Note on the Convergence Analysis of Homotopy Perturbation Method for Fractional Partial Differential Equations

Abstract

We apply the homotopy perturbation method to obtain the solution of partial differential equations of fractional order. This method is powerful tool to find exact and approximate solution of many linear and nonlinear partial differential equations of fractional order. Convergence of the method is proved and the convergence analysis is reliable enough to estimate the maximum absolute truncated error of the series solution. The fractional derivatives are described in the Caputo sense. Some examples are presented to verify convergence hypothesis and simplicity of the method.

1. Introduction

Recently, the partial differential equations of fractional order have attracted much attention. This is mostly due to their frequent appearance in many applications in fluid mechanics, viscoelastic, biology, engineering, and physics [1, 2].

Most of partial differential equations of fractional order do not have exact analytical solution, so approximations and numerical techniques must be used. Some of these methods are series solution methods which include Adomain decomposition method [3], homotopy analysis method [4, 5], variationalmiteration method [6], and homotopy perturbation method [7–9]. The homotopy perturbation method [10] proposed by He in 1998. This method is useful tool for obtaining exact and approximate solution of linear and nonlinear partial differential equations of fractional order. There is no need for a small parameter or linearization, the solution procedure is very simple, and only few iterations lead to high accurate solutions which are valid for the all solution domains. The solution is expressed as the summation of an infinite series which is supposed to be convergent to the exact solution. This method has been used to solve effectively, easily, and accurately many types of fractional equations of linear and nonlinear problems with approximations. For example, [11] applied HPM to solve a class of initialboundary value problems of fractional partial differential equations over finite domain. [12] used HPM for solving the Klein-Gordon partial differential equations of fractional order. Furthermore, many authors applied HPM for solving and investigating linear and nonlinear partial differential equations of fractional ordering; see [13, 14]. For more details about homotopy perturbation method and its applications, we refer to [15, 16].

Our aim in this study is to extend the applications of HPM to obtain approximate solution of some partial differential equations of fractional order such as Burgers’ equation of fractional order and fractional fourth-order parabolic partial differential equation and obtain the convergence of this method.

The paper is organized as follows. In Section 2, some basic definitions and properties of fractional calculus theory are given. In Section 3, the basic idea of HPM is presented. In Section 4, analysis of HPM is given. Some examples are given in Section 5. Concluding remarks are listed in Section 6.

2. Preliminaries

In this section, we give some basic definitions and properties of fractional calculus theory which are used in this paper.

Definition 1. A real function 𝑓(𝑥), 𝑥>0 is said to be in space 𝐶𝜇, 𝜇∈𝑅 if there exists a real number 𝑝>𝜇, such that 𝑓(𝑥)=𝑥^𝑝𝑓₁(𝑥) where 𝑓₁(𝑥)∈𝐶(0,∞), and it is said to be in the space 𝐶^𝑛_𝜇 if 𝑓^𝑛∈𝐶_𝜇, 𝑛∈𝑁.

Definition 2. The Riemann-Liouville fractional integral operator of order 𝛼≥0 of a function 𝑓∈𝐶𝜇, 𝜇≥−1 is defined as 𝐽^𝛼𝑓(𝑥)=1Γ(𝛼)^𝑥₀(𝑥−𝑡)^𝛼−1𝑓(𝑡)𝑑𝑡,𝛼>0,𝑡>0 (1) in particular 𝐽⁰𝑓(𝑥)=𝑓(𝑥).
For 𝛽≥0 and 𝛾≥−1. The operator 𝐽^𝛼 has the following properties(1)𝐽^𝛼𝐽^𝛽𝑓(𝑥)=𝐽^𝛼+𝛽𝑓(𝑥), (2)𝐽^𝛼𝐽^𝛽𝑓(𝑥)=𝐽^𝛽𝐽^𝛼𝑓(𝑥), (3)𝐽^𝛼𝑥^𝛾=Γ(𝛾+1)/Γ(𝛼+𝛾+1)^{𝑥𝛼+𝛾}.

Definition 3. The Caputo fractional derivative of 𝑓∈𝐶^𝑚₋₁, 𝑚∈𝑁, is defined as 𝐷^𝛼𝑓(𝑥)=1Γ(𝑚−𝛼)^𝑥₀(𝑥−𝑡)^{𝑚−𝛼−1}𝑓^𝑚(𝑡)𝑑𝑡,𝑚−1<𝛼≤𝑚. (2)

Lemma 4. If 𝑚−1<𝛼≤𝑚, 𝑚∈𝑁, 𝑓∈𝐶^𝑚_𝜇, 𝜇>−1, then the following two properties hold(1)𝐷^𝛼[𝐽^𝛼𝑓(𝑥)]=𝑓(𝑥), (2)𝐽^𝛼[𝐷^𝛼𝑓(𝑥)]=𝑓(𝑥)−∑^𝑚−1_𝑘=1𝑓^𝑘(0)(𝑥^𝑘/𝑘!).

3. Homotopy Perturbation Method

To illustrate the basic idea of this method, we consider the following nonlinear differential equation: 𝐴(𝑢)−𝑓(𝑟)=0,𝑟∈Ω (3) with boundary conditions 𝐵𝑢,𝜕𝑢𝜕𝑛=0,𝑟∈Γ, (4) where 𝐴 is a general differential operator, 𝐵 is a boundary operator, 𝑓(𝑟) is a known analytic function, and Γ is the boundary of the domain Ω.

In general, the operator 𝐴 can be divided into two parts 𝐿 and 𝑁, where 𝐿 is linear, while 𝑁 is nonlinear. Equation (3) therefor can be rewritten as follows: 𝐿(𝑢)+𝑁(𝑢)−𝑓(𝑟)=0. (5) By the homotopy technique [10, 17] we construct a homotopy 𝑣(𝑟,𝑝)∶Ω×[0,1]→𝑅 which satisfies 𝐻𝑣,𝑝=1−𝑝𝐿(𝑣)−𝐿𝑢₀𝐴(𝑣)(𝑟[]+𝑝−𝑓)=0𝑝∈0,1,𝑟∈Ω (6) or 𝐻𝑣,𝑝=𝐿(𝑣)−𝐿𝑢₀+𝑝𝐿𝑢₀+𝑝𝑁(𝑣)−𝑓(𝑟)=0, (7) where 𝑝∈[0,1]is an embedding parameter and 𝑢₀ is an initial approximation of (3) which satisfies the boundary conditions.

From (6) and (7), we have 𝐻(𝑣,0)=𝐿(𝑣)−𝐿𝑢₀𝐻()(𝑣)(𝑟)=0,𝑣,1=𝐴−𝑓=0. (8) The change in the process of 𝑝 from zero to unity is just that of 𝑣(𝑟,𝑝) from 𝑢₀(𝑟) to 𝑢(𝑟). In topology, this is called deformation and 𝐿(𝑣)−𝐿(𝑢₀), and 𝐴(𝑣)−𝑓(𝑟) are called homotopic.

Now, assume that the solution of (6) and (7) can be expressed as 𝑣=𝑣₀+𝑝𝑣₁+𝑝²𝑣₂+⋯. (9) The approximate solution of (3) can be obtained by setting 𝑝=1: 𝑢=lim_𝑝→1𝑣=𝑣₀+𝑣₁+𝑣₂+⋯. (10)

4. Analysis on Convergence and Solution

Consider the following fractional partial differential equations: ^𝑐𝐷^𝛼_𝑡𝑢(𝑡)=𝑓𝑡,𝑢(𝑡),𝐷^𝑛1𝑢(𝑡),𝐷^𝑛2𝑢(𝑡),…,𝐷^𝑛𝑞𝑢(𝑡,[]𝑢)𝑡∈0,𝑇_𝑘(0)=𝑏^𝑘,𝑢(𝑥,𝑡)=𝑔(𝑥,𝑡),𝑘=0,1,2,…, (11)(12) where 𝐷^𝛼_𝑡=𝜕^𝛼/𝜕𝑡^𝛼 is Caputo fractional derivative of order 𝛼, 𝑚−1≤𝛼≤𝑚, 𝑛_𝑖∈ℕ for every 𝑖. Consider that 𝑓∶[0,𝑇]×𝑅×𝑅×⋯×𝑅→𝑅 is a continuous mapping. 𝑓(𝑡,𝑢₁,𝑢₂,…,𝑢_𝑛) exists with continuous and bounded derivatives 𝜕𝑓/𝜕𝑢_𝑖 satisfing the Lipschitz condition 𝑓𝑡,𝑢₁(𝑡),𝐷^𝑛1𝑢₁(𝑡),…,𝐷^𝑛𝑞𝑢₁(𝑡)−𝑓𝑡,𝑢₂(𝑡),𝐷^𝑛1𝑢₂(𝑡),…,𝐷^𝑛𝑞𝑢₂(𝑡)≤𝐿𝑢₁,𝐷^𝑛1𝑢₁,𝐷^𝑛2𝑢₁,…,𝐷^𝑛𝑞𝑢₁−𝑢₂,𝐷^𝑛1𝑢₂,𝐷^𝑛2𝑢₂,…,𝐷^𝑛𝑞𝑢₂,𝑡≥0, (13) where 𝐿 is Lipschitz constant.

To illustrate the basic concepts of HPM for fractional partial differential equation (11) with the initial conditions (12), we construct the following homotopy for (11): 1−𝑝^𝑐_𝑎𝐷^𝛼_𝑡𝑢(𝑥,𝑡)+𝑝^𝑐_𝑎𝐷^𝛼_𝑡𝑢(𝑥,𝑡)−𝑓(𝑡,𝑢(𝑡),𝐷^𝑛1𝐷𝑢,_𝑛2𝑢(𝑡),…,𝐷^𝑛𝑞𝑢(𝑡)=0 (14) or ^𝑐_𝑎𝐷^𝛼_𝑡𝑢()𝑓((𝑡)𝑥,𝑡=𝑝𝑡,𝑢,𝐷^𝑛1𝑢(𝑡),𝐷^𝑛2𝑢(𝑡),…,𝐷^𝑛𝑞𝑢(𝑡). (15) Substituting (9) into (15) and equating the terms with having identical power of 𝑝, we obtain the following series of equations: 𝑝⁰∶^𝑐_𝑎𝐷^𝛼_𝑡𝑢₀()(),𝑝𝑥,𝑡=𝑓𝑥,𝑡₁∶^𝑐_𝑎𝐷^𝛼_𝑡𝑢₁(𝑥,𝑡)=𝑓𝑡,𝑢₀,𝐷^𝑛1𝑢₀(𝑡),𝐷^𝑛2𝑢₀(𝑡),…,𝐷^𝑛𝑞𝑢₀(𝑡),𝑝²∶^𝑐_𝑎𝐷^𝛼_𝑡𝑢₂(𝑥,𝑡)=𝑓𝑡,𝑢₁,𝐷^𝑛1𝑢₁(𝑡),𝐷^𝑛2𝑢₁(𝑡),…,𝐷^𝑛𝑞𝑢₁(𝑡),⋮𝑝^𝑛∶^𝑐_𝑎𝐷^𝛼_𝑡𝑢_𝑛(𝑥,𝑡)=𝑓𝑡,𝑢_𝑛−1,𝐷^𝑛1𝑢_𝑛−1(𝑡),𝐷^𝑛2𝑢_𝑛−1(𝑡),…,𝐷^𝑛𝑞𝑢_𝑛−1(𝑡)⋮ (16) Operating with Riemann-Liouville fractional operator 𝐽^𝛼, which is the inverse operator of Caputo derivative ^𝐶𝐷^𝛼_𝑎 in both sides of (16) the solution 𝑢₀(𝑥,𝑡)=^𝑛−1_𝑘=0𝑏^𝑘𝑡^𝑘𝑘!+𝐽^𝛼𝑓(,𝑢𝑥,𝑡)₁(𝑥,𝑡)=𝐽^𝛼𝑓𝑡,𝑢₀,𝐷^𝑛1𝑢₀(𝑡),𝐷^𝑛2𝑢₀(𝑡),…,𝐷^𝑛𝑞𝑢₀(𝑡)𝑢₂(𝑥,𝑡)=𝐽^𝛼𝑓𝑡,𝑢₁,𝐷^𝑛1𝑢₁(𝑡),𝐷^𝑛2𝑢₁(𝑡),…,𝐷^𝑛𝑞𝑢₁(𝑡),⋮𝑢_𝑛(𝑥,𝑡)=𝐽^𝛼𝑓𝑡,𝑢_𝑛−1,𝐷^𝑛1𝑢_𝑛−1(𝑡),𝐷^𝑛2𝑢_𝑛−1,…,𝐷^𝑛𝑞𝑢_𝑛−1(𝑡)⋮ (17) The solution of (11) in series form is given by 𝑢(𝑥,𝑡)=𝑢₁(𝑥,𝑡)+𝑢₂(𝑥,𝑡)+𝑢₃(𝑥,𝑡)+⋯. (18) Define that (𝐶[0,𝑇],‖⋅‖) is the Banach space, the space of all continuous functions on [0,𝑇] with the norm 𝑓(𝑡)=max_{[]∀𝑡∈0,𝑇}𝑓(𝑡). (19)

4.1. Existence and Uniqueness of Solutions

Theorem 5. Let 𝑓 satisfy the Lipschitz condition (13) and then the problem (11) has unique solution 𝑢(𝑥,𝑡), whenever 0<𝛾<1.

Proof. Let 𝑦 and 𝑧 be two different solutions of (11), and for All 𝑡∈[0,𝑇] and 𝜏∈[0,𝑡] is bounded. Let 𝑀=max_{0≤𝜏≤𝑡,0≤𝑡≤𝑇}|(𝑡−𝜏)^𝛼−1|; then, 𝑦−𝑧=𝐽^𝛼𝑓(𝑡),𝑦(𝑡),𝐷^𝑛1𝑦(𝑡),𝐷^𝑛2𝑦(𝑡),…,𝐷^𝑛𝑞𝑦(𝑡)−𝐽^𝛼𝑓(𝑡),𝑧(𝑡),𝐷^𝑛1𝑧(𝑡),𝐷^𝑛2𝑧(𝑡),…,𝐷^𝑛𝑞𝑧(𝑡),1𝑦−𝑧=Γ(𝛼)^𝑡₀(𝑡−𝜏)^𝛼−1𝑓𝜏,𝑦(𝑡),𝐷^𝑛1𝑦(𝑡),𝐷^𝑛2𝑦,…,𝐷^𝑛𝑞𝑦(𝑡−1)𝑑𝜏Γ(𝛼)^𝑡₀(𝑡−𝜏)^𝛼−1×𝑓𝜏,𝑧(𝑡),𝐷^𝑛1𝑧(𝑡),𝐷^𝑛2𝑧(𝑡),…,𝐷^𝑛𝑞𝑧(𝑡)=1𝑑𝜏,𝑦−𝑧Γ(𝛼)^𝑡₀(𝑡−𝜏)^𝛼−1𝑓𝜏,𝑦(𝑡),𝐷^𝑛1𝑦(𝑡),𝐷^𝑛2𝑦,…,𝐷^𝑛𝑞𝑦(𝑡)−1𝑑𝜏Γ(𝛼)^𝑡₀(𝑡−𝜏)^𝛼−1𝑓𝜏,𝑧(𝑡),𝐷^𝑛1𝐷𝑧,_𝑛2𝑧(𝑡),…,𝐷^𝑛𝑞𝑧(𝑡)≤𝐿𝑑𝜏Γ(𝛼)^𝑡₀(𝑡−𝜏)^𝛼−1≤𝐿𝑦−𝑧𝑑𝜏,max𝑦−𝑧Γ(𝛼)max^𝑡₀(𝑡−𝜏)^𝛼−1≤𝑦−𝑧𝑑𝜏,𝑦−𝑧𝐿𝑀𝑇Γ(𝛼)^𝑛⋅1,𝑛!𝑦−𝑧≤𝛾𝑦−𝑧1−𝛾𝑦−𝑧≤0. (20) Since 1−𝛾≠0, then ‖𝑦−𝑧‖=0; therefore, 𝑦=𝑧, and this completes the proof.

4.2. Proof of the Convergence

Theorem 6. Let 𝑢_𝑛(𝑥,𝑡) and 𝑢(𝑥,𝑡) be defined in Banach space (𝐶[0,𝑇],‖⋅‖). Then the series solution {𝑢_𝑛(𝑥,𝑡)}^∞_𝑛=1 defined by (18) converges to the solution of (11), if 0<𝛾<1.

Proof. Suppose that {𝑠_𝑛} is the sequence of partial sums of the series (18) and we need to show that 𝑠_𝑛(𝑡) is a Cauchy sequence in Banach space (𝐶[0,𝑇,‖⋅‖]). For this, we consider 𝑠_𝑛+1(𝑡)−𝑠_𝑛(𝑡)=𝑢_𝑛+1(𝑡𝑢)≤𝛾_𝑛(𝑡)≤𝛾²𝑢_𝑛−1(𝑡)≤⋯≤𝛾^𝑛+1𝑢₀(𝑡). (21) Now, for every 𝑛,𝑚∈𝑁, 𝑛≥𝑚, there are two arbitrary partial sums 𝑠_𝑛 and 𝑠_𝑚; by using (21) and triangle inequality successively, we have 𝑠_𝑛−𝑠_𝑚=𝑠_𝑛(𝑡)−𝑠_𝑛−1(𝑡)+𝑠_𝑛−1(𝑡)−𝑠_𝑛−2(𝑡)+⋯+𝑠_𝑚+1(𝑡)−𝑠_𝑚(𝑡)≤𝑠_𝑛−𝑠_𝑛−1+𝑠_𝑛−1−𝑠_𝑛−2+⋯+𝑠_𝑚+1−𝑠_𝑚≤𝛾^𝑛+𝛾^𝑛−1+𝛾^𝑛−2+⋯+𝛾^𝑚+1×𝑢₀(𝑡)≤𝛾^𝑚+1𝛾^{𝑛−𝑚−1}+𝛾^{𝑛−𝑚−2}+⋯+𝛾+1𝑢₀(𝑡)≤𝛾^𝑚+11−𝛾^𝑛−𝑚𝑢1−𝛾⁰(𝑡). (22) Since 0<𝛾<1, we have (1−𝛾^𝑛−𝑚)<1; then, 𝑠_𝑛−𝑠_𝑚≤𝛾^𝑚+11−𝛾max_{[]∀𝑡∈0,𝑇}𝑢₀(𝑡). (23) Since 𝑢₀ is bounded, lim_{𝑛,𝑚→∞}𝑠_𝑛(𝑡)−𝑠_𝑚(𝑡)=0. (24) Therefore, 𝑠_𝑛(𝑡) is a Cauchy sequence in 𝐶[0,𝑇], so the series converges and the proof complete.

4.3. Error Estimate

Theorem 7. The maximum absolute truncation error of the series solution (18) of the problem (11) is estimated to be 𝑢(𝑥,𝑡)−^𝑚_𝑖=0𝑢_𝑖(≤𝛾𝑥,𝑡)^𝑚+1𝑢1−𝛾⁰(𝑡). (25)

Proof. From Theorem 6 and inequality (22), we have 𝑢(𝑡)−𝑠_𝑚≤𝛾^𝑚+11−𝛾^𝑛−𝑚𝑢1−𝛾⁰(𝑡). (26) Since 0<𝛾<1, we have 1−𝛾^𝑛−𝑚<1; thus, we get the formula (25) 𝑢(𝑥,𝑡)−^𝑚_𝑖=0𝑢_𝑖(≤𝛾𝑥,𝑡)^𝑚+1𝑢1−𝛾⁰(𝑡). (27)
This completes the proof.

5. Applications

Example 8. Consider the following fractional partial differential equations with initial condition: 𝜕^𝛼𝑢𝜕𝑡^𝛼+𝑢𝜕𝑢=𝜕𝜕𝑥²𝑢𝜕𝑥²,()1𝑥,𝑡∈𝑅×0,2𝑢(𝑥,0)=2𝑥 (28)(29) with the exact solution at special case 𝛼=1𝑢(𝑥,𝑡)=2𝑥.1+2𝑡 (30) To solve (28) with initial condition (29), according to the homotopy perturbation technique, we construct the following homotopy: 𝜕1−𝑝^𝛼𝑢𝜕𝑡^𝛼−𝜕^𝛼𝑢₀𝜕𝑡^𝛼𝜕=𝑝²𝑢𝜕𝑥²−𝑢𝜕𝑢−𝜕𝜕𝑥^𝛼𝑢𝜕𝑡^𝛼 (31) or 𝜕^𝛼𝑢𝜕𝑡^𝛼−𝜕^𝛼𝑢₀𝜕𝑡^𝛼𝜕=𝑝²𝑢𝜕𝑥²−𝑢𝜕𝑢−𝜕𝜕𝑥^𝛼𝑢₀𝜕𝑡^𝛼. (32) Substituting of (9) in (32) and then equating the terms with same powers of 𝑝, we get the series 𝑝⁰∶𝜕^𝛼𝑢₀𝜕𝑡^𝛼𝑝=0,₁∶𝜕^𝛼𝑢₁𝜕𝑡^𝛼=𝜕²𝑢₀𝜕𝑥²−𝑢₀𝜕𝑢₀−𝜕𝜕𝑥^𝛼𝑢₀𝜕𝑡^𝛼,𝑝²∶𝜕^𝛼𝑢₂𝜕𝑡^𝛼=𝜕²𝑢₁𝜕𝑥²−𝑢₁𝜕𝑢₀𝜕𝑥−𝑢₀𝜕𝑢₁,𝑝𝜕𝑥₃∶𝜕^𝛼𝑢₃𝜕𝑡^𝛼=𝜕²𝑢₂𝜕𝑥²−𝑢₂𝜕𝑢₀𝜕𝑥−𝑢₁𝜕𝑢₁𝜕𝑥−𝑢₀𝜕𝑢₂,⋮𝑝𝜕𝑥_𝑖∶𝜕^𝛼𝑢_𝑖𝜕𝑡^𝛼=𝜕²𝑢_𝑖−1𝜕𝑥²−^𝑖−1_𝑗=0𝑢_𝑗𝜕𝑢_{𝑖−𝑗−1}.𝜕𝑥 (33) Operating with Riemann-Liouville fractional operator 𝐽^𝛼, which is the inverse operator of Caputo derivative ^𝑐𝐷^𝑎_𝑎 in both sides of (28), the solution reads 𝑢₀()𝑢𝑥,𝑡=2𝑥,₁(𝑥,𝑡)=−4𝑥𝑡^𝛼Γ(),𝑢𝛼+1₂(𝑥,𝑡)=16𝑥𝑡^2𝛼Γ(),𝑢2𝛼+1₃(𝑥,𝑡)=−16𝑥𝑡^3𝛼Γ()Γ()23𝛼+14+2𝛼+1Γ²(),⋮𝛼+1 (34) Since 𝑡≤𝛾/2, 0<𝛾<1 and 𝛼=1, and according to Theorem 6, we have 𝑠₁−𝑠₀=−4𝑥𝑡^𝛼Γ()=)𝛼+1(2𝑥−2𝑡^𝛼Γ()𝛾𝛼+1≤42‖2𝑥‖=𝛾𝑢₀,𝑠₂−𝑠₀≤𝑠₂−𝑠₁+𝑠₁−𝑠₀=16𝑥𝑡^2𝛼Γ()+2𝛼+1−4𝑥𝑡^𝛼Γ()=𝛼+1(2𝑥)8𝑡^2𝛼Γ()+2𝛼+(2𝑥)−2𝑡^𝛼Γ()=‖‖8𝛾𝛼+12𝑥2²𝛾+22𝛾≤162²‖2𝑥‖=𝛾²𝑢₀,𝑠₃−𝑠₁≤𝑠₃−𝑠₂+𝑠₂−𝑠₁=−16𝑥𝑡^3𝛼Γ()Γ()3𝛼+14+2𝛼+1Γ²()+𝛼+116𝑥𝑡^2𝛼Γ()=)2𝛼+1(2𝑥−8𝑡^3𝛼Γ()Γ()3𝛼+4+2𝛼Γ²()+𝛼+1(2𝑥)8𝑡^2𝛼Γ()≤8𝛾3𝛼2³𝛾+82²𝛾≤322³‖2𝑥‖=𝛾³𝑢₀. (35)
And so on, finally we get 𝑠_𝑛−𝑠_𝑚≤𝑠_𝑛−𝑠_𝑛−1+𝑠_𝑛−1−𝑠_𝑛−2+𝑠_𝑚+1−𝑠_𝑚=^𝑛_𝑗=0𝑢_𝑗−^𝑛−1_𝑖=0𝑢_𝑖+^𝑛−1_𝑗=0𝑢_𝑗−^𝑛−2_𝑖=0𝑢_𝑖+⋯+^𝑚+1_𝑗=0𝑢_𝑗−^𝑚_𝑖=0𝑢_𝑖=𝑢_𝑛+𝑢_𝑛−1+⋯+𝑢_𝑚≤2^2𝑛+2𝛾2^𝑛+1𝑢₀=𝛾^𝑛𝑢₀. (36) Therefore, lim_{𝑛,𝑚→∞}‖𝑠_𝑛−𝑠_𝑚‖≤lim_𝑛→∞𝛾^𝑛‖𝑢₀‖=0; that is, 𝑠_𝑛 is, a Cauchy sequence.
Then the approximate solution in a series form is 𝑢(𝑥,𝑡)=𝑢₀(𝑥,𝑡)+𝑢₁(𝑥,𝑡)+𝑢₂(𝑥,𝑡)+𝑢₃(𝑥,𝑡)+⋯=2𝑥−4𝑥𝑡^𝛼Γ()+𝛼+116𝑥𝑡^2𝛼Γ()−2𝛼+116𝑥𝑡^3𝛼Γ()×Γ()3𝛼+14+2𝛼+1Γ²(𝛼+1)+⋯. (37)

Example 9. Consider the fourth-order parabolic partial differential equation of fractional order 𝜕^𝛼𝑢𝜕^𝛼𝑡+𝑦+𝑧𝜕2cos𝑥−1⁴𝑢𝜕𝑥⁴+𝑧+𝑥𝜕2cos𝑦−1⁴𝑢𝜕𝑦⁴+𝑥+𝑦𝜕2cos𝑧−1⁴𝑢𝜕𝑧⁴𝜋=0,0<𝑥,𝑦,𝑧<3,𝑡∈[0,1],1<𝛼≤2 (38) subject to the initial conditions 𝑢𝑥,𝑦,𝑧,0=−𝜕𝑢.𝜕𝑡𝑥,𝑦,𝑧,0=𝑥+𝑦+𝑧−cos𝑥+cos𝑦+cos𝑧 (39) The exact solution for special case 𝛼=2 is 𝑢𝑥,𝑦,𝑧,𝑡=𝑥+𝑦+𝑧−cos𝑥−cos𝑦−cos𝑧𝑒^−𝑡. (40) For solving (38), according the homotopy perturbation method, we have 𝜕^𝛼𝑢₀𝜕𝑡^𝛼𝜕=0,_𝛼𝑢₁𝜕𝑡^𝛼=−𝑦+𝑧𝜕2cos𝑥−1⁴𝑢₀𝜕𝑥⁴−𝑧+𝑥𝜕2cos𝑦−1⁴𝑢₀𝜕𝑦⁴−𝑥+𝑦𝜕2cos𝑧−1⁴𝑢₀𝜕𝑧⁴−𝜕^𝛼𝑢₀𝜕^𝛼𝑡,𝜕^𝛼𝑢₂𝜕𝑡^𝛼=−𝑦+𝑧𝜕2cos𝑥−1⁴𝑢₁𝜕𝑥⁴−𝑧+𝑥𝜕2cos𝑦−1⁴𝑢₁𝜕𝑦⁴−𝑥+𝑦𝜕2cos𝑧−1⁴𝑢₁𝜕𝑧⁴⋮ (41) By applying 𝐽^𝛼 in both sides of (41), we got 𝑢₀),𝑢𝑥,𝑦,𝑧,𝑡=𝑥+𝑦+𝑧−cos𝑥+cos𝑦+cos𝑧(1−𝑡₁×𝑡𝑥,𝑦,𝑧,𝑡=𝑥+𝑦+𝑧−cos𝑥+cos𝑦+cos𝑧_𝛼Γ()−𝑡𝛼+1Γ(^𝛼+1),𝑢𝛼+2₁×𝑡𝑥,𝑦,𝑧,𝑡=𝑥+𝑦+𝑧−cos𝑥+cos𝑦+cos𝑧_2𝛼Γ()−𝑡2𝛼+1Γ(^2𝛼+1),⋮𝑢2𝛼+2_𝑛×𝑡𝑥,𝑦,𝑧,𝑡=𝑥+𝑦+𝑧−cos𝑥+cos𝑦+cos𝑧_𝑛𝛼Γ()−𝑡𝑛𝛼+1Γ(^𝑛𝛼+1)⋮𝑛𝛼+2 (42)
According to Theorem 6, we have 𝑠₂−𝑠₁≤𝑠₂−𝑠₁+𝑠₁−𝑠₀≤𝑡^2𝛼Γ()−𝑡2𝛼+1Γ(^2𝛼+1)+𝑡2𝛼+2Γ(^𝛼)−𝑡𝛼+1Γ(^𝛼+1)≤)𝑡𝛼+(1−𝑡^2𝛼/Γ(2𝛼+1)−𝑡^(2𝛼+1)/Γ(2𝛼+2)+)𝑡1−𝑡(1−𝑡^𝛼/Γ(𝛼+1)+𝑡^(𝛼+1)/Γ(𝛼+2).1−𝑡 (43) But for all 𝑡∈[0,1] and 𝛼=2, we have 𝑡^2𝛼/Γ(2𝛼+1)−𝑡^(2𝛼+1)/Γ(2𝛼+2)𝑡1−𝑡≤𝛾=0.0047<1_𝑎/Γ(𝛼+1)−𝑡^(𝛼+1)/Γ(𝛼+2)1−𝑡≤𝛾=0.21<1. (44) Thus, ‖𝑠₂−𝑠₁‖≤𝛾²‖𝑢₀‖𝑠₃−𝑠₁≤𝑠₃−𝑠₂+𝑠₂−𝑠₁≤𝑡^3𝛼Γ()−𝑡3𝛼+1Γ(^3𝛼+1)+𝑡3𝛼+2Γ(^2𝛼)−𝑡2𝛼+1Γ(^2𝛼+1)≤𝑡2𝛼+2(1−𝑡)^3𝛼/Γ(3𝛼+1)−𝑡^(3𝛼+1)/Γ(3𝛼+2)+)𝑡1−𝑡(1−𝑡^2𝛼/Γ(2𝛼+1)−𝑡^(2𝛼+1)/Γ(2𝛼+2)≤‖‖𝑡1−𝑡1−𝑡^3𝛼/Γ(3𝛼+1)−𝑡^(3𝛼+1)/Γ(3𝛼+2)+𝑡1−𝑡_2𝛼/Γ(2𝛼+1)−𝑡^(2𝛼+1)/Γ(2𝛼+2).1−𝑡 (45) Since, for all 𝑡∈[0,1], 0<𝛾<1, and 𝛼=2, we have 𝑡^3𝛼/Γ(3𝛼+1)−𝑡^(3𝛼+1)/Γ(3𝛼+2)𝑡1−𝑡≤𝛾=0.0004<1,_2𝛼/Γ(2𝛼+1)−𝑡^(2𝛼+1)/Γ(2𝛼+2)1−𝑡≤𝛾=0.0047<1. (46) Thus 𝑠₃−𝑠₁≤𝛾³𝑢₀, (47) where 𝛾³=((1−𝛾²)/(1−𝛾))𝛾².
And so on, finally we get𝑠_𝑛−𝑠_𝑚≤𝑠_𝑛−𝑠_𝑛−1+𝑠_𝑛−1−𝑠_𝑛−2+⋯+𝑠_𝑚+1−𝑠_𝑚=𝑡^𝑛𝛼Γ()−𝑡𝑛𝛼+1Γ(^𝑛𝛼+1)+𝑡𝑛𝛼+2_{(𝑛−1)𝛼}Γ))−𝑡((𝑛−1𝛼+1Γ)^{(𝑛−1)𝛼+1})𝑡((𝑛−1𝛼+2+⋯+^(𝑚+1)𝛼Γ))−𝑡((𝑚+1𝛼+1Γ)^{(𝑚+1)𝛼+1})≤)𝑡((𝑚+1𝛼+2(1−𝑡^𝑛𝛼/Γ(𝑛𝛼+1)−𝑡^𝑛𝛼+1/Γ(𝑛𝛼+2)+()𝑡1−𝑡1−𝑡^{(𝑛−1)𝛼}/Γ((𝑛−1)𝛼+1)−𝑡1−𝑡_{(𝑛−1)𝛼+1}/Γ((𝑛−1)𝛼+2)()𝑡1−𝑡+⋯+1−𝑡^(𝑚+1)𝛼/Γ((𝑚+1)𝛼+1)−𝑡1−𝑡_{(𝑚+1)𝛼+1}/Γ((𝑚+1)𝛼+2)≤‖‖𝑡1−𝑡1−𝑡^𝑛𝛼/Γ(𝑛𝛼+1)−𝑡^𝑛𝛼+1/Γ(𝑛𝛼+2)+𝑡1−𝑡_{(𝑛−1)𝛼}/Γ((𝑛−1)𝛼+1)−𝑡1−𝑡_{(𝑛−1)𝛼+1}/Γ((𝑛−1)𝛼+2)𝑡1−𝑡+⋯+^(𝑚+1)𝛼/Γ((𝑚+1)𝛼+1)−𝑡1−𝑡_{(𝑚+1)𝛼+1}/Γ((𝑚+1)𝛼+2)1−𝑡 (48) Since, for all 𝑡∈[0,1], 0<𝛾<1, and 𝛼=2, we have 𝑡^𝑛𝛼/Γ(𝑛𝛼+1)−𝑡^𝑛𝛼+1/Γ(𝑛𝛼+2)1−𝑡≤𝛾^𝑛𝑡<1,_{(𝑛−1)𝛼}/Γ((𝑛−1)𝛼+1)−𝑡^{(𝑛−1)𝛼+1}/Γ((𝑛−1)𝛼+2)1−𝑡≤𝛾^𝑛−1⋮𝑡<1,_(𝑚+1)𝛼Γ))−𝑡((𝑚+1𝛼+1Γ((𝑚+1)𝛼^{(𝑚+1)𝛼+1}+2)≤𝛾^𝑚+1<1. (49) Thus, 𝑠_𝑛−𝑠_𝑚≤𝛾^𝑛𝑢₀, (50) where 𝛾^𝑛=((1−𝛾^𝑛−𝑚)/(1−𝛾))𝛾^𝑚+1. Therefore, lim𝑠_𝑛−𝑠_{𝑚𝑛,𝑚→∞}≤𝛾^𝑛𝑢₀=0. (51) That is {𝑠_𝑛} is a Cauchy sequence.
The series of the solution is 𝑢=×𝑡𝑥,𝑦,𝑧,𝑡𝑥+𝑦+𝑧−cos𝑥−cos𝑦−cos𝑧1−𝑡+^𝛼Γ()−𝑡𝛼+1Γ(^𝛼+1)+𝑡𝛼+2_2𝛼Γ()−𝑡2𝛼+1Γ(^2𝛼+1).2𝛼+2+⋯ (52)

6. Conclusion

In this paper, we applied the HPM for fractional partial differential equations and obtained highly approximate solutions with few iterations. Further, we introduce the study problem of convergence of HPM. The sufficient condition for convergence of this method has been presented. In this paper, we studied the convergence analysis of homotopy perturbation method for fractional partial differential equations, Further, we consider the convergence analysis of homotopy perturbation method for fractional integro-differential equations as future work.

The convergence analysis is reliable enough to estimate the maximum absolute truncation error of the series solution.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are very grateful to the referees for their valuable suggestions and comments that improved the paper.