Abstract
In this paper, the local fractional version of homotopy perturbation method (HPM) is established for a new class of local fractional integral-differential equation (IDE). With the embedded homotopy parameter monotonously changing from 0 to 1, the special easy-to-solve fractional problem continuously deforms to the class of local fractional IDE. As a concrete example, an explicit and exact Mittag–Leffler function solution of one special case of the local fractional IDE is obtained. In the process of solving, two initial solutions are selected for the iterative operation of local fractional HPM. One of the initial solutions has a critical condition of convergence and divergence related to the fractional order, and the other converges directly to the real solution. This paper reveals that whether the sequence of approximate solutions generated by the iteration of local fractional HPM can approach the real solution depends on the selection of the initial approximate solutions and sometimes also depends on the fractional order of the selected initial approximate solutions or the considered equations.
1. Introduction
Fractals, solitons, and chaos together constitute three important branches of nonlinear sciences. In fractal space, there exist some magical functions which are continuous everywhere but nondifferentiable everywhere. The local fractional calculus [1] developed in recent years provides a powerful mathematical tool to handling with such type of nondifferentiable functions. Fractional calculus, which is widely believed to have originated more than 300 years ago, has attracted much attention [2–17]. It is of theoretical and practical value to solve fractional differential equations (DEs) directly connecting with fractional dynamical processes in a great many fields. For this reason, people often construct exact solutions of fractional DEs to obtain useful clues in these fractional dynamical processes for specific applications.
With the development of fractional calculus, many numerical and analytical methods for fractional DEs have been developed, such as integral transform method [1], series expansion method [3], Adomian decomposition method [4], Fan subequation method [5], variational iteration method (VIM) [6], variable separation method [7], finite difference method [8], homotopy perturbation method (HPM) [9], combined the HPM with Laplace transform [10], exp-function method [11], and Hirota bilinear method [12]. The HPM proposed by He [18] couples the homotopy method and the perturbation technique, which needs no the small parameters embedded in differential equations. More importantly, it is indicated in [19] that the HPM can truly eliminate the limitations existing in traditional perturbation methods.
One of the advantages of local fractional derivative is that it has been successfully used to describe some nondifferential problems appearing in science and engineering [1]. The concept of local fractional derivative, which is based on Riemann-Liouville fractional derivative, can be retrospected to Kolwankar and Gangal’s pioneering work [13]. More specifically, if the following limit exists and is finite for a given continuous function , then is called -order fractional derivative of at the point . Later, inspired by the relation of Jumarie [20], the local fractional derivative of a local fractional continuous but nondifferentiable function is defined as another form ([4], see Definition 1). Recently, the theory of local fractional calculus has been significantly developed. Yang and his collaborators [1, 3, 17] have made a series of achievements in the development of local fractional calculus. Benefiting from the graceful properties of local fractional calculus, some existing methods like those [5, 11, 12], originally proposed for DEs with integer orders, have successfully been extended to fractional DEs and many methods are meeting more and more challenges for solving fractional DEs.
The paper is aimed at establishing the local fractional HPM for a new class of local fractional IDEs: where , , , and are the local fractional continuous but nondifferentiable functions, and represent the local fractional derivative and integral [1], respectively, and is the well-known Euler’s Gamma function:
Considering a concrete application of the established local fractional HPM, we would like to solve a special case of Equation (2): where the Mittag–Leffler function which is defined on a fractal set [1].
The organization of the rest of this paper is as follows. Section 2 recalls the local fractional derivative and integral and some basic properties. Section 3 establishes the local fractional HPM for the class of local fractional IDEs (2). Section 4 takes the local fractional IDE (4), a special case of Equation (2), to test the established local fractional HPM and discuss the influence of not only the initial approximate solutions but also the fractional order on whether the sequence of approximate solutions can approach the real solution. Section 5 employs He-Laplace method coupling the HPM with Laplace transform to solve the local fractional IDE (4) and compares the obtained results. Section 6 summarizes the whole paper.
2. Local Fractional Derivative and Integral and Some Basic Properties
In this section, we recall the local fractional derivative and integral and some basic properties.
Definition 1 (see [1]). Let be a local fractional continuous but nondifferentiable function; then, -order local fractional derivative of at the point reads
where
The local fractional derivative has some basic properties [1]:
where , , , and are constants and is an integer, while and .
Definition 2 (see [1]). Let function ; then, the definition of -order local fractional integral of in the integral is as follows:
where with .
The local fractional integral has some basic properties [1]:
3. Local Fractional HPM for the Class of Local Fractional IDEs
In this section, we establish the local fractional HPM for the class of IDEs (2). For convenience, we rewrite Equation (2) as follows: where represents the local fractional operator.
In view of the local fractional HPM [17], we construct the local fractional homotopy , , and by the following: with the embedded parameter which monotonously changes from to leads to the result that the easy-to-solve equation continuously deforms to the original equation Using the constructed homotopy , we can continuously trace a curve which is implicitly defined from a starting point to a solution function
From the perspective of topology, the above changing process is called a deformation. In this deformation, and are homotopic.
Thus, the fractional homotopy in Equation (11) can be written as below:
Substituting the series expanded by the fractional homotopy parameter into Equation (14) and comparing the coefficients of the same power of , we obtain a set of fractional equations:
Here is assumed to be an initial approximate solution of Equation (2). Generally, the initial approximation or can be freely chosen. Solving above set of fractional equations, we can obtain solutions , , , , and so on.
Setting and using Equation (15), we finally arrive at an approximate solution of Equation (2).
As pointed by He [18], the series (17) has convergence in most cases and the convergent rate is determined by when . For the case of convergence, the series (17) can reach a closed form solution.
4. A Concrete Example
In this section, we apply the previously established local fractional HPM to the local fractional IDE (4).
Firstly, we construct such a fractional homotopy:
Secondly, we substitute Equation (15) into Equation (18), and then, the comparison of the coefficients of gives a system of fractional equations:
In view of the arbitrariness of or , here we set and then, from Equations (20)–(23), we have namely, namely, namely, namely,
Finally, we obtain an approximate solution of Equation (4). which can be simplified as follows:
Obviously, the th-order approximate solution of Equation (4) has the following form:
When the condition holds, the limit of Equation (30) exists. In this case, we obtain
In Figure 1, we show the curve of the condition with fractional order , where the dashed line represents . We can see from Figure 1 that there exists a unique value such that . At the same time, with the help of Mathematica, we have
Thus, we can more accurately determine the range of as
This tells that if and only if .
In Figure 2, the initial solution and 5th-order and 8th-order approximate solutions , , and and the exact solution are shown by constraining them to a Cantor set with dimension which does not satisfy the convergence condition . In this case, , , , and form a sequence of divergent approximate solutions.
Since the initial approximation or possesses arbitrariness as mentioned earlier, if we set again then the similar process yields from which we finally reach solution (33). So, for any , the solution (33) is always the exact solution of Equation (4). That is to say, if we chose an appropriate initial approximation or , then the operation can be considerably simplified. More importantly, the convergence condition that the sequence of approximate solutions depends on fractional order can be removed.
It should be noted that when and , , and are all the continuous and differentiable functions, solution (33) reduces to which is the known exact solution of the following IDE [19]:
5. He-Laplace Method and Comparison
As Deng and Ge [21] pointed out, He-Laplace method has a simple and reliable algorithm and it can be coupled with the HPM or the VIM for solving various nonlinear models, shedding a bright light on fractal calculus. A newest typical example of He-Laplace method to illustrate its simplicity, directness, strength, and great prospects can be found in [22]. In what follows, we employ the local fractional version of He-Laplace method [23] to solve the local fractional IDE (4).
Taking the local fractional Laplace transform on Equation (4), we can gain
Then, the inverse local fractional Laplace transform of Equation (42) gives where has been assumed and and are the local fractional Laplace transform operator and inverse operator [1], respectively.
Dealing Equation (43) with the local fractional HPM, we introduce and compare the coefficients of the same powers of ; then, He’s polynomials can be obtained:
We therefore obtain an approximate solution of Equation (4): which is the same as solution (29). It is not difficult to see that solution (46) has the same nth-order approximate solution (31), and when the condition (32) holds, the limit of solution (46) gives the same exact solution (33).
Through comparison, we find that the computational difficulty of solving the local fractional IDE (4) by the above two methods is about the same. When using the local fractional HPM, we need to introduce an appropriate initial approximate solution while the local fractional He-Laplace method uses a known initial value . Generally speaking, it is more difficult to choose an initial approximate solution than to find an initial value For the latest valuable work on the modified HPM, we suggest to refer to Refs. [24, 25].
6. Conclusion
In summary, we have established the local fractional HPM for the class of local fractional IDEs (1). Based on the established local fractional HPM, an explicit and exact Mittag–Leffler function solution (33) of the local fractional IDE (4) is obtained by selecting two different initial solutions (24) and (37). The comparison shows that the local fractional He-Laplace method [23] can also obtain the same solution (33), but when the initial approximate solution is not easy to choose, the local fractional He-Laplace method [23] has certain advantages over the method used in this paper. The obtained results show that if we choose the approximate solutions appropriately, for example solution (37), then the calculation can be considerably simplified and that the sequence of approximate solutions generated by the local fractional HPM can directly approach the real solution without the influence of fractional order. However, for the selected initial approximate solution (24), we obtained a sequence of approximate solutions converging the real solution (33) in a certain range of the fractional order , i.e., . At the same time, there is a divergence interval which depends on the fractional order . Here, , and an approximate range of is . That is to say, is the critical value of convergence and divergence related to fractional order of the obtained sequence of approximate solutions. This is different from the HPM for integer-order DEs, all the sequences of approximate solutions of which either converge or diverge, and there is no such a critical value of convergence and divergence. When the th-order approximate solution (31) is constrained to a Cantor set with dimension , this paper shows in Figure 2 a sequence of divergent approximate solutions. This paper fails to describe a sequence of convergent approximate solutions of (31), which is due to the complexity of the numerical simulation of fractal set. How to constrain solution (31) to other fractal sets and show some sequences of convergent approximate solutions? This is a question worth exploring. Besides, the research on qualitative behaviors of Equation (4) and other fractional IDEs is worth discussing. Some recent meaningful results of this research can be found in [26–28]. In 2007, Wang and He [29] took three concrete IDEs as examples to illustrate the effectiveness of the VIM for various IDEs. Based on this fact, we conclude that the local fractional version of VIM can also solve the local fractional IDE (4). In fact, the main steps of the local fractional VIM for Equation (4) can be summarized as follows: (i) identifying Lagrange multiplier , (ii) determining the iterative formula of solution: with and (iii) selecting the initial approximate solution to obtain the exact solution by using the determined iterative formula (47). Nevertheless, it is still worth trying to find an appropriate initial approximate solution and get the approximate solution (29) or (46) by Equation (47).
Data Availability
The data in the manuscript are available from the corresponding author upon reasonable request.
Conflicts of Interest
The authors declare that there is no conflict of interest regarding the publication of this article.
Acknowledgments
The authors B. Xu and S. Zhang are grateful to Professor Xiao-Jun Yang of China University of Mining and Technology for helpful discussion. This work was supported by the Liaoning BaiQianWan Talents Program of China (grant No. LRS2020[78]), the National Natural Science Foundation of China (grant No. 11547005), the Natural Science Foundation of Education Department of Liaoning Province of China (grant No. LJ2020002), and the Natural Science Foundation of Xinjiang Autonomous Region of China (grant No. 2020D01B01).