A reliable analytical approach for a fractional model of advection-dispersion equation

Jagdev Singh; Aydin Secer; Ram Swroop; Devendra Kumar

doi:10.1515/nleng-2018-0027

Article Open Access

A reliable analytical approach for a fractional model of advection-dispersion equation

Jagdev Singh , Aydin Secer , Ram Swroop and Devendra Kumar

Published/Copyright: June 4, 2018

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From the journal Nonlinear Engineering Volume 8 Issue 1

Abstract

Empirical investigations of solute fate and carrying in streams and rivers often contain inventive liberate of solutes at an upstream perimeter for a finite interval of time. An analysis of various worth references on surface-water-grade mathematical formulation reveals that the logical solution to the continual-parameter advection- dispersion problem for this type of boundary state has been generally missed. In this work, we study the q-fractional homotopy analysis transform method (q-FHATM) to find the analytical and approximate solutions of space-time arbitrary order advection-dispersion equations with nonlocal effects. The diagrammatical representation is done by using Maple package, which enhance the discretion and stability of family of q-FHATM series solutions of fractional advection-dispersion equations. The efficiency of the applied technique is demonstrated by using three numerical examples of space- and time-fractional advection-dispersion equations.

Keywords: Fractional advection-dispersion problem; Laplace transform method; q-fractional homotopy analysis transform technique; Mittage-Leffler function

1 Introduction

Numerous modern water-quality related problems contain usability and importance of the advection dispersion problems. The advection-dispersion problems involving particular initial and boundary limitations narrate spatial and temporal differences in solute concentration. An elementary and general form of the governing equation familiar as the advection-dispersion equation may be obtained for the instance of stable, consistent flow and spatially constant model specification. Fractional calculus has been succeeding in attaining continuous popularity among researchers, due to its applications in engineering, science, finance and other fields, after describing the half order (α = 1/2) derivative by Leibniz in 30 September 1695 [1]. Number of scientists investigated the concept of derivatives and integrals of arbitrary order for instance Caputo, Liouville, Grunewald, Letnikov, Riemann, etc. An excellent literature and approaches of fractional operators for differentiation and integration can be found in [2, 3]. In recent years differential equations of fractional order have gained popularity due to its significant applications [4, 5, 6, 7, 8, 9, 10, 11, 12]. The fractional advection-dispersion equation is investigated by number of methods which are the special case of q-FHATM solution such as fractional variational iteration technique [13], Adomian decomposition approach [14], homotopy perturbation algorithm [15, homotopy analysis technique [16], etc.

In this work, we discuss the existence of an efficient technique called q-fractional homotopy analysis transform method (q-FHATM), is a powerful composition of q-HAM with well-known Laplace transform to analyze space- and time- fractional advection-dispersion equations (STFADE). The coupling of q-HAM with Laplace transform giving time consuming con-sequences and less C.P.U time (Processor 2.65 GHz or more and RAM-1 GB or more) to obtain the numerical solution for nonlinear problems of integer or fractional orders. The advantage of this q-FHATM is its capability to obtaining the series solution of STFADE’s at large domain by choosing the approximate values of parameters ℏ and n. The q-HAM was proposed by El-Tavil and Huseen [17, 18] is more general than classical homotopy analysis method (HAM). The HAM was introduced and implied by Liao [19, 20] is an analytical approach to investigate several kinds of problems occurring in engineering, science and finance such as nonlinear equations occurring in heat transfer [21], fractional differential-difference equation [22], etc. In recent years analytical techniques are also combined with Laplace transform to study the various nonlinear problems such as Blasius flow equations [23], fractional biological population model [24], fractional Lotka-Volterra equation arising in biological systems [25], Klein-Gordon equations [26], singular system of transistor circuits [27], fractional coupled Burgers equations [28], fractional Rosenau-Hyman equation [29], linear differential equations of fractional orders [30], etc. In addition to this there are numerous methods have been employed to handle various problems of integer or fractional order [31, 32, 33, 34, 35, 36, 37, 38, 39].

In this work, we examine the space- and time- fractional advection-dispersion problem characterized as

∂αc∂tα=−w∂βc∂xβ+k∂2βc∂x2β,t>0,x>0,0<α,β≤1(1)

having the initial conditions

c(0,t)=ξ1(t),cx(0,t)=ξ2(t),c(x,0)=f(x),(2)

here c= c(x,t) is standing for the concentration of contaminant, x is indicating the spatial space region, t is denoting time, α, β are representing the parameters narrating the order of the time- and space- arbitrary ordered differential coefficients, respectively, w is indicating the average fluid velocity and k is standing for the dispersion coefficient.

2 Basic definitions

Definition 1

The fractional derivative of ϕ(t) defined by Caputo [40] is represented as:

Dβϕ(t)=Jm−βDmϕ(t)=1Γ(m−β)∫0t(t−τ)m−β−1ϕm(τ)dτ,(3)

for m – 1 < β ≤ m, m ∈ N, t > 0.

Definition 2

The Laplace transform (LT) of the fractional ordered differential coefficient in Caputo sense is defined as follows

L[Dβϕ(t)]=sβL[ϕ(t)]−∑r=0n−1sβ−r−1ϕ(r)(0+),n−1<β≤n.(4)

Definition 3

The LT of the fractional integral in Riemann-Liouville sense is represented as

L[Itβϕ(t)]=s−βϕ¯(s).(5)

Definition 4

The well known Mittag-Leffler (M-L) function is given as [41]:

Eβ(z)=∑k=0∞zkΓ(βk+1)(β∈C,Re(β)>0).(6)

3 Basic idea of q-FHATM

To demonstrate the fundamental structure of proposed algorithm, we discuss a nonlinear partial differential equation of fractional order of the form:

Dtβc(x,t)+Rc(x,t)+Nc(x,t)=h(x,t),n−1<β≤n(7)

In the above fractional differential equation Dtβc is denoting the fractional differential coefficient termed in Caputo sense of the function c,R represents the linear differential operator, N indicates the general nonlinear differential operator and h(x,t) shows the term occurring from source.

By employing the LT operator on Equation (7), we get

L[Dtβc]+L[Rc]+L[Nc]=L[h(x,t)].(8)

Now, applying the differentiation formula of the LT operator, we get

sβL[c]−∑k=0n−1sβ−k−1c(k)(x,0)+L[Rc]+L[Nc]=L[h(x,t)].(9)

On simplifying

L[c]−1sβ∑k=0n−1sβ−k−1c(k)(x,0)+1sβ[L[Rc]+L[Nc]−L[h(x,t)]]=0.(10)

We present the nonlinear operator in the following way

N[Ψ(x,t;q)]=L[Ψ(x,t;q)]−1sβ∑k=0n−1sβ−k−1Ψ(k)(x,t;q)(0+)+1sβ[L[RΨ(x,t;q)]+L[NΨ(x,t;q)]−L[h(x,t)]].(11)

In the above equation q belongs in the closed interval [0, 1/n] and Ψ(x,t;q) indicates a real function, which depends upon the variables x,t and q. We define a homotopy in the following way

(1−nq)L[Ψ(x,t;q)−c0(x,t)]=ℏqN[c(x,t)].(12)

In the above homotopy equation L indicates the LT operator, n ≥ 1, ℏ ≠ 0 stands for an auxiliary parameter, c₀(x, t) is an initial guess of c(x, t) and Ψ(x, t; q) is a unknown function. It is obvious that, when we set the value of embedding parameter q = 0 and q = 1n it yields the following results

Ψ(x,t;q)=c0(x,t),Ψ(x,t;1n)=c(x,t),(13)

respectively. Hence, when q enhances from 0 to 1n , the solution Ψ(x, t; q) changes from the initial guess c₀(x, t) to the solution c(x, t). Enlarging Ψ(x, t; q) in series form using Taylor’s Theorem about q, we get

Ψ(x,t;q)=∑m=0∞cm(x,t)qm.(14)

In the Eq. (14), the value of c_m(x, t) is obtained from

cm(x,t)=1m∂mΨ(x,t;q)∂qm|q=0.(15)

After properly choosing the value of initial guess c₀(x, t), the parameters n and ℏ the series given in Eq. (14) converges at q = 1n then we get

c(x,t)=∑m=0∞cm(x,t)1nm.(16)

The original nonlinear equations must contain Eq. (16) as one of its solution. The governing equation can be obtained from Eq. (12) on consideration of Eq. (16).

Set the vectors

c→m={c0,c1,…,cm}.(17)

On differentiation of Eq. (12) m-times about q and dividing by m!, and putting q = 0, the deformation equation of mth-order is obtained as

L[cm(x,t)−kmcm−1(x,t)]=ℏRm(c→m−1).(18)

Applying the inverse LT operator, we get

cm(x,t)=kmcm−1(x,t)+ℏL−1[Rm(c→m−1)].(19)

In the above equation the value of 𝓡_m (C⃗_m–1) is derived as follows

Rm(c→m−1)=1(m−1)!∂m−1N[Ψ(x,t;q)]∂qm−1|q=0,(20)

and

km=0,m≤1,n,m>1.(21)

From study it should be analyzed that in special case n = 1, q-FHATM directly convert to classical homotopy analysis method (HAM) i.e. HAM (q-FHATM, n = 1).

3.1 Convergence analysis

To prove the convergence of the solution, first of all we need to give some conditions that are needed to verify the convergence of the series (16). These are discussed in [18] via the following stated theorem.

Theorem 1

Let us suppose that the components c₀, c₁, c₂,… of the solution can be written as given in Eq. (18). In the series solution ∑m=0∞cm1nm presented in Eq. (16), converges if there exists 0 < λ < n s. t. ‖c_m+1 ‖ ≤ λn ‖c_m‖ for all m ≥ m₀ for some m₀ ∈ N.

Additionally, the estimation error is given by

c−∑m=0Mcm1nm≤λnM+11−λn||c0||.

4 Numerical examples

In this portion, we examine the space- and time- fractional advection dispersion problems with the aid of aforesaid reliable q-FHATM algorithm in a realistic and convenient way.

Example 1

We discuss the advection dispersion equation of fractional order given as follows

∂αc∂tα=−w∂c∂x+k∂2c∂x2,t>0,x>0,0<α≤1(22)

having the initial conditions

c(x,0)=sin⁡(x).(23)

Exerting LT operator on Eq. (22) and by using Eq. (23), we get the following result

L[c(x,t)]−1ssin⁡(x)−1sαL−w∂c∂x+k∂2c∂x2=0.(24)

We represent the nonlinear operator given as follows

N[Ψ(x,t;q)]=L[Ψ(x,t;q)]−1ssin⁡(x)−1sαL−w∂Ψ(x,t;q)∂x+k∂2Ψ(x,t;q)∂x2,(25)

and thus

Rm(c→m−1)=L(cm−1)−(1−kmn)1ssin⁡(x)−1sαL−w∂cm−1∂x+k∂2cm−1∂x2.(26)

The deformation equation of m^th-order is represented in the following manner

L[cm(x,t)−kmcm−1(x,t)]=ℏRm(c→m−1).(27)

By using the inverse LT operator, we get

cm(x,t)=kmcm−1(x,t)+ℏL−1[Rm(c→m−1)].(28)

Now on putting m = 1, 2, 3,… in Eq. (28) and solving it, we get the following iterates

c0(x,t)=sin⁡(x),c1(x,t)=ℏ(ksin⁡(x)+wcos⁡(x))tαΓ(α+1),c2(x,t)=ℏ(ℏ+n)(ksin⁡(x)+wcos⁡(x))tαΓ(α+1)+ℏ2((k2−w2)sin⁡(x)+2kwcos⁡(x))t2αΓ(2α+1),c3(x,t)=ℏ(ℏ+n)2(ksin⁡(x)+wcos⁡(x))tαΓ(α+1)+2ℏ2(ℏ+n)((k2−w2)sin⁡(x)+2kwcos⁡(x))t2αΓ(2α+1)ℏ3((k3−3kw2)sin⁡(x)+(3k2w−w3)cos⁡(x))t3αΓ(3α+1),⋮(29)

In the similar way, the remaining values of c_m(x, t) for m > 3 can be attained and the obtained series solution is represented in the following equation

c(x,t)=∑m=0∞cm(x,t)1nm,(30)

c(x,t)=sin⁡(x)+ℏn(ksin⁡(x)+wcos⁡(x))tαΓ(α+1)+ℏ(ℏ+n)n2(ksin⁡(x)+wcos⁡(x))tαΓ(α+1)+ℏ2n2((k2−w2)sin⁡(x)+2kwcos⁡(x))t2αΓ(2α+1)+ℏ(ℏ+n)2n3(ksin⁡(x)+wcos⁡(x))tαΓ(α+1)+2ℏ2(ℏ+n)n3((k2−w2)sin⁡(x)+2kwcos⁡(x)t2αΓ(2α+1)+ℏ3n3((k3−3kw2)sin⁡(x)+(3k2w−w3)cos⁡(x))t3αΓ(3α+1)+⋯(31)

If we set ℏ = –1 and n = 1 in (31), then we arrive at the results obtained by using HPM, DTM, RDTM, HPTM, LDM and ADM. The advantages of q-HATM over HAM, HPM, DTM, RDTM, HPTM, LDM and ADM is that q-HATM contains two parameters namely asymptotic parameter n and auxiliary parameter ℏ therefore we can easily settle and restrict the region of convergence of series solution derived with the aid of q-HATM by choosing suitable values of ℏ and n in a large domain.

Fig. 1 represents the behavior of the fourth order approximate q-FHATM surface solution of Eq. (22) verses x and time t. In Fig. 2 comparisons are made with Brownian motion i.e. α = 0.9, 0.8, 0.7 to standard motion i.e. α = 1 and show the continuity of fractional order derivatives for Eq. (22). Figs. 3(a)-(b) present the ℏ-curves. From Figs. 3(a)-3b) we can see that the value of ℏ is selected corresponding to n from the convergence region according to the ℏ-curve. The ℏ-curves contain a horizontal line segment that indicates the bounded range of ℏ which gives the guarantee of the convergence for Eq. (30).

Fig. 1

Approximate solution c for Eq. (22) at k = w = 0.8 and at (ℏ, n, α) = (–1,1,1).

Fig. 2

Fourth order approximation q-FHATM solution c verses time t for Eq. (22) at x = 0.5, k = w = 0.8 and different values of α.

Fig. 3

ℏ-curves for fourth order approximation q-FHATM solution of Eq. (22) for different value of α; (a) when n = 1, (b) or n = 5.

Example 2

Now, we examine the nonhomogeneous space-fractional advection-dispersion equation given as follows

∂2βc∂x2β−∂c∂x=∂c∂t+(2−2t−2x),t>0,0<β≤1,(32)

having the boundary and initial conditions

c(0,t)=t2,cx(0,t)=0,c(x,0)=x2.(33)

Employing Laplace transform on Eq. (32) and using Eq. (33), we get

L[c(x,t)]−1st2−2s2β+1+2s2β+1t+2s2β+2−1s2βL∂c∂x+∂c∂t=0.(34)

We construct the nonlinear operator in the following way

N[Ψ(x,t;q)]=L[Ψ(x,t;q)]−t2s−2s2β+1+2ts2β+1+2s2β+2−1s2βL[∂Ψ(x,t;q)∂x+∂Ψ(x,t;q)∂t],(35)

and thus

R(c→m−1)=L(cm−1)−1−kmnt2s+2s2β+1−2s2β+1t−2s2β+2−1s2βL[∂cm−1∂x+∂cm−1∂t].(36)

The deformation equation of m^th-order is represented by

L[cm(x,t)−kmcm−1(x,t)]=ℏRm(c→m−1).(37)

By using the inverse LT operator on Eq. (37), we obtain

cm(x,t)=kmcm−1(x,t)+ℏL−1[Rm(c→m−1)].(38)

For m = 1, 2, 3 … on solving Eq. (38), we have

c0(x,t)=t2+(2−2t)x2βΓ(2β+1)−2x2β+1Γ(2β+2),c1(x,t)=−2ℏtx2βΓ(2β+1)+2ℏx4βΓ(4β+1)−ℏ(2−2t)x4β−1Γ(4β)+2ℏx4βΓ(4β+1),(39)

In the similar way, the remaining values of c_m(x, t) for m > 1 can be gained and the q-FHATM solution in series form is obtained as

c(x,t)=∑m=0∞cm(x,t)1nm.(40)

If we put ℏ = –1 and n = 1 in (40), then we arrive at the results obtained by HPM, DTM, RDTM, HPTM, LDM and ADM. The advantages of q-HATM over HAM, HPM, DTM, RDTM, HPTM, LDM and ADM is that q-FHATM contains two parameters namely asymptotic parameter n and auxiliary parameter ℏ therefore we can easily settle and restrict the region of convergence of series solution derived with the aid of q-FHATM by choosing suitable values of ℏ and n in a large domain. When we put β = 1, ℏ = –1 and n = 1 in Eq. (40) then it converges to very fast to the exact solution c(x, t) = t² + x². The numerical results for HAM (q-FHATM, n = 1) are shown by Fig. 4(a)-(c) and it can be observed that c(x, t) increases with increase in both the variables x and t. On computation of the further components of c(x, t) it is verified that the accuracy and efficiency of the proposed technique can be dramatically improved when the q-FHATM is used and converge to the absolute error → 0. Fig. 5 shows that the Brownian motion also a continuous and increasing function of space x at t = 0.5. Figs. 6(a)-(b) represent the ℏ-curves. The horizontal line segment in the ℏ-curves denotes the valid range of ℏ which gives guarantee for the convergence of series (40).

Fig. 4

The surfaces show solution c(x, t) for Eq. (32) at ℏ = –1, n = 1 and β = 1: (a) Exact solution; (b) 3rd order q-FHATM approximate solution; (c) 3rd order absolute error.

Fig. 5

3^rd order approximation HAM (q-FHATM, n = 1) solution c(x, t) verses space x for Eq. (40) at t = 0.5, ℏ = –1 with different values of β.

Fig. 6

ℏ-curves at x = 0.02, t = 0.5 and β = 1; (a) for n = 1, (b) for n = 200.

Example 3

Finally, consider the following time FADE subject to the initial condition is

∂αc∂tα=k∂2c∂x2−∂c∂x,0<α≤1,t>0.(41)

having the in itial condition

c(x,0)=e−x.(42)

Solving the above equations in the similar way, we get the various components as follows

c0(x,t)=e−x,c1(x,t)=−ℏ(1+k)tαΓ(α+1)−x,c2(x,t)=−ℏ(ℏ+n)(1+k)tαΓ(α+1)e−x+ℏ2(1+k)2t2αΓ(2α+1)e−x,c3(x,t)=−ℏ(ℏ+n)2(1+k)tαΓ(α+1)e−x+2ℏ2(ℏ+n)(1+k)2t2αΓ(2α+1)e−x−ℏ3(1+k)3t3αΓ(3α+1)e−x,⋮(43)

In the similar way, the remaining values of c_m(x, t) for m > 3 can be attained and the series solution is obtained as follow

c(x,t)=∑m=0∞cm(x,t)1nm.(44)

Eq. (44) represents the family of series solution of Eq. (41).

If we select ℏ = –1 and n = 1 in (44), then we arrive at the results obtained with the help of HPM, DTM, RDTM, HPTM, LDM and ADM. The advantages of q-FHATM over HAM, HPM, DTM, RDTM, HPTM, LDM and ADM is that q-FHATM contains two parameters namely asymptotic parameter n and auxiliary parameter ℏ, therefore we can easily settle and restrict the region of convergence of series solution derived with the aid of q-FHATM by choosing suitable values of ℏ and n in a large domain.

If we select ℏ = –1 and n = 1, we get

c(x,t)=limN→∞∑m=0Ncm(x,t)=e−xlimN→∞∑r=0N(1+k)rΓ(rα+1)trα=e−xEα((1+k)tα),(45)

where E_α((1 + k)t^α) is the well known Mittag-Leffler function [41].

Fig. 7 presents the surface of the q-FHATM solution. In Fig. 8, q-FHATM solution is presented for different values of different values of ℏ and n. Figs. 9(a)-(d) show the ℏ-curves. Table 1 presents the absolute converge range for different values of n. From Figs.9(a)-(d), we can see that the valid range of convergence increases with increasing the value of n.

Fig. 7

q-FHATM solution c(x, t) verses x and time t at k = 0.03 for Eq. (43) at (ℏ, n, α) = (−1,1,1).

Fig. 8

q-FHATM solution c(x,t) for Eq. (43) when k = 0.03, x = 0.5 , α = 1 for different values of ℏ and n.

Fig. 9

ℏ-curves for fourth order approximation q-FHATM solution of Eq. (43) at x = 0.5, t = 0.02 and k = 0.03, for (a) n = 1, (b) n = 5, (c) n = 100, (d) n = 110.

Table 1

The fourth order absolute converge range of Eq. (44) decided from ℏ-curves at x = 0.5, t = 0.02, α =1 and k = 0.03.

n	Range of ℏ
2	–2.4 < ℏ < –1.68
5	–6.5 ≤ ℏ < –3.8
8	–9.9 < ℏ < –6.2
10	–12 ≤ ℏ < –8
20	–24 < ℏ ≤ – 17
50	–60 < ℏ ≤ – 40.01
80	–98 ≤ ℏ < –62
100	–120 ≤ ℏ < –82 [Fig. 9(c)]
110	–132 ≤ ℏ < –90 [Fig. 9(d)]

5 Explanation of ℏ-curves

The ℏ-curve and n-curve describe the q-FHATM series solution of space- and time- fractional advection-dispersion problems with all possible acceptable convergence range of ℏ corresponding to n and the central point of ℏ-curves interval i.e. ℏ = –n is a suitable choice, at this point the numerical solution convergences to series solution. Figs. 3(a)-(b) and Figs. 9(a)-(b) show that convergence range of ℏ decreases when α converge from their standard motion (α = 1) to Brownian motion (α = 0.9, 0.8) for STFADE’s. We have analyzed that convergence range ℏ increase/decrease with increase/decrease of arbitrary selected auxiliary parameter n(n ≥ 1) and ℏ-curve seem to the same value of q-FHATM solution c(x, t) for same value of α with different kind of n.

6 Conclusions

In this article, q-FHATM is effectively and successfully used to obtain a family of series solution for the STFADE’s by just utilizing the initial condition. Basic difference q-FHATM to HAM in their solution procedure is that HAM have the limitation for embedding parameter i.e. q = 1 but in q-FHATM, q is defined more general i.e. q = 12 , n ≥ 1. The most important advantages of q-FHATM over HAM, HPM, DTM, RDTM, HPTM, LDM and ADM is that q-FHATM contains two parameters namely asymptotic parameter n and auxiliary parameter ℏ, therefore we can easily settle and restrict the region of convergence of series solution derived with the aid of q-FHATM by choosing suitable values of ℏ and n in a large domain. To control and insure the convergence of series for generalized approach, we have noticed that the q-FHATM provides many more option. Hence, we can conclude that the suggested algorithm is very useful for examining the STFADE’s and similar kind of problems.

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Received: 2018-01-26

Revised: 2018-03-13

Accepted: 2018-04-07

Published Online: 2018-06-04

Published in Print: 2019-01-28

This work is licensed under the Creative Commons Attribution 4.0 Public License.

Articles in the same Issue

https://doi.org/10.1515/nleng-2018-0027

Keywords for this article

Fractional advection-dispersion problem; Laplace transform method; q-fractional homotopy analysis transform technique; Mittage-Leffler function

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