Abstract
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 fractional-order 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.
1. Introduction
The concept of fractional derivatives can be traced back to a question raised by Marquis de L’Hopital to Gottfried Wilhelm Leibniz three hundred years ([1–3]) 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. Therefore, 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 [4–8], and so on. These 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]. This method has been successfully applied to solve various linear or nonlinear problems [8, 14–18]. The 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 [26–28].
In this paper, we use the HAM to solve fractional Cauchy–Riemann equations [29] and fractional acoustic wave equations [30], which are given bywhere and and are positive numbers and to solve the partial differential equation with a time-fractional-order of the following form:with a time-fractional order , and is the Caputo fractional derivative.
By setting and , the models (1) and (2) were transformed into Cauchy–Riemann equations and acoustic wave equations, respectively. Suppose that and satisfy the Cauchy–Riemann equations in an open subset of , and consider the vector field . In fluid dynamics, such a vector field is a potential flow. In the second model, and 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.
The 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.
2. Definitions and Lemmas
Definition 1. (see [3]). A real function , , is said to be in the space , , if there exists a real number , such that , where , and it is said to be in the space , if and only if , .
Definition 2. (see [3]). The Riemann–Liouville fractional integral of order , of a function is defined as
Definition 3. (see [3]). The Riemann–Liouville fractional derivative of order , on the usual Lebesgue space , is given bywhere .
Definition 4. (see [3]). The Caputo fractional derivative of , is defined as
Definition 5. (see [3]). The classical Mittag-Leffler function is defined byThe generalized Mittag-Leffler function is defined by
Definition 6. (see [3]). The functions , , , and () are defined byObviously, Euler’s equations have the following forms:In particular, when , we haveand when , we have
Definition 7. (see [3]). The functions , , , and () are defined byObviously, the hyperbolic sine’s and cosine’s equations have the following forms:In particular, when , we haveWhen , we have
Lemma 1. If , , , and are defined as in Definition 6, then
Proof. We denote the beta function by , and then, according to the definition of the Riemann–Liouville fractional integral, we haveSimilarly, we obtain equation (18). Next, we will prove equation (19). From the definition of the Riemann–Liouville fractional derivative, we getSimilarly, we obtain equation (20). In particular, when , , we haveWhile replacing the Riemann–Liouville fractional derivative with the Caputo derivative, we get the following forms:
Lemma 2. If , , , and are defined as in Definition 7, then
Proof. Similar to the proof of Lemma 1, we obtain equations (25)–(28). In particular, when and ,While, replacing the Riemann–Liouville fractional derivative with the Caputo derivative, we get the following forms:
3. HAM
In this section, we consider a linear or nonlinear equation in a general form:where is an unknown function and and are independent variables. Let denote an initial approximation of the solution of equation (31), a nonzero auxiliary parameter, a nonzero auxiliary function, and is an auxiliary linear operator. Then, we construct the HAM deformation equation in the following form:where is an embedding parameter. Obviously, when and , the abovementioned HAM deformation equation (32) has the solutionsrespectively. Thus, as increases from 0 to 1, varies from the initial guesses to the solution of equation (31). Expanding in Taylor’s series with respect to , we havewhere
For brevity, we define a vector
Differentiating the HAM deformation equation (32) times with respect to , then setting , and finally dividing it by , we obtain the mth-order deformation equation:where , and
Operating the inverse operator of on both sides of equation (37), we have
In this way, it is easy to obtain one after another, and then, we get an exact solution of the original equation (31) of the series form:
4. 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], [13–16].
4.1. The 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 , we construct the zeroth-order deformation equations (ZDE):
Thirdly, differentiating the ZDEs times with respect to , then setting , and dividing it by , we get the mth-order deformation equations:where
Finally, operating the operators and on both the sides of equations (44) and (45), respectively, we have
From Lemma 1 and calculating one by one, we get
By repeating this procedure for and noting Definition 5, we obtain the exact solution:
More interestingly, when and ,are the exact solutions of Cauchy–Riemann equationswith initial conditions and . We draw the computer graphics of the exact solutions and by using MATLAB software; see Figures 1–4. Figures 5–8 show the approximate solution obtained by the HAM with intercepting 15 items, and the absolute errors of exact solutions and approximate solutions are shown in Figures 9–12, with different values and . In the process of compiling the program, we calculate and by the formulas and , respectively. The MATLAB code for the Mittag-Leffler function was provided by Podlubny [31].
4.2. Fractional Acoustic Wave Equations
In this section, we consider fractional acoustic wave equation (2) with initial conditions:
First, the choose two linear fractional order operators:
Secondly, we define two linear operators as
Using the abovementioned definitions and Lemma 2, similar to the method of the fractional Cauchy–Riemann equations, we get
By repeating this procedure for , noting Definitions 6 and 7, we obtain the exact solutions:
More interestingly, when and ,are the exact solutions of the following equations:with initial conditions , . In Figures 13–16, the absolute errors of exact solutions and approximate solutions for and with differential fractional order and are shown, which indicates the high efficiency and precision of this method. In the process of compiling the program, we calculate and by the formulas and , respectively.
4.3. 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 conditionsand boundary conditionswhere . First, we choose the linear fractional order operator
Secondly, we define the linear operator as
Using the abovementioned definitions and with assumption , we construct the zeroth-order deformation equation(ZDE):
Obviously, when and , it holds
Thirdly, differentiating the ZDE times with respect to , then setting , and dividing it by , we get the mth-order deformation equation:where
Finally, operating the operator on both the sides of equation (67), we have
Calculating one by one, we get
By repeating this procedure for , we get the exact solution,for example 3. We omit the images of example 3.
5. 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. The 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
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was partly supported by NSF of China under contract no. 11771265, NSF of Guang-dong Provincial Department of Education under contract nos. 2017KQNCX130, 2018KQNCX156, and 2020KZDZX1147, and Chaozhou Science and Technology Plan Project no. 2019GY02.