Review Article | Open Access
Atimad Harir, Said Melliani, Lalla Saadia Chadli, "Solving Higher-Order Fractional Differential Equations by the Fuzzy Generalized Conformable Derivatives", Applied Computational Intelligence and Soft Computing, vol. 2021, Article ID 5571818, 8 pages, 2021. https://doi.org/10.1155/2021/5571818
Solving Higher-Order Fractional Differential Equations by the Fuzzy Generalized Conformable Derivatives
In this paper, the generalized concept of conformable fractional derivatives of order for fuzzy functions is introduced. We presented the definition and proved properties and theorems of these derivatives. The fuzzy conformable fractional differential equations and the properties of the fuzzy solution are investigated, developed, and proved. Some examples are provided for both the new solutions.
A closed-form solution for nonlinear fractional differential equations (FDEs) plays a significant role in understanding the qualitative as well as quantitative features of complex physical phenomena. The nonlinear appear in different sciences and engineering problems such as control theory, signal processing, finance, electricity, mechanics, plasma physics, stochastic dynamical system, economics, and electrochemistry [1–7]. A fuzzy fractional differentiation and fuzzy integration operators have different kinds of definitions that we can mention, the fuzzy Riemann–Liouville definition [8, 9], the fuzzy Caputo definition [9, 10], and so on. Lately, Khalid et al.  introduced a new simple definition of the fractional derivative named the conformable fractional derivative, which can redress shortcomings of the other definitions, and this new definition satisfies formulas of derivative of product and quotient of two functions [12, 13]. Harir et al.  introduced the fuzzy generalized conformable fractional derivative, which generalized and extended the concept of Hukuhara differentiability for set-valued mappings to the class of fuzzy mapping [15, 16].
Our objective of this article is to present a generalized concept of conformable fractional derivative of order for fuzzy functions. Then, we have investigated in more detail some new properties of these derivatives and we have proved some useful related theorems. We interpret fuzzy conformable fractional differential equations using this concept. We introduce new definitions of solutions. Two examples are provided.
Let us denote by the class of fuzzy subsets of the real axis satisfying the following properties [13, 17]:(i) is normal, i.e., there exists an such that .(ii) is fuzzy convex, i.e., for and ,(iii) is upper semicontinuous.(iv) is compact.
Then, is called the space of fuzzy numbers. Obviously, . For denoting , then from (i) to (iv), it follows that the -level set for all is a closed bounded interval which is denoted by . By , we denote the family of all nonempty compact convex subsets of and define the addition and scalar multiplication in as usual.
Theorem 1. (see ). If , then(i) for all .(ii) for all .(iii) is a nondecreasing sequence which converges to ; then,
Conversely, if is a family of closed real intervals verifying (i) and (ii), then is defined as a fuzzy number such that for and .
Lemma 1 (see ). Let be the fuzzy sets. Then, if and only if for all .
The following arithmetic operations on fuzzy numbers are well known and frequently used below . If , then
Definition 1. Let . If there exists such as , then is called the -difference of and it is denoted as .
Theorem 2. (see )(i)Let we denote then is a neutral element with respect to , i.e., .(ii)With respect to , none of has opposite in .(iii)For any with or and any , we have ; for general , the above property does not hold.(iv)For any and any , we have .(v)For any and any , we have .
Define by the following equation:where is the Hausdorff metric:
It is well known that is a complete metric space. We list the following properties of :for all and .
Let be a sequence in converging to . Then, Theorem in  gives us an expression for the limit.
Theorem 3. (see ). If as , thenLet be an interval. We denote by the space of all continuous fuzzy functions on which is a complete metric space with respect to the metric
3. Generalized Fuzzy Conformable Fractional Derivatives
Definition 2. (see ). Let be a fuzzy function. order “fuzzy conformable fractional derivative” of is defined byfor all . If is -differentiable in some , and exists, thenand the limits exist (in the metric d).
Remark 1. (see ). From the definition, it directly follows that if is -differentiable, then the multivalued mapping is -differentiable for all andHere, is denoted as the conformable fractional derivative of of order . The converse result does not hold, since the existence of Hukuhara differences , does not imply the existence of -difference .
We consider the following definition .
Definition 3. Let be a fuzzy function and . One says, is -differentiable at point if there exists an element such that for all sufficiently near to 0, there exist and the limits (in the metric d) is -differentiable at if for all sufficiently near to 0, there exist If is -differentiable at , we denote its -derivatives, for .
Definition 4. Let be a fuzzy function and now we introduce definitions and theorems for for some natural number . For the sake of convenience, we concentrate on case be -differentiable at , where . Then the fuzzy conformable fractional derivative of of order is defined bywhere and is the smallest integer greater than or equal to and the limits exist (in the metric d).
Theorem 4. Let and and . If is -differentiable and is -differentiable, then
Remark 2 (see ). is -differentiable on , if exists on and it is -differentiable on . The second derivatives of are denoted by for .
Proof. We present the details only for , since the other case is analogous. Let in Definition 4, then . Therefore, if and , we haveDividing by , we haveand passing to the limitSimilarly, we obtainand passing to the limit and gives .
Theorem 5. Let , and be a fuzzy function 2 times differentiable on an open real interval , then the fuzzy conformable derivative obeys to the followingwhere .
Lemma 2. Let and be a fuzzy function 2 times differentiable on an open real interval , then the fuzzy conformable derivative obeys the followingwhere .
Theorem 6. Let and . Then the fuzzy conformable fractional derivative of order , where exists, is defined bywhere .
Remark 3. is -differentiable on , if exists on and it is -differentiable on and . The -differentiable (conformable fractional derivatives of order ) of is denoted by for .
Theorem 7. Let and . Then the fuzzy conformable fractional derivative of order , where or exists, where , is defined by(i)If is-differentiable, then and are -differentiable and(ii)If is -differentiable, then and are -differentiable and(iii)If is -differentiable, then and are -differentiable and(iv)If is -differentiable, then and are -differentiable and
Proof. We present the details only for , since the other cases are analogous. If and , we haveDividing by , we haveDividing by , we haveSimilarly, we obtainand passing to the limit, we haveand using Theorem 6 gives the theorem.
4. Fuzzy Conformable Fractional Differential Equations of Order
In this section, we study the fuzzy conformable fractional differential equations of order :where , and is a continuous fuzzy function on some interval . We give the following definition for the solutions of (41).
Definition 5. Let and . y is a solution (n, m), for problem (41) on , if (or and ) exist on and
Therefore, since the fuzzy conformable derivatives of fuzzy processesprovided these two intervals define fuzzy numbers and in , otherwise we apply Definition 2 and Theorem 7, then we have one of the following cases for and :(i)System (1, 1)(ii)System (1, 2)(iii)System (2, 1)(iv)System (2, 2)where and .
Theorem 8. Let and be a solution (n, m) for problem (41) on . Then and solve the associated system (n, m).
Proof. By using Theorem 4 and Remark 4 and suppose is the solution (n, m) of problem (41), according to Definition 5, then and or and exist and satisfy problem (41). By Theorem 6 in , (21) and (25) and substituting and their conformable fractional derivatives in problem (41), we get the system (n, m) corresponding to solution (n, m). This completes the proof.
Theorem 9. Let and and solve the system (n, m) on , for all . Let . If has valid cut sets on and exists, then is a solution (n, m) for fuzzy problem (41).
Proof. Let and is -differentiable fuzzy function, let , so by using Theorems 6 and 7 and Theorem 6 in  and Remark 4, we can compute and according to . Due to the fact that and solve system (n, m), from Definition 5, then is a solution (n, m) for equation (41).
Example 1. We consider a conformable fractional ordinary differential equation :where are the triangular fuzzy numbers having -cuts . If is solution (1, 1) for problem (49) and , then andand they satisfy system (1, 1) associated with equation (41). Using Theorems 6 and 4, so the conformable fractional system (1, 1) has only the following solution:Then, has valid -cuts for . By Theorem 2.3 in , is a conformable fractional derivative of order for . So defines a solution (1, 1) for . For solution (1, 2), we deducewhere has valid -cuts for and is a conformable fractional derivative of order for . Hence, gives us a solution on . For solution (1, 2), we getwhere has valid -cuts for . We can see is a solution (1, 2) on . Finally, system (2, 2) giveswhere has valid -cuts for all and defines a solution (2, 2) on . Then, we have an example of a fuzzy conformable fractional ordinary differential equation with four solutions.
Example 2. Given a conformable fractional ordinary differential equation [13, 21]where are the triangular fuzzy numbers having -cutsTo find solution (1, 1), we havewhere has valid -cuts for and . From Theorem 2.3 in , is a conformable fractional derivative of order for . So defines a solution (1, 1) for . For solution (1, 2), we deduceWe see that has valid alpha-cut and . From Theorem 2.3 in , is a conformable fractional derivative of order for . Since system (1, 2) has only the above solution, then solution (1, 2) does not exist. For solution (2, 1), we get has valid -cuts and is a conformable fractional derivative of order for . Then, solution (2, 1) does not exist. For solution (2, 2), we deduceWe see that has valid -cut and . From Theorem 2.3 in , is a conformable fractional derivative of order for all . Then defines a solution for . Then, we have a fuzzy conformable fractional ordinary differential equation and two solutions.
By using the concept of conformable generalized derivative and its extension to fractional derivatives of order , we show that we have several possibilities or types to define fractional derivatives of order of fuzzy-number-valued functions. Then, we propose a new method to solve fuzzy fractional differential equations based on the selection of conformable derivative types covering all former solutions. With these ideas, the selection of conformable derivative type in each step of deprivation plays a crucial role.
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
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