+x1x2ψ0x2x2+x2x1ψ0x1x1tdp1dp2=L3−11sζ∫0p1∫0p2L34x1x2tζ−1Ξ1,ζtdp1dp2=L3−14p1p2s2ζ−1s−1ψ1=4t2ζ−1Ξ1,2ζt.
In the same way, we receive that(52)ψ2=L3−11sζ∫0p1∫0p2L3x2x1ψ1x1x1+x1x2ψ1x2x2+x2x1ψ1x1x1tdp1dp2=L3−10=0,(53)ψ3=0,ψ4=0,⋯;by adding all the terms, we have
(54)ψx1,x2,t=ψ0+ψ1+ψ2+⋯;therefore, the approximation solution of Equation (47) is denoted by
(55)ψx1,x2,t=x12−x22+x12tζΞ1,ζ+1t−x22tζΞ1,ζ+1t−4tζΞ1,ζ+1t+4t2ζ−1Ξ1,2ζt.
By usingζ=1 and Equations (13) and (14), the approximation solution becomes
(56)ψx1,x2,t=x12−x22et.Figures1(a) and 1(b) show the approximate and exact solutions of Equation (47); at t=1 and α=1, we obtain the exact solution of Equation (47); by taking different values of α such as α=0.85, α=0.90, and α=0.95, we get the approximate solution. Figures 1(c) and 1(d) show the plot of function ψx1,x2,t in three dimensions.Figure 1(a)
The approximate solutions ofψx1,x2,t for Equation (47) with x2=0, t=1, and α=1, 0.85, 0.90, and 0.95(b)
The approximate solutions ofψx1,x2,t for Equation (47) with x1=0, t=1, and α=1, 0.85, 0.90, and 0.95(c)
The surface showsψx1,x2,t for Equation (47) with x2=0(d)
The surface showsψx1,x2,t for Equation (47) with x1=0In the next problem, we apply the previous method.Problem 13.
Consider the next nonlinear singular2+1-D pseudoparabolic equation:
(57)Dtζψ−1x1x1∂∂x1ψx1−1x2x2∂∂x2ψx2−1x1∂2∂x1∂tx1∂∂x1ψx1=−2μx1ψx1ψx1x1+νx1ψx12+fx1,x2,t,0<ζ≤1,0≤x1,x2,t<∞,(58)subjecttoψx1,x2,0=f1x1,x2.
By applying the previous technique, the first approximation is given by(59)ψ0=f1x1,x2+L3−11sζ∫0p1∫0p2∂2∂p1∂p2L3fx1,x2,tdp1dp2,and the rest of the terms are given by
(60)ψm+1=L3−11sζ∫0p1∫0p2L3x2x1ψmx1x1+x1x2ψmx2x2+x2x1ψmx1x1tdp1dp2−L3−11sζ∫0p1∫0p2L3x1x2μx1Am−x1x2νx1Bmdp1dp2,where nonlinear terms Am and Bm are decomposed as
(61)Am=∑m=0∞ψmx1ψmx1x1,Bm=∑m=0∞ψmx12.
The nonlinear termsψx1ψx1x1 and ψx12 are denoted by
(62)A0=ψ0x1ψ0x1x1,(63)A1=ψ0x1ψ1x1x1+ψ1x1ψ0x1x1,(64)A2=ψ0x1ψ2x1x1+ψ1x1ψ1x1x1+ψ2x1ψ0x1x1,(65)A3=ψ0x1ψ3x1x1+ψ1x1ψ2x1x1+ψ2x1ψ1x1x1+ψ3x1ψ0x1x1,(66)B0=ψ0x12,(67)B1=2ψ0x1ψ1x1,(68)B2=2ψ0x1ψ2x1+ψ1x12,(69)B3=2ψ0x1ψ3x1+2ψ1x1ψ2x1.To demonstrate this technique for a nonlinear problem, we examine the following example.Example 14.
Consider the following nonlinear pseudoparabolic equation:(70)Dtζψ−1x1x1∂∂x1ψx1−1x2x2∂∂x2ψx2−1x1x1∂∂x1ψx1x1t=−x1ψx1ψx1x1+ψx12+2x12−x22e2t−8e2t,0≤x1,x2,t<∞,0<ζ≤1,(71)subjecttoψx1,x2,0=x12−x22.
By using the mentioned 3-DLADM and Theorem8, we have
(72)ψ0=x12−x22+x12tζΞ1,ζ+12t−x22tζΞ1,ζ+12t−8tζΞ1,ζ+12t,(73)ψm+1=L3−11sζ∫0p1∫0p2L3x2x1ψmx1x1+x1x2ψmx2x2+x2x1ψmx1x1tdp1dp2−L3−11sζ∫0p1∫0p2L3x12x2Am−x1x2Bmdp1dp2,where Am and Bm are defined in Equations (65) and (69). The subsequent terms are introduced by
(74)ψ1=L3−11sζ∫0p1∫0p2L3x2x1ψ0x1x1+x1x2ψ0x2x2+x2x1ψ0x1x1tdp1dp2−L3−11sζ∫0p1∫0p2L3x12x2A0−x1x2B0dp1dp2=L3−11sζ∫0p1∫0p2L38x1x2tζ−1Ξ1,ζ2tdp1dp2=L3−11sζ∫0p1∫0p28p12p22sζ−1s−2dp1dp2=L3−18p1p2s2ζ−1s−2ψ1=8t2ζ−1Ξ1,2ζ2t.
Following in a similar manner, we have(75)ψ2=0,ψ3=0,ψ4=0,⋯.
Hence, according to Equation (43), we have
(76)ψx1,x2,t=x12−x22+x12tζΞ1,ζ+12t−x22tζΞ1,ζ+12t−8tζΞ1,ζ+12t+8t2ζ−1Ξ1,2ζ2t;if we set ζ=1 and Equations (13) and (14), then the exact solutions of Equation (70) are presented by
(77)ψx1,x2,t=x12−x22e2t.Figures2(a) and 2(b) show the approximate and exact solutions of Equation (70); at t=1 and α=1, we get the exact solution of Equation (70). For the different values of α such as α=0.85, α=0.90, and α=0.95, we obtain the approximate solution. Figures 2(c) and 2(d) represent the surface of the function ψx1,x2,t.Figure 2(a)
The approximate solutions ofψx1,x2,t for Equation (70) with x2=0, t=1, and α=1, 0.85, 0.90, and 0.95(b)
The approximate solutions ofψx1,x2,t for Equation (70) with x1=0, t=1, and α=1, 0.85, 0.90, and 0.95(c)
The surface showsψx1,x2,t for Equation (70) with x2=0(d)
The surface showsψx1,x2,t for Equation (70) with x1=0


## 5. Singular 2-D Coupled Pseudoparabolic Equation and 3-DLADM

The aim of this part is devoted to establishing the solution of the coupled singular2+1-D pseudoparabolic equation by applying 3-DLADM.The coupled singular 2-D pseudoparabolic equation of fractional order is given by(78)Dtζψ−1x1x1∂∂x1ψx1−1x2x2∂∂x2ψx2−1x1x1∂∂x1ψx1t−μx1ψx1+λφ=fx1,x2,t,(79)Dtζφ−1x1x1∂∂x1φx1−1x2x2∂∂x2φx2−1x1x1∂∂x1φx1t−γx1φx1+λψ=gx1,x2,t,(80)subjecttoψx1,x2,0=f1x1,x2,φx1,x2,0=g1x1,x2,where fx1,x2,t, gx1,x2,t,f1x1,x2, and g1x1,x2 are given functions and λ is the coupling parameter. One can obtain the solution of Equation (79) by using 3-DLADM. This method contains the following steps.
(1)
Multiplying both sides of Equation (79) by x1x2 leads to the following equation:(81)x1x2Dtβψ−x2x1∂∂x1ψx1−x1x2∂∂x2ψx2−x2x1∂∂x1ψx1t−μx2ψx1+λx1x2φ=x1x2fx1,x2,t,x1x2Dtβφ−x2x1∂∂x1φx1−x1x2∂∂x2φx2−x2x1∂∂x1φx1t−γx2φx1+λx1x2ψ=x1x2gx1,x2,t.(2)
We apply 3-DLT and both sides of Equation (81) and 2-DLT for Equation (80); we get L3x1x2Dtζψ=L3x2x1∂/∂x1ψx1+x1x2∂/∂x2ψx2+x2x1∂/∂x1ψx1t+L3μx2ψx1−λx1x2φ+L3x1x2fx1,x2,t,(82)L3x1x2Dtζφ=L3x2x1∂∂x1φx1+x1x2∂∂x2φx2+x2x1∂∂x1φx1t+L3γx2φx1−λx1x2ψ+L3x1x2gx1,x2,t,on using Theorem 8 and 2-DLT for condition, we obtain
(83)∂2∂p1∂p2Ψp1,p2,s=1sζ∂2∂p1∂p2Lx1Lx2f1x1,x2+1sζ∂2∂p1∂p2L3fx1,x2,t+1sζL3x2x1∂∂x1ψx1+x1x2∂∂x2ψx2+x2x1∂∂x1ψx1t+1sζL3μx2ψx1−λx1x2φ,∂2∂p1∂p2Φp1,p2,s=1sζ∂2∂p1∂p2Lx1Lx2g1x1,x2+1sζ∂2∂p1∂p2L3gx1,x2,t+1sζL3x2x1∂∂x1φx1+x1x2∂∂x2φx2+x2x1∂∂x1φx1t+1sζL3γx2φx1−λx1x2ψ.(3)
By integrating Equation (83) from 0 to p1 and 0 to p2 with respect to p1 and p2, respectively, we have(84)Ψp1,p2,s=1sζ∫0p1∫0p2∂2∂p1∂p2Lx1Lx2f1x1,x2dp1dp2+1sζ∫0p1∫0p2∂2∂p1∂p2L3fx1,x2,tdp1dp2+1sζ∫0p1∫0p2L3ϕdp1dp2,(85)Φp1,p2,s=1sζ∫0p1∫0p2∂2∂p1∂p2Lx1Lx2g1x1,x2dp1dp2+1sζ∫0p1∫0p2∂2∂p1∂p2L3gx1,x2,tdp1dp2+1sζ∫0p1∫0p2L3χdp1dp2.Finally, by using the inverse of 3-DLT, we can evaluateψx1,x2,t and φx1,x2,t as follows:
(86)ψx1,x2,t=f1x1,x2+L3−11sζ∫0p1∫0p2∂2∂p1∂p2L3fx1,x2,tdp1dp2+L3−11sζ∫0p1∫0p2L3ϕdp1dp2,(87)φx1,x2,t=g1x1,x2+L3−11sζ∫0p1∫0p2∂2∂p1∂p2L3gx1,x2,tdp1dp2+L3−11sζ∫0p1∫0p2L3χdp1dp2.The 3-DLADM decomposes the unknown functionsψx1,x2,t and φx1,x2,t by the infinite series of components as
(88)ψx1,x2,t=∑m=0∞ψmx1,x2,t,φx1,x2,t=∑m=0∞φmx1,x2,t.By substituting Equation (88) into Equations (86) and (87), we get
(89)∑m=0∞ψmx1,x2,t=f1x1,x2+L3−11sζ∫0p1∫0p2∂2∂p1∂p2L3fx1,x2,tdp1dp2+L3−11sζ∫0p1∫0p2L3∑m=0∞ϕmdp1dp2,(90)∑m=0∞φmx1,x2,t=g1x1,x2+L3−11sζ∫0p1∫0p2∂2∂p1∂p2L3gx1,x2,tdp1dp2+L3−11sζ∫0p1∫0p2L3∑m=0∞χmdp1dp2,where
(91)ϕm=x2∂∂x1x1ψmx1+x1∂∂x2x2ψmx2+x2∂2∂x1∂tx1ψmx1+μx2ψmx1−λx1x2φm,(92)χm=x2∂∂x1x1φmx1+x1∂∂x2x2φmx2+x2∂2∂x1∂tx1φmx1+μx2φmx1−λx1x2ψm.Our method recommends that the zeroth componentsψ0 and φ0 are determined by the initial conditions and nonhomogeneous parts:
(93)ψ0=f1x1,x2+L3−11sζ∫0p1∫0p2∂2∂p1∂p2L3fx1,x2,tdp1dp2,(94)φ0=g1x1,x2+L3−11sζ∫0p1∫0p2∂2∂p1∂p2L3gx1,x2,tdp1dp2.The remainder terms are given by(95)ψm+1=L3−11sζ∫0p1∫0p2L3∑n=0∞ϕmdp1dp2,(96)φn+1=L3−11sζ∫0p1∫0p2L3∑n=0∞χmdp1dp2.In order to check the applicability of our method for solving the fractional coupled pseudoparabolic equation, the next example has been considered.Example 15.
Time fractional coupled pseudoparabolic equations are given by(97)Dtζψ=1x1x1∂∂x1ψx1+1x2x2∂∂x2ψx2+1x1x1∂∂x1ψx1t+2x1ψx1−2x2ψx2−2φ,(98)Dtζφ=1x1x1∂∂x1φx1+1x2x2∂∂x2φx2+1x1x1∂∂x1φx1t+2x1φx1−2x2φx2−2ψ,where
(99)0≤x1,x2,t<∞,1,0<ζ≤1,with initial condition
(100)ψx1,x2,0=x12−x22,φx1,x2,0=x12−x22.Using the 3-DLADM procedure Equations (94), (95), and (96), we obtain the following components:
(101)ψ0=x12−x22,φ0=x12−x22,at m=0,(102)ψ1=L3−11sζ∫0p1∫0p2L3x2x1∂∂x1ψ0x1+x1x2∂∂x2ψ0x2+x2x1∂∂x1ψ0x1tdp1dp2+L3−11sζ∫0p1∫0p2L32x2ψ0x1−2x1ψ0x2−2x1x2φ0dp1dp2=L3−11sζ∫0p1∫0p2L38x1x2−2x13x2+2x1x23dp1dp2=L3−18p1p2sζ+1−4p13p2sζ+1+4p1p23sζ+1,ψ1=8tζΓζ+1−2tζx12Γζ+1+2tζx22Γζ+1,(103)φ1=L3−11sζ∫0p1∫0p2L3x2x1∂∂x1φ0x1+x1x2∂∂x2φ0x2+x2x1∂∂x1φ0x1tdp1dp2+L3−11sζ∫0p1∫0p2L32x2φ0x1−2x1φ0x2−2x1x2ψ0dp1dp2=L3−18p1p2sζ+1−4p13p2sζ+1+4p1p23sζ+1,φ1=8tζΓζ+1−2tζx12Γζ+1+2tζx22Γζ+1;by the same way, at m=1, we have
(104)ψ2=−8t2ζ−1Γ2ζ−32t2ζΓ2ζ+1+4t2ζx12Γ2ζ+1−4t2ζx22Γ2ζ+1,(105)φ2=−8t2ζ−1Γ2ζ−32t2ζΓ2ζ+1+4t2ζx12Γ2ζ+1−4t2ζx22Γ2ζ+1;at n=2,
(106)ψ3=32t3ζ−1Γ3ζ+96t3ζΓ3ζ+1−8t3ζx12Γ3ζ+1+8t3ζx22Γ3ζ+1,(107)φ3=32t3ζ−1Γ3ζ+96t3ζΓ3ζ+1−8t3ζx12Γ3ζ+1+8t3ζx22Γ3ζ+1;at n=3,
(108)ψ4=−96t4ζ−1Γ4ζ−256t4ζΓ4ζ+1+16t4ζx12Γ4ζ+1−16t4ζx22Γ4ζ+1,(109)φ4=96t4ζ−1Γ4ζ−256t4ζΓ4ζ+1+16t4ζx12Γ4ζ+1−16t4ζx22Γ4ζ+1,thus for remaining elements. Using Equation (88), therefore, the approximate solutions are determined by
(110)ψx1,x2,t=x12−x22−2tζx12Γζ+1+2tζx22Γζ+1+4t2ζx12Γ2ζ+1−4t2ζx22Γ2ζ+1−8t3ζx12Γ3ζ+1+8t3ζx22Γ3ζ+1+16t4ζx12Γ4ζ+1−16t4ζx22Γ4ζ+1+8tζΓζ+1−8t2ζ−1Γ2ζ−32t2ζΓ2ζ+1+32t3ζ−1Γ3ζ+96t3ζΓ3ζ+1−96t4ζ−1Γ4ζ−256t4ζΓ4ζ+1,(111)φx1,x2,t=x12−x22−2tζx12Γζ+1+2tζx22Γζ+1+4t2ζx12Γ2ζ+1−4t2ζx22Γ2ζ+1−8t3ζx12Γ3ζ+1+8t3ζx22Γ3ζ+1+16t4ζx12Γ4ζ+1−16t4ζx22Γ4ζ+1+8tζΓζ+1−8t2ζ−1Γ2ζ−32t2ζΓ2ζ+1+32t3ζ−1Γ3ζ+96t3ζΓ3ζ+1−96t4ζ−1Γ4ζ−256t4ζΓ4ζ+1.We setζ=1, the fractional solution becomes
(112)ψx1,x2,t=ψ0+ψ1+ψ2+ψ3+⋯=1−2t+2t22!−2t33!+2t44!−⋯x12−x22,φx1,x2,t=φ0+φ1+φ2+φ3+⋯=1−2t+2t22!−2t33!+2t44!−⋯x12−x22.Hence, the exact solution becomes(113)ψx1,x2,t=x12−x22e−2t,φx1,x2,t=x12−x22e−2t.Figures3(a)–3(d) show the approximate and exact solutions of Equation (98); put t=1 and α=1, we have the exact solution of Equation (98); by using different values of α such as α=0.85, α=0.90, and α=0.95, we get the approximate solution. The approximate solutions of the functions ϕx1,x2,t and ψx11,x2,t are given by Figures 3(e)–3(h).Conclusion 16.
In the current study, a powerful method called the 3-DLADM was approved for finding approximate and exact solution of the time-fractional singular 2-D pseudoparabolic equation. The suggested method is easier in its precept and active in solving linear and nonlinear singular two-dimensional pseudoparabolic equation. Therefore, we conclude that the 3-DLADM is very effective and more precise for any fractional order partial differential equations.Figure 3(a)
The approximate solutions ofϕx1,x2,t for Equation (98) with x2=0, t=1, and α=1, 0.85, 0.90, and 0.95(b)
The approximate solutions ofψx1,x2,t for Equation (98) with x2=0, t=1, and α=1, 0.85, 0.90, and 0.95(c)
The approximate solutions ofϕx1,x2,t for Equation (98) with x1=0, t=1, and α=1, 0.85, 0.90, and 0.95(d)
The approximate solutions ofψx1,x2,t for Equation (98) with x1=0, t=1, and α=1, 0.85, 0.90, and 0.95(e)
The surface showsϕx1,x2,t for Equation (98) with x2=0(f)
The surface showsψx1,x2,t for Equation (98) with x2=0(g)
The surface showsϕx1,x2,t for Equation (98) with x1=0(h)
The surface showsψx1,x2,t for Equation (98) with x1=0



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