Homotopy Perturbation Method for Fractional Gas Dynamics Equation Using Sumudu Transform

Abstract

A user friendly algorithm based on new homotopy perturbation Sumudu transform method (HPSTM) is proposed to solve nonlinear fractional gas dynamics equation. The fractional derivative is considered in the Caputo sense. Further, the same problem is solved by Adomian decomposition method (ADM). The results obtained by the two methods are in agreement and hence this technique may be considered an alternative and efficient method for finding approximate solutions of both linear and nonlinear fractional differential equations. The HPSTM is a combined form of Sumudu transform, homotopy perturbation method, and He’s polynomials. The nonlinear terms can be easily handled by the use of He’s polynomials. The numerical solutions obtained by the proposed method show that the approach is easy to implement and computationally very attractive.

1. Introduction

Fractional calculus is a field of applied mathematics that deals with derivatives and integrals of arbitrary orders. During the last decade, fractional calculus has found applications in numerous seemingly diverse fields of science and engineering. Fractional differential equations are increasingly used to model problems in fluid mechanics, acoustics, biology, electromagnetism, diffusion, signal processing, and many other physical processes [1–19].

There exists a wide class of literature dealing with the problems of approximate solutions to fractional differential equations with various different methodologies, called perturbation methods. The perturbation methods have some limitations; for example, the approximate solution involves series of small parameters which poses difficulty since the majority of nonlinear problems have no small parameters at all. Although appropriate choices of small parameters sometimes lead to ideal solution, in most of the cases unsuitable choices lead to serious effects in the solutions. Therefore, an analytical method is welcome which does not require a small parameter in the equation modeling the phenomenon.

Recently, there is a very comprehensive literature review in some new asymptotic methods for the search for the solitary solutions of nonlinear differential equations, nonlinear differential-difference equations, and nonlinear fractional differential equations; see [20]. The homotopy perturbation method (HPM) was first introduced by He [21]. The HPM was also studied by many authors to handle linear and nonlinear equations arising in various scientific and technological fields [22–32]. The Adomian decomposition method (ADM) [33] and variational iteration method (VIM) [34] have also been applied to study the various physical problems.

In a recent paper, Singh et al. [35] have paid attention to study the solutions of linear and nonlinear partial differential equations by using the homotopy perturbation Sumudu transform method (HPSTM). The HPSTM is a combination of Sumudu transform, HPM, and He’s polynomials and is mainly due to Ghorbani and Saberi-Nadjafi [36] and Ghorbani [37].

In this paper, we consider the following nonlinear time-fractional gas dynamics equation of the form 𝐷^𝛼_𝑡1𝑈+2𝑈²_𝑥−𝑈(1−𝑈)=0,𝑡>0,0<𝛼≤1, (1) with the initial condition 𝑈(𝑥,0)=𝑒^−𝑥, (2) where 𝛼 is a parameter describing the order of the fractional derivative. The function 𝑈(𝑥,𝑡) is the probability density function, 𝑡 is the time, and 𝑥 is the spatial coordinate. The derivative is understood in the Caputo sense. The general response expression contains a parameter describing the order of the fractional derivative that can be varied to obtain various responses. In the case of 𝛼=1 the fractional gas dynamics equation reduces to the classical gas dynamics equation. The gas dynamics equations are based on the physical laws of conservation, namely, the laws of conservation of mass, conservation of momentum, conservation of energy, and so forth. The nonlinear fractional gas dynamics has been studied previously by Das and Kumar [38].

Further, we apply the HPSTM and ADM to solve the nonlinear time-fractional gas dynamics equation. The objective of the present paper is to extend the application of the HPSTM to obtain analytic and approximate solutions to the time-fractional gas dynamics equation. The advantage of the HPSTM is its capability of combining two powerful methods for obtaining exact and approximate analytical solutions for nonlinear equations. It provides the solutions in terms of convergent series with easily computable components in a direct way without using linearization, perturbation, or restrictive assumptions. It is worth mentioning that the HPSTM is capable of reducing the volume of the computational work as compared to the classical methods while still maintaining the high accuracy of the numerical result; the size reduction amounts to an improvement of the performance of the approach.

2. Sumudu Transform

In the early 90’s, Watugala [39] introduced a new integral transform, named the Sumudu transform and applied it to the solution of ordinary differential equation in control engineering problems. The Sumudu transform, is defined over the set of functions 𝐴=𝑓(𝑡)∣∃𝑀,𝜏₁,𝜏₂>0,𝑓(𝑡)<𝑀𝑒^{|𝑡|/𝜏𝑗},if𝑡∈(−1)^𝑗×[0,∞) (3) by the following formula: 𝑓(𝑢)𝑓(𝑡==𝑆)_∞0𝑓(𝑢𝑡)𝑒^−𝑡𝑑𝑡,𝑢∈−𝜏₁,𝜏₂. (4)

Some of the properties were established by Weerakoon in [40, 41]. In [42], by Aşiru, further fundamental properties of this transform were also established. Similarly, this transform was applied to the one-dimensional neutron transport equation in [43] by Kadem. In fact it was shown that there is a strong relationship between Sumudu and other integral transforms; see Kılıçman et al. [44]. In particular the relation between Sumudu transform and Laplace transforms was proved in Kılıçman and Gadain [45].

Further, in Eltayeb et al. [46], the Sumudu transform was extended to the distributions and some of their properties were also studied in Kılıçman and Eltayeb [47]. Recently, this transform is applied to solve the system of differential equations; see Kılıçman et al. in [48].

Note that a very interesting fact about Sumudu transform is that the original function and its Sumudu transform have the same Taylor coefficients except the factor 𝑛; see Zhang [49]. Thus if 𝑓(𝑡)=∑^∞_𝑛=0𝑎_𝑛𝑡^𝑛then 𝐹(𝑢)=∑^∞_𝑛=0𝑛!𝑎_𝑛𝑢^𝑛; see Kılıçman et al. [44]. Similarly, the Sumudu transform sends combinations, 𝐶(𝑚,𝑛), into permutations, 𝑃(𝑚,𝑛), and hence it will be useful in the discrete systems.

3. Basic Definitions of Fractional Calculus

In this section, we mention the following basic definitions of fractional calculus which are used further in the present paper.

Definition 1. The Riemann-Liouville fractional integral operator of order 𝛼>0, of a function 𝑓(𝑡)∈𝐶_𝜇, and 𝜇≥−1 is defined as [5] 𝐽^𝛼𝑓(𝑡)=1Γ(𝛼)^𝑡₀(𝑡−𝜏)^𝛼−1𝑓(𝜏)(),𝐽𝑑𝜏,𝛼>0₀𝑓(𝑡)=𝑓(𝑡). (5)(6) For the Riemann-Liouville fractional integral, we have 𝐽^𝛼𝑡^𝛾=Γ𝛾+1Γ𝑡𝛾+𝛼+1^𝛼+𝛾. (7)

Definition 2. The fractional derivative of 𝑓(𝑡) in the Caputo sense is defined as [10] 𝐷^𝛼_𝑡𝑓(𝑡)=𝐽^𝑚−𝛼𝐷^𝑛𝑓(𝑡)=1Γ(𝑛−𝛼)^𝑡₀(𝑡−𝜏)^{𝑚−𝛼−1}𝑓^(𝑚)(𝜏)𝑑𝜏, (8) for 𝑚−1<𝛼≤𝑚, 𝑚∈𝑁,  𝑡>0.
For the Riemann-Liouville fractional integral and the Caputo fractional derivative, we have the following relation: 𝐽^𝛼_𝑡𝐷^𝛼_𝑡𝑓(𝑡)=𝑓(𝑡)−^𝑚−1_𝑘=0𝑓^(𝑘)()𝑡0+^𝑘.𝑘! (9)

Definition 3. The Sumudu transform of the Caputo fractional derivative is defined as follows [50]: 𝑆𝐷^𝛼_𝑡𝑓(𝑡)=𝑢^−𝛼𝑆𝑓(𝑡−_{𝑚−1𝑘=0})𝑢^−𝛼+𝑘𝑓^(𝑘)(0+),(𝑚−1<𝛼≤𝑚). (10)

4. Solution by Homotopy Perturbation Sumudu Transform Method (HPSTM)

4.1. Basic Idea of HPSTM

To illustrate the basic idea of this method, we consider a general fractional nonlinear nonhomogeneous partial differential equation with the initial condition of the form 𝐷^𝛼_𝑡𝑈()()()(),𝑈()(𝑥),𝑥,𝑡+𝑅𝑈𝑥,𝑡+𝑁𝑈𝑥,𝑡=𝑔𝑥,𝑡𝑥,0=𝑓 (11)(12) where 𝐷^𝛼_𝑡𝑈(𝑥,𝑡) is the Caputo fractional derivative of the function 𝑈(𝑥,𝑡), 𝑅 is the linear differential operator, 𝑁 represents the general nonlinear differential operator, and 𝑔(𝑥,𝑡) is the source term.

Applying the Sumudu transform (denoted in this paper by 𝑆) on both sides of (11), we get 𝑆𝐷^𝛼_𝑡𝑈(𝑥,𝑡)+𝑆[𝑅𝑈(𝑥,𝑡)]+𝑆[𝑁𝑈(𝑥,𝑡)]=𝑆𝑔(𝑥,𝑡). (13) Using the property of the Sumudu transform, we have 𝑆[𝑈(𝑥,𝑡)]=𝑓(𝑥)+𝑢^𝛼𝑆𝑔(𝑥,𝑡)−𝑢^𝛼𝑆[𝑅𝑈(𝑥,𝑡)+𝑁𝑈(𝑥,𝑡)]. (14) Operating with the Sumudu inverse on both sides of (14) gives 𝑈(𝑥,𝑡)=𝐺(𝑥,𝑡)−𝑆⁻¹𝑢^𝛼𝑆[𝑅𝑈(𝑥,𝑡)+𝑁𝑈(𝑥,𝑡)], (15) where 𝐺(𝑥,𝑡) represents the term arising from the source term and the prescribed initial conditions. Now we apply the HPM: 𝑈(𝑥,𝑡)=^∞_𝑛=0𝑝^𝑛𝑈_𝑛(𝑥,𝑡), (16) and the nonlinear term can be decomposed as 𝑁𝑈(𝑥,𝑡)=^∞_𝑛=0𝑝^𝑛𝐻_𝑛(𝑈), (17) for some He’s polynomials 𝐻_𝑛(𝑈) [37] that are given by 𝐻_𝑛𝑈₀,𝑈₁,…,𝑈_𝑛=1𝜕𝑛!𝜕𝑝^𝑛𝑛𝑁^∞_𝑖=0𝑝^𝑖𝑈_𝑖𝑝=0,𝑛=0,1,2,…. (18) Substituting (16) and (17) in (15), we get ^∞_𝑛=0𝑝^𝑛𝑈_𝑛()()𝑆𝑥,𝑡=𝐺𝑥,𝑡−𝑝⁻¹𝑢^𝛼𝑆𝑅^∞_𝑛=0𝑝^𝑛𝑈_𝑛(𝑥,𝑡)+^∞_𝑛=0𝑝^𝑛𝐻_𝑛(𝑈), (19) which is the coupling of the Sumudu transform and the HPM using He’s polynomials. Comparing the coefficients of like powers of 𝑝, the following approximations are obtained: 𝑝⁰∶𝑈₀()(),𝑝𝑥,𝑡=𝐺𝑥,𝑡₁∶𝑈₁(𝑥,𝑡)=−𝑆⁻¹𝑢^𝛼𝑆𝑅𝑈₀(𝑥,𝑡)+𝐻₀(𝑈,𝑝₂)∶𝑈₂(𝑥,𝑡)=−𝑆⁻¹𝑢^𝛼𝑆𝑅𝑈₁(𝑥,𝑡)+𝐻₁(𝑈,𝑝₃)∶𝑈₃(𝑥,𝑡)=−𝑆⁻¹𝑢^𝛼𝑆𝑅𝑈₂(𝑥,𝑡)+𝐻₂(𝑈,⋮) (20) Proceeding in this same manner, the rest of the components 𝑈_𝑛(𝑥,𝑡) can be completely obtained and the series solution is thus entirely determined. Finally, we approximate the analytical solution 𝑈(𝑥,𝑡) by truncated series: 𝑈(𝑥,𝑡)=lim^𝑁_{𝑁→∞𝑛=0}𝑈_𝑛(𝑥,𝑡). (21) The above series solutions generally converge very rapidly.

4.2. Solution of the Problem

Consider the following nonlinear time-fractional gas dynamics equation: 𝐷^𝛼_𝑡1𝑈+2𝑈²_𝑥−𝑈(1−𝑈)=0,0<𝛼≤1, (22) with the initial condition 𝑈(𝑥,0)=𝑒^−𝑥. (23) Applying the Sumudu transform on both sides of (22), subject to the initial condition (23), we have 𝑆[𝑈(𝑥,𝑡)]=𝑒^−𝑥−𝑢^𝛼𝑆12𝑈²_𝑥−𝑈(1−𝑈). (24) The inverse Sumudu transform implies that 𝑈(𝑥,𝑡)=𝑒^−𝑥−𝑆⁻¹𝑢^𝛼𝑆12𝑈²_𝑥−𝑈(1−𝑈). (25) Now applying the HPM, we get ^∞_𝑛=0𝑝^𝑛𝑈_𝑛(𝑥,𝑡)=𝑒^−𝑥−𝑝𝑆⁻¹𝑢^𝛼𝑆12^∞_𝑛=0𝑝^𝑛𝐻_𝑛(𝑈)−^∞_𝑛=0𝑝^𝑛𝑈_𝑛()+𝑥,𝑡_∞𝑛=0𝑝^𝑛𝐻^′_𝑛(𝑈), (26) where 𝐻_𝑛(𝑈) and 𝐻^′_𝑛(𝑈) are He’s polynomials [37] that represent the nonlinear terms. So, the He’s polynomials are given by ^∞_𝑛=0𝑝^𝑛𝐻_𝑛(𝑈)=𝑈²_𝑥. (27) The first few components of He’s polynomials are given by 𝐻₀(𝑈)=𝑈²_0𝑥,𝐻₁(𝑈)=2𝑈₀𝑈_1𝑥,𝐻₁(𝑈)=𝑈²₁+2𝑈₀𝑈_2𝑥,⋮ (28) and for 𝐻^′_𝑛(𝑈), we find that ^∞_𝑛=0𝑝^𝑛𝐻^′_𝑛(𝑈)=𝑈²,𝐻^′₀(𝑈)=𝑈²₀,𝐻^′₁(𝑈)=2𝑈₀𝑈₁,𝐻^′₂(𝑈)=𝑈²₁+2𝑈₀𝑈₂,⋮ (29) Comparing the coefficients of like powers of 𝑝, we have 𝑝⁰∶𝑈₀(𝑥,𝑡)=𝑒^−𝑥,𝑝¹∶𝑈₁(𝑥,𝑡)=−𝑆⁻¹𝑢^𝛼𝑆12𝐻₀(𝑈)−𝑈₀+𝐻^′₀(𝑈)=𝑒^−𝑥𝑡^𝛼Γ(),𝑝𝛼+1₂∶𝑈₂(𝑥,𝑡)=−𝑆⁻¹𝑢^𝛼𝑆12𝐻₁(𝑈)−𝑈₁+𝐻^′₁(𝑈)=𝑒^−𝑥𝑡^2𝛼Γ(),𝑝2𝛼+1₃∶𝑈₃(𝑥,𝑡)=−𝑆⁻¹𝑢^𝛼𝑆12𝐻₂(𝑈)−𝑈₂+𝐻^′₂(𝑈)=𝑒^−𝑥𝑡^3𝛼Γ(),⋮3𝛼+1 (30) Therefore, the series solution is 𝑈(𝑥,𝑡)=𝑒^−𝑥𝑡1+^𝛼Γ()+𝑡𝛼+1Γ(^2𝛼)+𝑡2𝛼+1Γ(^3𝛼).3𝛼+1+⋯ (31) Setting 𝛼=1 in (31), we reproduce the solution of the problem as follows: 𝑈(𝑥,𝑡)=𝑒^−𝑥𝑡1+𝑡+²+𝑡2!³.3!+⋯ (32) This solution is equivalent to the exact solution in closed form: 𝑈(𝑥,𝑡)=𝑒^𝑡−𝑥. (33) Now, we calculate numerical results of the probability density function 𝑈(𝑥,𝑡) for different time-fractional Brownian motions 𝛼 = 1/3, 2/3, 1 and for various values of 𝑡 and 𝑥. The numerical results for the approximate solution (31) obtained by using HPSTM and the exact solution (33) for various values of 𝑡,𝑥, and 𝛼 are shown in Figures 1(a)–1(d) and those for different values of 𝑡 and 𝛼 at 𝑥=1 are depicted in Figure 2.

(a)

(b)

(c)

(d)

Figure 1 

The behaviour of the 𝑈(𝑥,𝑡) w.r.t. 𝑥 and 𝑡 are obtained when (a) 𝛼=1/3, (b) 𝛼=2/3, (c) 𝛼=1, and (d) exact solution.

Figure 2 

Plots of 𝑈(𝑥,𝑡) versus 𝑡 at 𝑥=1 for different values of 𝛼.

It is observed from Figures 1 and 2 that 𝑈(𝑥,𝑡) increases with the increase in 𝑡 and decreases with the increase in 𝛼. Figures 1(c) and 1(d) clearly show that, when 𝛼=1, the approximate solution (31) obtained by the present method is very near to the exact solution. It is to be noted that only the third-order term of the HPSTM was used in evaluating the approximate solutions for Figures 1 and 2. It is evident that the efficiency of the present method can be dramatically enhanced by computing further terms of 𝑈(𝑥,𝑡) when the HPSTM is used.

5. Solution by Adomian Decomposition Method (ADM)

5.1. Basic Idea of ADM

To illustrate the basic idea of ADM [51, 52], we consider a general fractional nonlinear nonhomogeneous partial differential equation with the initial condition of the form 𝐷^𝛼_𝑡𝑈(𝑥,𝑡)+𝑅𝑈(𝑥,𝑡)+𝑁𝑈(𝑥,𝑡)=𝑔(𝑥,𝑡), (34) where 𝐷^𝛼_𝑡𝑈(𝑥,𝑡) is the Caputo fractional derivative of the function 𝑈(𝑥,𝑡), 𝑅 is the linear differential operator, 𝑁 represents the general nonlinear differential operator, and 𝑔(𝑥,𝑡) is the source term.

Applying the operator 𝐽^𝛼_𝑡 on both sides of (34) and using result (9), we have 𝑈(𝑥,𝑡)=^𝑚−1_𝑘=0𝜕^𝑘𝑈𝜕𝑡^𝑘_𝑡=0𝑡^𝑘𝑘!+𝐽^𝛼_𝑡𝑔(𝑥,𝑡)−𝐽^𝛼_𝑡[𝑅𝑈(𝑥,𝑡)+𝑁𝑈(𝑥,𝑡)]. (35) Next, we decompose the unknown function 𝑈(𝑥,𝑡) into sum of an infinite number of components given by the decomposition series 𝑈=^∞_𝑛=0𝑈_𝑛, (36) and the nonlinear term can be decomposed as 𝑁𝑈=^∞_𝑛=0𝐴_𝑛, (37) where 𝐴_𝑛 are Adomian polynomials that are given by 𝐴_𝑛=1𝑑𝑛!𝑑𝜆^𝑛𝑛𝑁^𝑛_𝑖=0𝜆^𝑖𝑈_𝑖𝜆=0,𝑛=0,1,2,… (38) The components 𝑈₀,𝑈₁,𝑈₂,… are determined recursively by substituting (36) and (37) into (34) leading to ^∞_𝑛=0𝑈_𝑛=^𝑚−1_𝑘=0𝜕^𝑘𝑈𝜕𝑡^𝑘_𝑡=0𝑡^𝑘𝑘!+𝐽^𝛼_𝑡𝑔(𝑥,𝑡)−𝐽^𝛼_𝑡𝑅^∞_𝑛=0𝑈_𝑛+^∞_𝑛=0𝐴_𝑛. (39) This can be written as 𝑈₀+𝑈₁+𝑈₂+⋯=^𝑚−1_𝑘=0𝜕^𝑘𝑈𝜕𝑡^𝑘_𝑡=0𝑡^𝑘𝑘!+𝐽^𝛼_𝑡𝑔(𝑥,𝑡)−𝐽^𝛼_𝑡𝑅𝑈₀+𝑈₁+𝑈₂+𝐴+⋯₀+𝐴₁+𝐴₂+⋯. (40) Adomian method uses the formal recursive relations as 𝑈₀=^𝑚−1_𝑘=0𝜕^𝑘𝑈𝜕𝑡^𝑘_𝑡=0𝑡^𝑘𝑘!+𝐽^𝛼_𝑡𝑔(),𝑈𝑥,𝑡_𝑛+1=−𝐽^𝛼_𝑡𝑅𝑈_𝑛+𝐴_𝑛,𝑛≥0. (41)

5.2. Solution of the Problem

Consider the following nonlinear time-fractional gas dynamics equation: 𝐷^𝛼_𝑡1𝑈+2𝑈²_𝑥−𝑈(1−𝑈)=0,0<𝛼≤1, (42) with the initial condition 𝑈(𝑥,0)=𝑒^−𝑥. (43) Applying the operator 𝐽^𝛼_𝑡 on both sides of (42) and using result (9), we have 𝑈=¹⁻¹_𝑘=0𝑡^𝑘𝐷𝑘!^𝑘𝑡𝑈_𝑡=0−𝐽^𝛼_𝑡12𝑈²_𝑥−𝑈+𝑈². (44) This gives the following recursive relations using (41): 𝑈₀=⁰_𝑘=0𝑡^𝑘𝐷𝑘!^𝑘𝑡𝑈_𝑡=0,𝑈_𝑛+1=−𝐽^𝛼_𝑡𝐴_𝑛−𝑈_𝑛,𝑛=0,1,2,…, (45) where 𝐴_𝑛=11𝑛!2𝜕𝑑𝜕𝑥+1^𝑛𝑑𝜆^𝑛𝑛_𝑖=0𝜆^𝑖𝑈_𝑖²_𝜆=0,𝑛=0,1,2,… (46) which using the results (7), (5), and (43) gives 𝑈₀(𝑥,𝑡)=𝑒^−𝑥,𝐴₀𝑈=0,₁(𝑥,𝑡)=𝑒^−𝑥𝑡^𝛼Γ(),𝐴𝛼+1₁𝑈=0,₂(𝑥,𝑡)=𝑒^−𝑥𝑡^2𝛼Γ(),𝐴2𝛼+1₂𝑈=0,₃(𝑥,𝑡)=𝑒^−𝑥𝑡^3𝛼Γ(),⋮3𝛼+1 (47) Therefore, the decomposition series solution is 𝑈(𝑥,𝑡)=𝑒^−𝑥𝑡1+^𝛼Γ()+𝑡𝛼+1Γ(^2𝛼)+𝑡2𝛼+1Γ(^3𝛼),3𝛼+1+⋯ (48) which is the same solution as obtained by using HPSTM.

From Table 1, it is observed that the values of the approximate solution at different grid points obtained by the HPSTM and ADM are close to the values of the exact solution with high accuracy at the third-term approximation. It can also be noted that the accuracy increases as the order of approximation increases.

Table 1 

Comparison study between HPSTM, ADM, and the exact solution when 𝛼 = 1.

The comparison between the third iteration solution of the HPSTM and the second iteration solution of the ADM is given in Figure 3.

Figure 3 

Comparison of the HPSTM and the ADM when 𝑥=1 and 𝛼=1.

It is observed that for 𝑥=1 and 𝛼=1, there is a good agreement between the two methods.

6. Conclusions

In this paper, the homotopy perturbation Sumudu transform method (HPSTM) and the Adomian decomposition method (ADM) are successfully applied for solving nonlinear time-fractional gas dynamics equation. The numerical solutions show that there is a good agreement between the two methods. Therefore, these two methods are very powerful and efficient techniques for solving different kinds of linear and nonlinear fractional differential equations arising in different fields of science and engineering. However, the HPSTM has an advantage over the ADM which is that it solves the nonlinear problems without using Adomian polynomials. In conclusion, the HPSTM and the ADM may be considered as a nice refinement in existing numerical techniques and might find the wide applications.

Acknowledgment

The authors are grateful to the referees for their invaluable suggestions and comments for the improvement of the paper.