Homotopy Analysis Method for Three Types of Fractional Partial Differential Equations

In this paper, three types of fractional order partial differential equations, including the fractional Cauchy–Riemann equation, fractional acoustic wave equation, and two-dimensional space partial differential equation with time-fractional-order, are considered, and these models are obtained from the standard equations by replacing an integer-order derivative with a fractionalorder derivative in Caputo sense. Firstly, we discuss the fractional integral and differential properties of several functions which are derived from the Mittag-Leffler function. Secondly, by using the homotopy analysis method, the exact solutions for fractional order models mentioned above with suitable initial boundary conditions are obtained. Finally, we draw the computer graphics of the exact solutions, the approximate solutions (truncation of finite terms), and absolute errors in the limited area, which show that the effectiveness of the homotopy analysis method for solving fractional order partial differential equations.


Introduction
e concept of fractional derivatives can be traced back to a question raised by Marquis de L'Hopital to Gottfried Wilhelm Leibniz three hundred years ( [1][2][3]) ago. Due to the lack of geometric or physical background support, fractional calculus has not entered the field of vision of most researchers. Until recent decades, some researchers have found that the fractional models are better than integer models in describing the chemical process, diffusion reaction, financial vibration, and other fields. erefore, the fractional order problems gradually attracted the interest of many researchers, and the application expanded to many scientific fields including fluid flow, rheology, dynamical processes in self-similar and porous structures, diffusive transport akin to diffusion, electrical networks, probability and statistics, control theory of dynamical systems, viscoelasticity, electrochemistry of corrosion, optics and signal processing [4][5][6][7][8], and so on. ese applications in interdisciplinary sciences motivate us to try to find out numerical or analytic solutions for the fractional differential equations. Now, many effective methods for fractional differential equations have been presented, such as the finite difference method [9], spectral method [5], matrix approach [10], homotopy analysis method (HAM) [11], and homotopy perturbation method [12]. Especially, the HAM was first proposed by Liao in [13]. is method has been successfully applied to solve various linear or nonlinear problems [8,[14][15][16][17][18]. e following is a brief survey.
In 2007, firstly, the HAM that was developed for an integer-order differential equation was directly extended to derive explicit and numerical solutions of nonlinear fractional differential equations by by Song and Zhang [19]. In 2008, Xu and Cang [20] employed the HAM to derive the solutions of the time fractional wave-like differential equations with a variable coefficient. In [21], HAM is applied to solve linear and nonlinear fractional initial-value problems by Hashim ea al\enleadertwodots. In 2010, Dehghan et al. [22] applied HAM to solve several fractional KDV equations and showed the high accuracy and efficiency of the proposed technique. In [23], Abbasbandy et al. used HAM to obtain approximate solutions of fractional integrodifferential equations and gave some examples to illustrate the high efficiency and precision of this method. Very recently, Morales-Delgado et al. [24] have presented an analysis based on a combination of the Laplace transform and homotopy methods in order to provide new analytical approximated solutions of the fractional partial differential equations in the Liouville-Caputo and Caputo-Fabrizio senses. In 2019, an optimal homotopy analysis approach [25] was proposed to deal with nonlinear fractional differential equations, and the corresponding optimal initial approximation was discussed. For further information about the HAM, refer to [26][27][28].
In this paper, we use the HAM to solve fractional Cauchy-Riemann equations [29] and fractional acoustic wave equations [30], which are given by where 0 < α, β ≤ 1 and b 0 and c 0 are positive numbers and to solve the partial differential equation with a time-fractionalorder of the following form: with a time-fractional order 1 < α ≤ 2, and c D α 0,t (·) is the Caputo fractional derivative.
By setting α � 1 and β � 1, the models (1) and (2) were transformed into Cauchy-Riemann equations and acoustic wave equations, respectively. Suppose that v and ρ satisfy the Cauchy-Riemann equations in an open subset of R 2 , and consider the vector field [v, ρ] T . In fluid dynamics, such a vector field is a potential flow. In the second model, b 0 and c 0 denote the medium density and the propagation velocity without an acoustic disturbance, respectively. Model (3) is a two-dimensional space and time-fractional-order model, which has important applications in many fields. e rest of this paper is organized as follows. In Section 2, the definitions and properties of fractional derivatives of some functions are introduced. Section 3 gives an introduction of the HAM which is used to solve fractional order differential equations. In Section 4, the analytic solutions for three types of fractional order differential equations were obtained by using the HAM, and the computer graphics of the exact solutions, the approximate solutions, and absolute errors were drawn in the limited area to clarify the effectiveness of the HAM. Finally, Section 5 offers some concluding remarks.

Definitions and Lemmas
Definition 1. (see [3]). A real function f(x), x > 0, is said to be in the space C μ , μ ∈ R, if there exists a real number p > μ, such that f(x) � x p f 1 (x), where f 1 (x) ∈ C (0,∞) , and it is said to be in the space C n μ , if and only if f n ∈ C μ , n ∈ N.

HAM
In this section, we consider a linear or nonlinear equation in a general form: where u(x, t) is an unknown function and x and t are independent variables. Let u 0 (x, t) denote an initial approximation of the solution of equation (31), h a nonzero auxiliary parameter, H(x, t) a nonzero auxiliary function, and L is an auxiliary linear operator. en, we construct the HAM deformation equation in the following form: where q ∈ [0, 1] is an embedding parameter. Obviously, when q � 0 and q � 1, the abovementioned HAM deformation equation (32) has the solutions respectively. us, as q increases from 0 to 1, Φ(x, t; q) varies from the initial guesses Φ(x, t; 0) to the solution Φ(x, t; 1) of equation (31). Expanding Φ(x, t; q) in Taylor's series with respect to q, we have where For brevity, we define a vector Differentiating the HAM deformation equation (32) m times with respect to q, then setting q � 0, and finally dividing it by m!, we obtain the mth-order deformation equation: where Operating the inverse operator of L on both sides of equation (37), we have In this way, it is easy to obtain u 1 (x, t), u 2 (x, t), . . . one after another, and then, we get an exact solution of the original equation (31) of the series form:

Applying HAM
In this section, we present three examples to illustrate the applicability of HAM for solving FCPDEs introduced in Section 3. For more details about the HAM, the reader can refer to [8], [13][14][15][16].

e Fractional Cauchy-Riemann Equations.
In this section, we consider the fractional Cauchy-Riemann equation (1) with the following initial conditions: First, we choose two linear fractional order operators: Secondly, we define two linear operators as Using the abovementioned definitions and with the assumption H(x, t) � 1, we construct the zeroth-order deformation equations (ZDE): irdly, differentiating the ZDEs m times with respect to q, then setting q � 0, and dividing it by m!, we get the mthorder deformation equations: where Finally, operating the operators L −1 1 and L −1 2 on both the sides of equations (44) and (45), respectively, we have v m (x, t) � χ m v m−1 (x, t) From Lemma 1 and calculating one by one, we get (49)

Fractional Acoustic Wave Equations.
In this section, we consider fractional acoustic wave equation (2) with initial conditions: (53) First, the choose two linear fractional order operators: Secondly, we define two linear operators as      Complexity (55) Using the abovementioned definitions and Lemma 2, similar to the method of the fractional Cauchy-Riemann equations, we get By repeating this procedure for h � −1, noting Definitions 6 and 7, we obtain the exact solutions: More interestingly, when α � 1 and are the exact solutions of the following equations: with initial conditions v(x, 0) � e x , ρ(x, 0) � sin x. In Figures 13-16, the absolute errors of exact solutions and approximate solutions for v(x, t) and ρ(x, t) with differential fractional order α and β are shown, which indicates the high efficiency and precision of this method. In the process of compiling the program, we calculate sinh α,β (x) and cosh α,β (x) by the formulas sinh α,β (Z) � ((E α,β (Z)− E α,β (−Z))/2) and cosh α,β (Z) � ((E α,β (Z) + E α,β (−Z))/2), respectively.

Two-Dimensional Space Partial Differential Equation with a Time-Fractional-Order.
In this section, we use the HAM to solve the two-dimensional space partial differential equation with a time-fractional-order of the following form: with initial conditions   10 Complexity and boundary conditions Using the abovementioned definitions and with assumption H(x, t) � 1, we construct the zeroth-order deformation equation(ZDE): Obviously, when q � 0 and q � 1, it holds irdly, differentiating the ZDE m times with respect to q, then setting q � 0, and dividing it by m!, we get the mthorder deformation equation: By repeating this procedure for h � −1, we get the exact solution, for example 3. We omit the images of example 3.

Conclusions
In this paper, using the HAM, we obtained exact solutions for the fractional Cauchy-Riemann equations, fractional acoustic wave equations, and partial differential equation with a time-fractional-order. e absolute errors of the approximate solutions obtained by the HAM show that the approximate solutions are in good agreement with the exact solutions. Fractional differential equations are the generalization of integral ones. In fact, fractional models provide us with an adjustable parameter, which is the order of derivatives.

Data Availability
Data can be provided by the corresponding author upon reasonable request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.