Abstract
This paper aims to present a novel optimization technique, the residual power series (RPS), for handling certain classes of fuzzy fractional differential equations of order under strongly generalized differentiability. The proposed technique relies on generalized Taylor formula under Caputo sense aiming at extracting a supportive analytical solution in convergent series form. The RPS algorithm is significant and straightforward tool for creating a fractional power series solution without linearization, limitation on the problem’s nature, sort of classification, or perturbation. Some illustrative examples are provided to demonstrate the feasibility of the RPS scheme. The results obtained show that the scheme is simple and reliable and there is good agreement with exact solution.
1. Introduction
Fuzzy fractional differential equation is hot and important branch of mathematics. It has attracted much attention recently due to potential applications in artificial intelligence, industrial engineering, physics, chemistry, and other fields of science. Parameters and variables in many of the nature studies and technological processes that were designed utilizing the fractional differential equation (FDE) are specific and completely defined. Indeed, such information may be vague and uncertain because of experimentation and measurement errors that then lead to uncertain models, which cannot handle these studies. The process of analyzing the relative influence of uncertainty in inputs information to outputs led us to study solutions to the qualitative behavior of equations. Therefore, it is necessary to obtain some mathematical tools to understand the complex structure of uncertainty models [1–5]. On the other hand, the theory of fractional calculus, which is a generalization of classical calculus, deals with the discussion of the integrals and derivatives of noninteger order, has a long history, and dates back to the seventeenth century [6–10]. Different forms of fractional operators are introduced to study FDEs such as Riemann–Liouville, Grunwald-Letnikov, and Caputo. Out of these forms, the Caputo concept is an appropriate tool for modeling practical situations due to its countless benefits as it allows the process to be performed based on initial and boundary conditions as is traditional and its derivative is zero for constant [11–17].
The residual power series (RPS) method developed in [18] is considered as an effective optimization technique to determine and define the power series solution’s values of coefficients of first- and second-order fuzzy differential equations [19–22]. Furthermore, the RPS is characterized as an applicable and easy technique to create power series solutions for strongly linear and nonlinear equations without being linearized, discretized, or exposed to perturbation [23–27]. Unlike the classical power series method, the RPS neither requires comparing the corresponding coefficients nor is a recursion relation needed as well. Besides that, it calculates the power series coefficients through chain of equations of one or more variables and offers convergence of a series solution whose terms approach quickly, especially when the exact solution is polynomial.
The remainder of this paper is organized as follows. In Section 2, essential facts and results related to the fuzzy fractional calculus will be shown. In Section 3, the concept of Caputo’s H-differentiability will be presented together with some closely related results. In Section 4, basic idea of the RPS method will be presented to solve the fuzzy FDEs of order . In Section 5, numerical application will be performed to show capability, potentiality, and simplicity of the method. Conclusions will be given in Section 6.
2. Preliminaries
In this section, necessary definitions and results relating to fuzzy fractional calculus are presented. For the fuzzy derivative concept, the strongly generalized differentiability will be adopted, which is considered H-differentiability modification.
A fuzzy set in a nonempty set is described by its membership function . So, for each the degree of membership of in is defined by .
Definition 1 ([28]). Suppose that is a fuzzy subset of ℝ. Then, is called a fuzzy number such that is upper semicontinuous membership function of bounded support, normal, and convex.
If is a fuzzy number, then , where and for each . The symbol is called the -level representation or the parametric form of a fuzzy number .
Theorem 2 ([29]). Suppose that satisfy the following conditions:(1) is a bounded nondecreasing function.(2) is a bounded nonincreasing function.(3).(4)for each , and .(5) and . Then given by is a fuzzy number with parameterization .
Definition 3 ([29]). Let . If there exists an element such that , then we say that is the Hukuhara difference (H-difference) of and , denoted by .
The sign stands always for Hukuhara difference. Thus, it should be noted that Normally, is denoted by . If the H-difference exists, then .
Definition 4 ([30]). The complete metric structure on is given by the Hausdorff distance mapping such thatfor arbitrary fuzzy numbers ( and .
Definition 5 ([30]). Let . Then the function is continuous at if for every , such that , for each , whenever .
Remark 6. If the function is continuous for each , where the continuity is one-sided at endpoints of , then is continuous function on . This means that is continuous on if and only if and are continuous on .
Definition 7 ([28]). For fixed and , the function is called a strongly generalized differentiable at , if there is an element such that either(i)the H-differences exist, for each sufficiently tends to 0 and , or(ii)the H-differences exist, for each sufficiently tends to 0 and ,
where the limit here is taken in the complete metric space
Theorem 8 ([31]). Suppose that , where , then(1)the functions and are two differentiable functions and , when is (1)-differentiable;(2)the functions and are two differentiable functions and , when is (2)-differentiable.
Definition 9 ([31]). Suppose that . One can say that is -differentiable at , if exists on a neighborhood of as a fuzzy function and it is -differentiable at . The second-order derivatives of at are indicated by for .
Theorem 10 ([32]). Let and , where for each :(1)If is (1)-differentiable, then and are differentiable functions and ,(2)If is (2)-differentiable, then and are differentiable functions and ,(3)If is (1)-differentiable, then and are differentiable functions and ,(4)If is (2)-differentiable, then and are differentiable functions and .
Definition 11 ([32]). Let and . One can say that is Caputo fuzzy -differentiable at when exists, where . Also, we say that is Caputo -differentiable if is (1)-differentiable and is Caputo differentiable if is (2)-differentiable, where and stand for the space of all continuous and Lebesque integrable fuzzy-valued functions on , respectively.
Theorem 12 ([33]). Let and Then, for each , the Caputo fuzzy fractional derivative exists on such thatfor (1)-differentiable andfor (2)-differentiable.
The next characterization theorem shows a way to convert the FFDEs into a system of ordinary fractional differential equations (OFDEs), ignoring the fuzzy setting approach.
Theorem 13 ([34]). Consider the below fuzzy fractional IVPssubject towhere such that
(i) .
(ii) for any there exist such that and , whenever and and are uniformly bounded on any bounded set.
(iii) there is a constant (say) such thatandTherefore, there are two systems of OFDEs that are equivalent to FFDEs (4) and (5) as follows:
Case 1. When is Caputo [(1)-]-differentiablewith , .
Case 2. When is Caputo [(2)-]-differentiablewith , .
3. Formulation of Fuzzy Fractional IVPs of Order
Consider the below fuzzy fractional differential equationsubject to fuzzy initial conditionswhere is a linear or nonlinear continuous fuzzy-valued function, is a continuous real valued function with nonnegative values on , and is unknown analytical fuzzy function to be determined. We assume that the fuzzy fractional IVPs (10) and (11) have unique smooth solution on the domain of interest.
Next, some theorems and definitions which are used later in this paper are presented.
Definition 14. Let be fuzzy function such that . Then, for , Caputo’s H-derivative of at is defined asAlso, we say that is Caputo -differentiable for , when exists, and is -differentiable.
Theorem 15. Let , such that . Caputo’s H-derivative of order exists on such that
(i) If is (1,1)-differentiable, then , .
(ii) If is (1,2)-differentiable, then .
(iii) If is (2,1)-differentiable, then .
(iv) If is (2,2)-differentiable, then , where .
The -solution of fuzzy fractional IVPs (10) and (11) is a function that has Caputo []-differentiable and satisfies the FFIVPs (10) and (11). To compute it, we firstly convert the fuzzy problem into equivalent system of second OFDEs, called correspondence -system, based upon the type of derivative chosen. Then, by utilizing the -cut representation of , , and the initial data in (11) such that , , , and , the following corresponding -systems will be hold:
(i) (1,1)-system such that
(ii) the (1,2)-system such that
(iii) the (2,1)-system such that
(iv) the (2,2)-system such thatsubject to initial conditions
Theorem 16 ([33]). Let and let be an -solution of FFIVPs (10) and (11) on . Then, and will be a solution to the associated -system.
Theorem 17 ([33]). Let and let and be the solution of -system for each . If has valid level sets and is Caputo -differentiable, then is an -solution of FFIVPs (10) and (11) on .
The aim of the next algorithm is to perform a strategy to solve the FFIVPs (10) and (11) in terms of its -cut representation form. Indeed, there are four cases that depend on type of differentiability.
Algorithm 18. To determine the solutions of FFIVPs (10) and (11), do the following:
Case (I). If is Caputo [(1,1)-]-differentiable and the FFIVPs (10) and (11) will be converted to crisp system described in (13) and (17), then do the following steps: Step 1: Solve the required system. Step 2: Ensure that and are valid level sets for each . Step 3: Construct (1,1)-solution whose -cut representation is .Case (II). If is Caputo [(1,2)-]-differentiable and the FFIVPs (10) and (11) will be converted to crisp system described in (14) and (17), then do the following steps: Step 1: Solve the required system. Step 2: Ensure that and are valid level sets for each . Step 3: Construct (1,2)-solution whose -cut representation is .Case (III). If is Caputo [(2,1)-]-differentiable and the FFIVPs (10) and (11) will be converted to crisp system described in (15) and (17), then do the following steps: Step 1: Solve the required system. Step 2: Ensure that and are valid level sets for each . Step 3: Construct (2,1)-solution whose -cut representation is .Case (IV). If is Caputo [(2,2)-]-differentiable and the FFIVPs (10) and (11) will be converted to crisp system described in (16) and (17), then do the following steps: Step 1: Solve the required system. Step 2: Ensure that and are valid level sets for each . Step 3: Construct (2,2)-solution whose -cut representation is .
4. Description of Fractional RPS Method
In this section, the RPS scheme is presented for constructing an analytical solution of FFIVPs (10) and (11) through substituting the expansion of fractional power series (FPS) among the truncated residual functions. In view of that, the resultant equation helps us to derive a recursion formula for the coefficients’ computation, where the coefficients can be computed recursively through the recurrent fractional differentiating of the truncated residual function.
Definition 19 ([35]). A fractional power series (FPS) representation at has the following form:where , , and ’s are the coefficients of the series.
Theorem 20 ([35]). Suppose that has the following FPS representation at :where and for ; then the coefficients will be in the form such that (-times).
Conveniently, for obtaining -solution of FFIVPs (10) and (11) utilizing the solution of the corresponding -system, we will explain the fashion to determine -solution equivalent to the solution for the system of OFDEs (13) and (17). Further, same manner can be applied to construct other type of -solutions. To achieve our goal, assume that the solution of OFDEs (13) and (17) at has the following form:
Since and satisfy the initial conditions in (17), then the following polynomials and will be the initial guesses for the system and the solutions can also be represented by
Consequently, the -truncated series solutions can be given by
The residual functions and are defined as follows:and the -residual functions and for are defined as follows:
From (23), we have and for and each , which leads to . Also, the fractional derivatives and are equivalent at for each , that is, . However, holds for .
Regarding employing the RPS algorithm to obtain the unknown coefficients, and , substitute the approximations and into the residual functions and of (24) such thatand based upon the facts , we have and . Therefore, the RPS approximate solutions can be written as
Currently, for the unknown coefficients, and substitute and into the residual functions, and of (24) such that
Then, by applying the fractional derivative on both sides of and , using the facts as well, the values of and will be given by
For the unknown coefficients, and substitute and into the residual functions, and of (24), and then by computing and and using the facts , the coefficients, and , will be given such that
Using similar argument, the unknown coefficients, and , will be given utilizing the facts . The same manner can be repeated until we obtain on the coefficients’ arbitrary order of the FPS solution for the OFDE (13).
5. Numerical Simulation and Discussion
This section aims to verify the efficiency and applicability of the proposed algorithm by applying the RPS method to a numerical example. Here, all necessary calculations and analysis are done using Mathematica 10.
For this purpose, let us consider the fuzzy fractional differential equationwith the fuzzy initial conditionswhere and are the fuzzy numbers whose -cut representation is .
Based on the type of differentiability, the FFIVPs (30) and (31) can be converted into one of the following systems.
Case 1. If is (1,1)-solution, then the corresponding (1,1)-system will beIf , then the exact solution of (32) is , . In finding the fuzzy (1,1)-solution of FFDEs (30), let be Caputo [(1,1)-]-differentiable. Sequentially, after selecting the initial guesses as and , the FPS expansion of solutions for OFDEs (32) can be represented as follows:To determine the RPS approximate solution for OFDEs (32), substitute the -truncated series and into the -residual functions and such that and . Thus, based upon the facts and , we have and . Hence, the RPS approximate solution for OFDEs (32) can be written in the form of Similarly, to find out the RPS approximate solution for OFDEs (32), substitute the truncated series and into the residual functions and such that and . Now, applying the fractional derivative on both sides of and yields the following: and . So, the unknown coefficients are and through using the facts Therefore, the RPS approximate solution for OFDEs (32) is given byAccordingly, the unknown coefficients and will be vanished for by continuing in the similar approach, that is, and . Hence, the RPS approximate solutions corresponding to (1,1)-system are coinciding well with the exact solutions and . Here, , , and are valid level sets for and . Moreover, is a (1,1)-solution for FFIVPs (30) and (31) on .
Case 2. If is (1,2)-solution, then the corresponding (1,2)-system will beIf , then the exact solution of (36) is , . In finding the fuzzy (1,2)-solution of FFDEs (30), let be Caputo [(1,2)-]-differentiable. Sequentially, after selecting the initial guesses as in case 1, the FPS expansion of solutions for OFDEs (36) can be represented byTo determine the RPS approximate solution for OFDEs (36), substitute the truncated series and into the residual functions and such that and Thus, based upon the facts , we have and . Hence, the RPS approximate solution for OFDEs (36) can be written in the form of Similarly, to find out the RPS approximate solution for OFDEs (36), substitute the truncated series and into the residual functions and such that and . Then, applying the fractional derivative on both sides of and yields the following: and . So, the unknown coefficients are and through using the facts Therefore, the RPS approximate solution for OFDEs (36) is given byBy continuing in the similar manner, the unknown coefficients and will be vanished for , that is, and . Hence, the RPS approximate solutions corresponding to (1,2)-system are coinciding well with the exact solutions and . Here, , , and are valid level sets for and . On the other hand, is a (1,2)-solution for FFIVPs (30) and (31) on .
Case 3. If is (2,1)-solution, then the corresponding (2,1)-system will beIf , then the exact solution of (40) is , . To obtain the fuzzy (2,1)-solution of FFDEs (30), let is Caputo [(2,1)-]-differentiable. By using the same manner in previous cases, the solutions for (2,1)-system can be obtained such as and . It is easy to check that and are also valid level sets for and . Thus, is a (2,1)-solution for FFIVPs (30) and (31) on .
Case 4. If is (2,2)-solution, then the corresponding (2,2)-system will beIf , then the exact solution of OFDEs (41) is , . Finally, to determine the fuzzy (2,2)-solution of FFDEs (30), let be Caputo [(2,2)-]-differentiable. By using the same manner in previous cases, the solutions for (2,2)-system can be obtained such as and . Here, and are also valid level sets for and . However, defines as a (2,2)-solution for FFIVPs (30) and (31) on .
To demonstrate the agreement between the exact and approximate solution, Table 1 shows the absolute error of the PRS approximate solution for FFIVPs (30) and (31) obtained for different values of -cut representations and nodes with fractional order . Some graphical results are also presented in Figures 1 and 2. The numerical results obtained indicate that the RPS approximate solutions are in good agreement with each other and with the exact solutions for all cases of differentiability.
(a) RPS solution of -system
(b) RPS solution of -system
(c) RPS solution of -system
(d) RPS solution of -system
(a) -levels of -system
(b) -levels of -system
(c) -levels of -system
(d) -levels of -system
6. Conclusion
In this paper, the RPS algorithm is successfully developed, investigated, and applied to solve the fuzzy differential equation of fractional order with fuzzy initial constraints under the fuzzy concept of Caputo H-differentiability. The fuzziness is represented using upper semicontinuous membership function of bounded support, convex, and normalized fuzzy numbers based on its single parametric form. The behavior of approximate solution for different values of fractional order is discussed quantitatively as well as graphically. The numerical results in this paper demonstrate the efficiency of the algorithm. We conclude that the proposed scheme is highly accurate in solving widely array of fuzzy fractional issues.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research was financially supported by the UKM (Grant no. GP-K007788 and GP-K006926).