Analysis of Homotopy Perturbation Method for Solving Fractional Order Differential Equations

Shumaila Javeed 1,*, Dumitru Baleanu 2,3, Asif Waheed 4 and Mansoor Shaukat Khan 1 and Hira Affan 5 1 Department of Mathematics, COMSATS University Islamabad, Park Road, Chak Shahzad Islamabad 45550, Pakistan; mansoorkhan@comsats.edu.pk 2 Department of Mathematics, Cankaya University, 06790 Ankara, Turkey; dumitru@cankaya.edu.tr 3 Institute of Space Sciences, 077125 Magurele-Bucharest, Romania 4 Department of Mathematics, COMSATS University Islamabad, Kamra Rd, Attock, Punjab 43600, Pakistan; asifwaheed8@gmail.com 5 Department of Physics, COMSATS University Islamabad, Park Road, Chak Shahzad Islamabad 45550, Pakistan; hira.affan@comsats.edu.pk * Correspondence: shumaila_javeed@comsats.edu.pk

Fractional differential equations (FDEs) involve real or complex order derivatives [11]. Various researchers contributed on fractional derivatives during the 18th and 19th centuries, for example, Abel [12], Caputo [13], Euler [14], Fourier [15], Laplace [16], Liouville [17], or Ross [18]. In 1974, Oldham and Spanier presented fractional operators with mass and heat transfer applications [19]. In this paper, the analytical technique to solve fractional order partial differential equations (PDEs) is described. In order to obtain analytical solution, HPM is used to solve fractional order PDEs. Fractional order PDEs do not have closed form exact solutions in most problems, therefore, it is required to develop efficient and accurate analytical and numerical methods. HPM is well known for its accuracy and simplicity [20,21]. HPM has been widely used to obtain approximate series solutions of fractional order linear and nonlinear PDEs [22][23][24][25][26][27].
The Burger Poisson equation is widely used to express different physical phenomena, for instance, mathematical models for shallow water and shock waves in a viscous fluid [28]. Tian and Gao proved the existence of the uni-dimensional viscous equation in 2009 [29]. Moreover, Abidi and Omrani obtained the solution of Burger Poisson equation using homotopy analysis method [30].
The obstacle problem plays a role of bridge in the field of variational inequalities and differential equations. It is originated from the study of elasticity theory. In elasticity theory, it is required to obtain the equilibrium position of elastic membrane with fixed boundaries. Obstacle problems occur in diffusion equation and signals processing while determining heat flux at the boundary of semi-infinite rod [31].
The focus of the paper is to generalize the convergence theorem, in the sense that the mapping is nonself and only Y is complete. Theorem is applied to the solution of FBP equation acquired by HPM. Furthermore, this work presents the method to get the solution of FPDEs, while the same PDE with ordinary derivative i.e., for α = 1 is not defined in the given domain. Moreover, the proposed HPM method is applied to complex obstacle BVP.
Beside the introduction, the distribution of the article is as under: Section 2 comprises of important definitions and properties, Section 3 contains the implementation of HPM to solve FPDEs and convergence theorem, Section 4 comprises of results and discussion of FBP, Section 5 contains solutions of obstacle BVP, and Section 6 includes the conclusion of the paper.

Preliminaries
Fractional calculus is a developing area in mathematical analysis. Several definitions of fractional operators have been propounded like Riemann-Liouvlle, Caputo, and Grunwald-Letnikov [13,17]. These definitions have some limitations, for instance D α a (1) = 0, do not satisfy the product, quotient and chain rules of derivatives. Recently, Khalil et al. published a definition uses for fractional derivatives called conformable [32,33]. Conformable is simpler and natural extension of the usual derivatives as it satisfies the aforementioned properties of derivatives.
Using the above mentioned definition given in Equation (1), we obtain the following useful results: Let g, f are α-differentiable and α ∈ (0, 1], then

Fractional integral:
The fractional integral of order α can be defined below: Here, J α is the Riemann improper integral.

Application of HPM to FBP Equation
Consider the time-dependent operator equation where B denotes a differential operator and w (s, t) is an unknown function. Moreover, we assume h(s, t) is an analytic function, we can decompose the operator B as; where L is linear and N is nonlinear or sometimes the complicated part to handle. Let Ω ⊆ R × R the homotopy H : Ω × [0, 1] → R, is defined as: Note that the function w 0 (s, t) is the initial guess which satisfies the given operator equation. The choice of p from zero to one provide us the deformation from w 0 (s, t) to the solution w (s, t). Clearly which is the given operator equation. Using perturbation method, we suppose the solution in power series as; is an approximate solution to the given operator equation.
Remark 1. The Banach contraction type theorem about the convergence of the solution stated in [34]. Here the generalized, corrected, and unified form is presented in which the completeness of X is not required and Y must be a subset of X. If Y ⊆ X, we cannot say about the sequence x n+1 = N n (x 0 ) = N (x n ) is contained in Y or not. On the basis of above discussion, a unified theorem is presented as follows: for some k ∈ [0, 1), then the sequence for any x 0 ∈ X converges to a unique fixed point of N.
Proof. We consider the picard sequence x n+1 = N (x n ) ⊆ Y, it will be shown that this sequence (x n ) is Cauchy in Y. For integers m ≥ n consider, Using the contractive condition (C) , and induction on n, is given as This shows that (x n ) is Cauchy sequence in Y, completeness of Y allows us to find z ∈ Y, such that Clearly, (C) ensures the continuity of N, thus This completes the proof, the uniqueness of z is obvious. The proof of Theorem 1 is similar to the proof given in [34,35], but our case is generalized, in the sense that the mapping is nonself and only Y is complete. Now, the extended HPM using conformable is presented to solve space-time FBP equation.

Test Problem 1
The partial differential FBP equation in unidirectional propagation water waves can be described as follows [36,37]: Afterwards, the fractional integral operator J α with conformable derivative definition (c.f. Equation (2)) is applied on both sides of Equation (6), we have . . .
In order to calculate next terms, we have If we define N : C (Ω) → C (Ω) , with iterative sequence as, Then for any w 0 ∈ C (Ω) , by Theorem 1, the sequence w l+1 = N (w l ) converges to the unique solution w of the given FBP.
The Equation (8) can be calculated with the help of symbolic softwares, for instance, Mathematica and Maple. The HPM solution is given below: The HPM solution of FBP equation when α = 1 is as follows: Remark 2. It is remarked that the exact and HPM solution of FBP equation when α = 1, given in Equation (12) does not exist at t = 1, while for any α ∈ (0, 1) , the solution given in Equation (10) of FBP equation exists. This shows the importance of fractional derivative and its way of dealing these types of the situations where solution of some ordinary PDEs fail to exist.

Convergence of solution:
The FBP equation is as follows: with w(x, 0) = −x.
The approximate first four term solution of FBP equation for p = 1 is given by where, The sequence generated by HPM will be regarded as We assume that V 0 = w 0 . According to theorem for non-linear mapping N, a sufficient condition for convergence of HPM is strictly contraction N. Therefore, we have Therefore, lim n→∞ V n − w γ n V 0 − w = 0, that is, w(x, t) = lim n→∞ V n = −xα+t α α−t α , which is an exact solution.

Discussion
In this section, results obtained by HPM are discussed. Analytical series solution of space-time FBP equation is given in Equation (9). In Figure 1, a solution is presented for different values of α, such as (0.1, 0.3, 0.5, 0.7). Figure 1 shows a big difference in smaller and larger values of α. For larger values of α, w(x, t) attains height and for values closer to 0, height of w(x, t) reduces. In Figure 2, results are presented for α = 0.8 and α = 0.9, respectively. Exact solution of FBP model for α = 1 is given in Equation (12). Equation (12) shows discontinuity at t = 1 which is clearly depicted in Figure 2. As value of α approaches to 1, the shock is produced in the vicinity of t = 1.

Test Problem 2
Consider the fractional second-order obstacle BVP: In order to apply HPM, we construct homotopy in three different domains: Here p is parameter that lies between 0 and 1. The solution v(x) is given as follows: Now, substitute Equation (3) in Equation (2), and comparing the coefficients of similar powers of p, we get Now, the fractional integral operator J α with comfortable derivative definition is applied on both sides of Equation (5) Case II: Now, substitute Equation (3) in Equation (7), and comparing the coefficients of similar powers of p, we get Now, the fractional integral operator J α with comfortable derivative definition is applied on both sides of Equation (8) Case III: 5 2 ≤ x ≤ 3 In this case, the constructed homotopy will be same as in Case I. After substituting Equation (3) in Equation (2) We calculate the results by taking α = 1.25, 1.5 and 1.75.

For
In similar manners, we can find the solution of problem mentioned in Equation (1) for α = 1.5 and α = 1.75.
2. For α = 1. 5 We get the following analytical solution of system given in Equation (1) 3. For α = 1. 75 We obtain the following analytical solution of the considered problem given in Equation (1) for 15 4 +0.1294835584x 11 4 , f or 19 4 −0.1417523126x 15 4 + 0.5286045753x 11 4 , f or Discussion  Figure 6 shows the comparison between different values of α. In Figure 6, obstacle achieves more height for larger values of α and vice versa.

Conclusions
The analysis of HPM for the solution of FPDEs was given. A unified convergence theorem was proved and results were validated for the solution of FBP equation. The method to solve FPDEs was presented, while the same partial differential equation with ordinary derivative i.e., for α = 1 fails to exist. This study demonstrated the importance of fractional derivative and the technique of dealing with these types of PDEs where solution of some ordinary PDEs does not exist. Moreover, HPM was applied to solve complex obstacle BVP. The suggested method can be applied to find solutions of other PDEs (both linear and nonlinear) of fractional order. The present study can be useful to analyze other traditional analytical techniques, such as Adomian Decomposition Method and Homotopy Analysis Method, to solve nonlinear differential equation of non-integer order. Furthermore, the present work may be extended to solve practical fractional models, for example wireless networks and nonlinear obstacle problems.