# Natural Transform along with HPM Technique for Solving Fractional ADE

Authors: N. Pareek; A. Gupta; G. Agarwal; D. L. Suthar
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

The authors of this paper solve the fractional space-time advection-dispersion equation (ADE). In the advection-dispersion process, the solute movement being nonlocal in nature and the velocity of fluid flow being nonuniform, it leads to form a heterogeneous system which approaches to model the same by means of a fractional ADE which generalizes the classical ADE, where the time derivative is substituted through the Caputo fractional derivative. For the study of such fractional models, various numerical techniques are used by the researchers but the nonlocality of the fractional derivative causes high computational expenses and complex calculations so the challenge is to use an efficient method which involves less computation and high accuracy in solving such models numerically. Here, in order to get the FADE solved in the form of convergent infinite series, a novel method NHPM (natural homotopy perturbation method) is applied which couples Natural transform along with the homotopy perturbation method. The homotopy peturbation method has been applied in mathematical physics to solve many initial value problems expressed in the form of PDEs. Also, the HPM has an advantage over the other methods that it does not require any discretization of the domains, is independent of any physical parameters, and only uses an embedding parameterp∈0,1. The HPM combined with the Natural transform leads to rapidly convergent series solutions with less computation. The efficacy of the used method is shown by working out some examples for time-fractional ADE with various initial conditions using the NHPM. The Mittag-Leffler function is used to solve the fractional space-time advection-dispersion problem, and the impact of changing the fractional parameter α on the solute concentration is shown for all the cases.


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## Body

## 1. Introduction and Preliminaries

Fractional calculus generalizes the integration and differentiation of integer order to arbitrary order that is being studied for the past 300 years. The growing interest of researchers in this field has led to solving the real-world issues in type of fractional differential equations due to their nonlocal behavior, and these equations are well suited to describe various phenomena in the field of engineering and science. Also, fractional derivatives are capable to model various processes mathematically which exhibit the memory and hereditary properties [1–5].The ADE arises in the study of transport of solute or Brownian motion of particles in a fluid occurring due to the simultaneous occurrence of advection and particle dispersion. Fractional advection-dispersion equation describes the phenomena of anomalous diffusion of the particles in the transport process in a better way; as in anomalous diffusion, the solute transport is quicker or speedier than time’s inferred square root given by Baeumer et al. [6]. The equation is used to study groundwater pollution, pollution of the atmosphere produced by smoke or dust, the spread of chemical solutes and contaminant discharges, etc. [7]. Hence, FADE has attracted the attention of many researchers. Hence, the interest of the researchers lies in solving the FADE to find out the solute concentration at a particular instant of time and space. Analytical solution of one-dimensional ADE was found by Jaiswal et al. [8]. Huang et al. [9] solve the one-dimensional fractional flux ADE and found the finite element solutions. The intermediate fractional ADE was studied by El-Sayed et al. [10]. To solve the space-time fractional ADE, Momani and Odibat [7] utilized the ADM and variational iteration approach. In this continuation, Yildirim and Koçak [11] solve the space-time fractional ADE by applying homotopy perturbation technique in Caputo sense and Hikal and Abu Ibrahim [12] solved it by the Adomian decomposition method. Alliche and Chikh [13] studied the nonpremixed chaotic fire of the hydrogen-air downward injector system using the generalized finite rate chemistry model. Liu et al. [14] applied numerical methods to study various advection-dispersion models. Rocca et al. [15] developed a general solution to the fractional diffusion-advection equation for solar cosmic-ray transport. Ramani et al. [16] explored the fractional decreased differential transform approach for revisiting the analytical-approximate formulation of the time-fractional Rosenau-Hyman problem. The extended differential transform approach was used by Garg and Manohar [17] to solve the space-time fractional Fokker Planck (FFP) equation analytically. Also, Habenom et al. [18] studied the formulation of FFP equation using fractional power series technique. The N-transform was used by Khan and Khan [19] to study the unsteady fluid flow over a plane wall, and N-transform of some functions along with the properties was presented. Belgacem and Silambarasan [20] renamed it as Natural transform which they used to solve Bessel’s differential equation with a polynomial coefficient and also Maxwell’s equation.In this article, first, we recall few concepts of fractional calculus, Natural transform, and HPM which have been used in our main findings, in Sections2, 3, and 4, respectively. Then, we gave a solution to the space-time ADE by the NHPM in Section 5, and at the last, Section 6 contains some related examples, which show the efficiency of this method. In Section 7, a conclusion has been discussed.


## 2. Basic Definitions

The Riemann-Liouville and Caputo-type fractional integral operator and its properties are discussed in this section. These definitions and properties (see detail [1–3]) will be used to get the main results.Definition 1.
Letfx with x>0 be a real-valued function. If there is a real number p>μ, it is said to be in the space C in R. Such that fx=xpf1x, where f1x∈C0,∞.Definition 2.
Letfx with x>0 be a real-valued function; then, it would seem to be in space Cμn,m∈N∪0,iffm∈Cμ.Definition 3.
For a functionf∈Cμ, where μ=1, the R-L fractional integral operator of order ϑ=0 is described as
(1)Jϑfz=1Γϑ∫0zz−tϑ−1ftdt;ϑ>0,z>0,J0fz=fz,JϑJϑfz=Jϑ+ϑfz,Jϑzϑ=Γϑ+1Γϑ+ϑ+1zϑ+ϑ.Definition 4.
In Caputo’s view, the fractional derivative off∈Cn−1 is expressed by
(2)Dϑfz=1Γn−ϑ∫0zz−τn−ϑ−1fnτdτ,n−1<ϑ<n,n∈N∗,dndtnfz,ϑ=n.
Also,(3)DϑK=0;Kbeingaconstant,Dϑzϑ=Γϑ+1Γϑ−ϑ+1zϑ−ϑ,ϑ>ϑ−1,0,ϑ=ϑ−1.Definition 5.
The two-parameter M-L function is described as follows:(4)Eϑ,ϑz=∑k=0∞zkΓϑk+ϑ.
Consequently, the one-parameter M-L function is described as follows:(5)Eϑz=Eϑ,1z=∑k=0∞zkΓϑk+1.


## 3. Natural Transform

Over the set, Natural transform is specified:(6)A=ft:∃M,τ1,τ2,vt<Met/τj,ift∈−1j×0,∞,(7)ℕft=Fs,u=1u∫0∞e−st/uftdt;u>0,s>0,t∈0,∞,where s and u denote the Natural transform variables [21, 22].Remark 6.
(i)
Ifu=1, (7) reduces to the Laplace transform(ii)
Ifs=1, (7) reduces to the Sumudu transform
ℕ-transforms of some elementary functions and the conversions to Sumudu and Laplace [19, 21–23] are given in Tables 1 and 2.Table 1
Relation betweenℕ-transforms, Sumudu transform, and Laplace transform.
ftℕftSftLft11s11stus2u1s2eat1s−au11−au1s−atn−1n−1!,n=1,2..un−1snun−11snsintus2+u2u1+u211+s2Table 2
Properties ofℕ-transforms.
FunctionNatural transform of the functionftFs,ufat1aFs,uf′tsuFs,u−1uf0f′′ts2u2Fs,u−su2f0−1uf′0aft±bgtaFs,u±bGs,u


## 4. The Homotopy Perturbation Method

The general form of the time-dependent differential equation (see [24–26]) can be written as
(8)Auς,t−fς,t=0,where A is the differential operator, uς,t is the unknown function, r is the independent variables for space, t is the independent variables for time, and fς,t is the analytic function.In general,A can be divided into L (linear) and ℕ (nonlinear) component s.t.:
(9)A=L+ℕ.By substituting the value ofA in (8),
(10)Luς,t+ℕuς,t−fς,t=0.Using the homotopy technique presented by Liao [27], a homotopy ∅r,t:p can be constructed which satisfies
(11)ℍ∅ς,t:p,p=1−pL∅ς,t:p−Lu0ς,t+pA∅ς,t:p−fς,t=0,where p∈0,1 is an embedding parameter and u0r,t is an initial guess for ur,t satisfying initial/boundary conditions. The homotopy equation (11) can be written in an equivalent form as
(12)L∅ς,t:p−Lu0ς,t+pN∅ς,t:p+Lu0ς,t−fς,t=0.Hence, whenp=0, we obtain
(13)ℍ∅ς,t:0,0=L∅ς,t:0−Lu0ς,t=0,and whenp=1, we get
(14)ℍ∅ς,t:1,1=A∅ς,t:1−fς,t=0.We observe thatuς,t is the solution of (14) as well as (8) and if L is taken to be linear, u0ς,t is the only solution of (13). So, we have
(15)∅ς,t:0=u0ς,t,∅ς,t:1=uς,t.Change inp from 0 to 1 is followed by change in ∅ς,t:p from u0ς,t to uς,t, termed as deformation. If the embedding parameter p1≥p≥0 is thought to be tiny, according to the classic perturbation technique, the solution to the given equation may be assumed as a power series in p, so
(16)∅ς,t:p=u0ς,t+pu1ς,t+p2u2ς,+⋯,forp=1,
(17)uς,t=u0ς,t+u1ς,t+u2ς,t+⋯,which gives the approximate solution of (8). The series in (17) converges in most of the cases and leads to the exact solution.


## 5. Solution of the Space-Time ADE by the NHPM

The classical one-dimensional ADE with constant parameters is of the form (see [14])
(18)∂C∂τ=D∂2C∂ξ2−V∂C∂ξ,where V is the drift velocity, ξ is the spatial coordinate, D>0 is the constant diffusivity, and Cξ,τ is the solute concentration.To write equation (18) in a simplified form by setting t=τV,x=ξ and replacing Cξ,τ by Cx,t, it reduced into
(19)Ct=μCxx−Cx,where μ=D/V.We write the general form of the space-time fractional ADE as(20)DtϑC=μDxpϑC−DxrϑC,with ϑ=1/m,0<ϑ≤1,1<pϑ≤2,0<rϑ≤1,m,p,rεN,0<x<L,Dxpϑ=DxϑDxϑ⋯Dxϑptimes,Dxrϑ=DxϑDxϑ⋯Dxϑrtimes,DtαandDxϑ are the Caputo fractional derivatives.The initial condition is(21)Cx,0=fx.ℕ-transform of (20) is written as
(22)sϑuϑCx,s,u−sϑ−1uϑCx,0=ℕ+μDxpϑC−DxrϑC.Now, by substituting initial condition from (21) in the above equation, we obtain
(23)Cx,s,u=fxs+uϑsϑℕ+μDxpϑC−DxrϑC.Applying the inverseℕ-transform on (23), we get
(24)Cx,t=fx+ℕ−1uϑsϑℕ+μDxpϑC−DxrϑC.By using the homotopy perturbation method, we can write(25)Cx,t=∑n=0∞pnCnx,t.Substituting (25) in (24),
(26)∑n=0∞pnCnx,t=fx+pℕ−1uϑsϑℕ+∑n=0∞μpnDxpϑCn−∑n=0∞pnDxrϑCn.Comparison of the coefficients of like powers ofp on both sides yields to the corresponding assumptions:
(27)p0:C0x,t=fx,p1:C1x,t=ℕ−1uϑsϑℕ+μDxpϑC0−DxrϑC0=ℕ−1uϑsϑℕ+μDxpϑfx−Dxrϑfx=μDxpϑfx−Dxrϑfxℕ−1uϑsϑ+1=tϑΓϑ+1μDxpϑ−Dxrϑfx,p2:C2x,t=ℕ−1uϑsϑℕ+μDxpϑC1−DxrϑC1=ℕ−1uϑsϑℕ+μDxpϑμDxpϑfx−DxrϑfxtϑΓϑ+1−DxrϑμDxpϑfx−DxrϑfxtϑΓϑ+1=μ2Dx2pϑfx−2μDxp+rϑfx+Dx2rϑfx×ℕ−1uϑsϑℕ+tϑΓϑ+1=μ2Dx2pϑfx−2μDxp+rϑfx+Dx2rϑfxℕ−1u2ϑs2ϑ+1=t2ϑΓ2ϑ+1μDxpϑ−Dxrϑ2fx.Similarly,(28)p3:C3x,t=t3ϑΓ3ϑ+1μDxpϑ−Dxrϑ3fx,and so on.The analytic series solution of (20) can be given as
(29)Cx,t=limN⟶∞∑n=0NCnx,t,Cx,t=C0x,t+C1x,t+C2x,t+C3x,t+⋯=fx+tαΓϑ+1μDxpϑ−Dxrϑfx+t2αΓ2ϑ+1μDxpϑ−Dxrϑ2fx+t3αΓ3ϑ+1μDxpϑ−Dxrϑ3fx⋯=1+tαΓϑ+1μDxpϑ−Dxrϑ+t2αΓ2ϑ+1μDxpϑ−Dxrϑ2+t3αΓ3ϑ+1μDxpϑ−Dxrϑ3+⋯fx=∑n=0∞tnαΓnϑ+1μDxpϑ−Dxrϑnfx,which can also be written as
(30)Cx,t=EϑtϑμDxpϑ−Dxrϑfx.where Eαx is the one-parameter Mittag-Leffler function.Remark 7.
Settingp=2,r=1, (20) reduces to space-time fractional ADE of the form
(31)DtϑC=μDx2ϑC−DxϑC,and the solution is
(32)Cx,t=EϑtϑμDx2ϑ−Dxϑfx,if fx=e−x, the solution is
(33)Cx,t=EϑtϑμDx2ϑ−Dxϑe−x=∑n=0∞tnϑΓnϑ+1∑k=0n−1knckμn−k∑s=2n−kϑ∞−1sxs−2n−kϑΓs−2n−kϑ+1,
where. denotes the ceiling function.This is the same as obtained by Hikal and Abu Ibrahim [12] using ADM.


## 6. Examples

Example 1.
Consider the time-fractional ADE (settingp=2,r=1,m=1 in (20)),
(34)DtαC=μDx2C−DxC,the initial condition being
(35)Cx,0=e−x.
Solution: by applying the NHPM,(36)p0:C0x,t=e−x,p1:C1x,t=ℕ−1uϑsϑℕ+μC0xx−C0x=μ+1tϑΓϑ+1e−x,p2:C2x,t=ℕ−1uϑsϑℕ+μC1xx−C1x=μ+12t2ϑΓ2ϑ+1e−x.
Similarly,(37)p3:C3x,t=μ+13t3ϑΓ3ϑ+1e−x,and so on. Thus, the analytic series solution is given by
(38)Cx,t=C0x,t+C1x,t+C2x,t+C3x,t+⋯=e−x1+μ+1tϑΓϑ+1+μ+12t2ϑΓ2ϑ+1+μ+13t3ϑΓ3ϑ+1+⋯=e−xEα1+μtϑ.
The solution converges to the exact solution of the ADE forα=1 as obtained by El-Sayed et al. [10]:
(39)Cx,t=e1+μt−x.
The result obtained for Example1 is presented in Figure 1.Example 2.
Equation (34) with the initial condition Cx,0=x3−x2.
Solution: by applying the NHPM, we obtain(40)p0:C0x,t=x3−x2,p1:C1x,t=−3x2+2x1+3μ−2μtϑΓϑ+1,p2:C2x,t=6x−2−12μt2ϑΓ2ϑ+1,p3:C3x,t=−6t3ϑΓ3ϑ+1,p4:C4x,t=0.
Thus, the analytic series solution is given by(41)Cx,t=C0x,t+C1x,t+C2x,t+C3x,t+⋯Cx,t=x3−x2+−3x2+2x1+3μ−2μtϑΓϑ+1+6x−2−12μt2ϑΓ2ϑ+1−6t3ϑΓ3ϑ+1.
The result obtained for Example2 is presented in Figure 2.Example 3.
Equation (34) with the initial condition Cx,0=cosx.
Solution: by applying the NHPM, we get(42)p0:C0x,t=cosx,p1:C1x,t=sinx−μcosxtϑΓϑ+1,p2:C2x,t=−cosx−2μsinx+μ2cosxt2ϑΓ2ϑ+1,p3:C3x,t=−sinx+3μcosx+3μ2sinx−μ3cosxt3ϑΓ3ϑ+1,and so on. Thus, the analytic series solution is given by
(43)Cx,t=C0x,t+C1x,t+C2x,t+C3x,t+⋯Cx,t=cosx+sinx−μcosxtϑΓϑ+1+−cosx−2μsinx+μ2cosxt2ϑΓ2ϑ+1+−sinx+3μcosx+3μ2sinx−μ3cosxt3ϑΓ3ϑ+1+⋯.
The result obtained for Example3 is presented in Figure 3.Remark 8.
The convergence of the series solutions obtained for the above cases can be proved by means of a comparison test using the Mittag-Leffler function.Figure 1
Variation of the concentrationCx,t with the fractional parameter α for μ=.01.
(a)
ϑ=0.25(b)
ϑ=0.5(c)
ϑ=1Figure 2
Variation of the concentrationCx,t with the fractional parameter α for μ=.01.
(a)
ϑ=0.25(b)
ϑ=0.5(c)
ϑ=1Figure 3
Variation of the concentrationCx,t with the fractional parameter α for μ=.01.
(a)
ϑ=0.25(b)
ϑ=0.5(c)
ϑ=1


## 7. Conclusion

In the present article, the NHPM is successfully applied to find the solution for the general form of the space-time fractional ADE and the analytic solution is found in terms of M-L function for different cases. The results found show the dependence of solute concentration on the fractional order of the derivative along with the space and time variables. Solutions obtained for the three examples are plotted with spatial and time coordinate for different values of the fractional parameterα using MATLAB R2015a. Figure 1 exhibits a decrease in solute concentration corresponding to the increment in the fractional parameter α. For the second case, the solution is in exact form and is plotted in Figure 2 showing that as the fractional parameter α increases, the solute concentration decreases, whereas it increases with the corresponding increment in α for the third cases as is shown in Figure 3. Hence, the NHPM is a powerful technique to solve various models in the form of linear or nonlinear partial differential equations appearing in the field of science and engineering. The approach should be expanded to solve the two or three dimensions of space-time fractional ADM.



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