Fuzzy Fractional Differential Equations Research Articles

Optimal homotopy asymptotic method for solving several models of first order fuzzy fractional IVPs

In this work, the Optimal Homotopy Asymptotic Method (OHAM) is prolifically implemented to find the optimal solutions of fractional order of fuzzy differential equations. We inspect the competence of the method by examining fractional order time-dependent first order fuzzy initial value problems in the Caputo derivative sense. The concepts of fuzzy sets theory and fractional calculus were used to present a new general fuzzification formulation of the OHAM method followed by fuzzy analysis for the proposed fractional problems are presented in detail. OHAM is recognized as a feasible and convergent method for solving linear and nonlinear models, and it provides a convenient technique for controlling the convergence of approximate solutions optimally. The method's capability is demonstrated by obtaining approximate solutions to linear and nonlinear fuzzy initial value problems in various orders of fractional derivative. The numerical results acknowledge that OHAM is a feasible and transformation mechanism for solving linear and nonlinear fuzzy fractional initial value problems. Also, the results ensure that the method used was succinct, efficient, and simple to use, to deal with a more comprehensive fuzzy fractional differential equation.

Analytic Fuzzy Formulation of a Time-Fractional Fornberg–Whitham Model with Power and Mittag–Leffler Kernels

This manuscript assesses a semi-analytical method in connection with a new hybrid fuzzy integral transform and the Adomian decomposition method via the notion of fuzziness known as the Elzaki Adomian decomposition method (briefly, EADM). Moreover, we use the aforesaid strategy to address the time-fractional Fornberg–Whitham equation (FWE) under gH-differentiability by employing different initial conditions (IC). Several algebraic aspects of the fuzzy Caputo fractional derivative (CFD) and fuzzy Atangana–Baleanu (AB) fractional derivative operator in the Caputo sense, with respect to the Elzaki transform, are presented to validate their utilities. Apart from that, a general algorithm for fuzzy Caputo and AB fractional derivatives in the Caputo sense is proposed. Some illustrative cases are demonstrated to understand the algorithmic approach of FWE. Taking into consideration the uncertainty parameter ζ∈[0,1] and various fractional orders, the convergence and error analysis are reported by graphical representations of FWE that have close harmony with the closed form solutions. It is worth mentioning that the projected approach to fuzziness is to verify the supremacy and reliability of configuring numerical solutions to nonlinear fuzzy fractional partial differential equations arising in physical and complex structures.

Controllability for Fuzzy Fractional Evolution Equations in Credibility Space

This article addresses exact controllability for Caputo fuzzy fractional evolution equations in the credibility space from the perspective of the Liu process. The class or problems considered here are Caputo fuzzy differential equations with Caputo derivatives of order β∈(1,2), 0CDtβu(t,ζ)=Au(t,ζ)+f(t,u(t,ζ))dCt+Bx(t)Cx(t)dt with initial conditions u(0)=u0,u′(0)=u1, where u(t,ζ) takes values from U(⊂EN),V(⊂EN) is the other bounded space, and EN represents the set of all upper semi-continuously convex fuzzy numbers on R. In addition, several numerical solutions have been provided to verify the correctness and effectiveness of the main result. Finally, an example is given, which expresses the fuzzy fractional differential equations.

Existence and Stability of Solutions of Fuzzy Fractional Stochastic Differential Equations with Fractional Brownian Motions

The existence, uniqueness, and stability of solutions to fuzzy fractional stochastic differential equations (FFSDEs) driven by a fractional Brownian motion (fBm) with the Lipschitzian condition are investigated. Finally, we investigate the exponential stability of solutions.

An estimation of effects of memory and learning experience on the eoq model with price dependent demand

In this article, an economic order quantity model has been studied in view of joint impacts of the memory and learning due to experiences on the decision-making process where demand is considered as price dependant function. The senses of memory and experience-based learning are accounted by the fractional calculus and dense fuzzy lock set respectively. Here, the physical scenario is mathematically captured and presented in terms of fuzzy fractional differential equation. The α-cut defuzzification technique is used for dealing with the crisp representative of the objective function. The main credit of this article is the introduction of a smart decision-making technique incorporating some advanced components like memory, self-learning and scopes for alternative decisions to be accessed simultaneously. Besides the dynamics of the EOQ model under uncertainty is described in terms of fuzzy fractional differential equation which directs toward a novel approach for dealing with the lot-sizing problem. From the comparison of the numerical results of different scenarios (as particular cases of the proposed model), it is perceived that strong memory and learning experiences with appropriate keys in the hand of the decision maker can boost up the profitability of the retailing process.

Fuzzy solution of system of fuzzy fractional problems using a reliable method

Under uncertainty, the analytical behavior of fractional differential equations is frequently puzzling and difficult to predict. As a result, it's necessary to develop a suitable, comprehensive, and highly effective theory to solve these challenges. The fuzzy fractional differential equation theory is a relatively recent concept that has applications in engineering, applied mathematics. The fractional differential transform method (FDTM) is utilized in this study to determine the analytical fuzzy solution of some fuzzy fractional differential equations in Caputo sense, which are used to analyze a range of physical models in many sciences and engineering disciplines. The developed FFDE solutions are more generalized and have a wider range of applications. A parametric characterization of the solutions is obtained by converting the fuzzy fractional differential equation to an equivalent crisp system of corresponding fractional differential equations. The numerical and graphical presentation shows the symmetry between lower and upper cut representations of the fuzzy solutions and may be useful in better understanding of automatic control models, artificial intelligence, image classification, computer science, medical science, quantum optics, measure theory, physics, biology, optimal control theory, mathematical economics, and other domains, and non-financial analysis.

Solution of fuzzy fractional order differential equations by fractional Mellin transform method

This paper deals with the application of fuzzy Mellin transform for fuzzy valued functions. It is an attempt to investigate fuzzy Mellin transform in fractional sense and is called fuzzy fractional Mellin transform. Some techniques are proposed for the solution of fuzzy fractional order differential equations by using left-sided Riemann–Liouville fractional derivative. Moreover, some illustrative examples are solved to show the capability and effectiveness of fuzzy Mellin transforms method by converting it into a general solution for an analogous equation with the left-sided Riemann–Liouville derivative and then presenting its explicit representation.

Existence, Uniqueness, and Eq–Ulam-Type Stability of Fuzzy Fractional Differential Equation

This paper concerns with the existence and uniqueness of the Cauchy problem for a system of fuzzy fractional differential equation with Caputo derivative of order q∈(1,2], 0cD0+qu(t)=λu(t)⊕f(t,u(t))⊕B(t)C(t),t∈[0,T] with initial conditions u(0)=u0,u′(0)=u1. Moreover, by using direct analytic methods, the Eq–Ulam-type results are also presented. In addition, several examples are given which show the applicability of fuzzy fractional differential equations.

Similarity solutions of fractional parabolic boundary value problems with uncertainty

We use similarity method to solve nonlinear fractional order fuzzy parabolic boundary value problem with Caputo fractional derivative. The fractional order fuzzy parabolic equation is converted into fuzzy fractional differential equation. Schauder’s fixed point theorem is used to derive the existence results. Finite difference method is used to obtain the numerical results. Examples are provided to justify the results.

Solving Higher-Order Fractional Differential Equations by the Fuzzy Generalized Conformable Derivatives

In this paper, the generalized concept of conformable fractional derivatives of order q ∈ n , n + 1 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.

Fuzzy fractional differential equations under the Mittag-Leffler kernel differential operator of the ABC approach: Theorems and applications

Abstract In this manuscript, we introduced, analyzed, and studied fuzzy fractional differential equations in terms of Atangana-Baleanu-Caputo differential operator equipped with uncertain constraints coefficients and initial conditions. To this end, we discussed both the fuzzy Atangana-Baleanu-Caputo fractional derivative and integral. Also, Newton-Leibniz fuzzy inversion formulas for both derivative and integral are proved. Using Banach fixed point theorem, existence and uniqueness results of solution are established by means of fuzzy strongly generalized differentiability of fuzzy fractional differential equation with Atangana-Baleanu fractional derivative under the Lipschitz condition. To achieve the above results, some prerequisite provisions for characterizing the solution in synonymous systems of crisp Atangana-Baleanu-Caputo fractional differential equations are argued. In this tendency, a new computational algorithm is proposed to obtain analytic solutions of the studied equations. To grasp the debated approach, some illustrative examples are provided and analyzed by the figures to visualize and support the theoretical results.

Computational algorithm for solving drug pharmacokinetic model under uncertainty with nonsingular kernel type Caputo-Fabrizio fractional derivative

This paper proposes an advanced numerical-analytical approach for handling a class of fuzzy fractional differential equations involving Caputo-Fabrizio derivative with a non-singular kernel arsing in the medical sector. The solution methodology relies on the reproducing-kernel algorithm to generate analytical solutions in the form of a uniformly convergent series in the direct sum of the desired Hilbert spaces. The effectiveness of the method is analyzed by studying some theoretical, analytical, and stability results of the derived solutions based on the reproducing kernel theory. Numerical simulations are also provided in tables and graphs to demonstrate the reliability of this algorithm in solving fuzzy models using the new Caputo-Fabrizio fractional operator, especially for the drug pharmacokinetic model. The obtained results show the ability of the applied algorithm to solve a wide range of nonlinear fractional models emerging in pharmacology, medicine, and biochemistry.

Approximate Solution for a Fuzzy Fractional Multi-term Differential Equation by Müntz Polynomials

In this study, we present a new approach for the numerical solution of a fuzzy fractional multi term differential equation under the Caputo fractional derivative which orders in [0, 1]. To do this, collocation method based on Muntz polynomials are used. As in the interval [0, 1] the introduced Muntz polynomials and their fractional derivatives of order $$0 \le \alpha \le 1$$ are not negative; therefore, they are suitable bases for fuzzy approximate. Finally, some examples were included to confirm the applicability, accuracy, and efficiency of the proposed method.

Reproducing kernel approach for numerical solutions of fuzzy fractional initial value problems under the Mittag–Leffler kernel differential operator

In this research study, fuzzy fractional differential equations in presence of the Atangana–Baleanu–Caputo differential operators are analytically and numerically treated using extended reproducing kernel Hilbert space technique. With the utilization of a fuzzy strongly generalized differentiability form, a new fuzzy characterization theorem beside two fuzzy fractional solutions is constructed and computed. To besetment the attitude of fuzzy fractional numerical solutions, analysis of convergence and conduct of error beyond the reproducing kernel theory are explored and debated. In this tendency, three computational algorithms and modern trends in terms of analytic and numerical solutions are propagated. Meanwhile, the dynamical characteristics and mechanical features of these fuzzy fractional solutions are demonstrated and studied during two applications via three‐dimensional graphs and tabulated numerical values. In the end, highlights and future suggested research work are eluded.

Homotopy analysis Shehu transform method for solving fuzzy differential equations of fractional and integer order derivatives

In this paper, we propose the fuzzy Shehu transform method (FSTM) using Zadeh’s decomposition theorem and fuzzy Riemann integral of real-valued functions on finite intervals. As an alternative to standard fuzzy Laplace transform and the fuzzy Sumudu integral transform, we established some potential useful (new or known) properties of the FSTM and validate their applications. Furthermore, the FSTM is coupled with the well-known homotopy analysis method to obtain the approximate and exact solutions of fuzzy differential equations of integer and non-integer order derivatives. The convergence analysis and the error analysis of the suggested technique are provided and supported by graphical solutions. Comparison of the numerical simulations of exact and approximate solutions of two fuzzy fractional partial differential equations are tabulated to further justify the reliability and efficiency of the proposed method.

Artificial neural network approach for solving fuzzy fractional order initial value problems under gH‐differentiability

This paper aims to solve the celebrated fuzzy fractional differential equations (FFDEs) using an artificial neural network (ANN) technique. To accomplish the aforementioned aim, the error back propagation algorithm and a multilayer feed forward neural architecture are utilized using the unsupervised learning in order to minimize the error function as well as the modification of the parameters such as weights and biases. By combining the initial conditions with the ANN output provides an appropriate approximate solution of the proposed FFDE. Then, two illustrative examples are solved to confirm the applicability of the concept as well as to demonstrate both the precision and effectiveness of the developed method. By comparing with some traditional methods, the obtained results reveal a close match that confirms both accuracy and correctness of the proposed method.

Fuzzy Conformable Fractional Differential Equations

In this study, fuzzy conformable fractional differential equations are investigated. We study conformable fractional differentiability, and we define fractional integrability properties of such functions and give an existence and uniqueness theorem for a solution to a fuzzy fractional differential equation by using the concept of conformable differentiability. This concept is based on the enlargement of the class of differentiable fuzzy mappings; for this, we consider the lateral Hukuhara derivatives of order q ∈ 0,1 .

On Z-fractional differential equations

-numbers is an ordered pair of fuzzy numbers used for formalization of imprecise and/or incomplete information. Basically, the concepts of -numbers determine the reliability of information where the possibility distribution functions are taken into account. This theory has proved its potential and efficiency in modeling uncertain and complex phenomena by dynamical systems acting on both fuzziness and randomness environments. In this paper, we study an extension of fuzzy fractional differential equations to the -numbers-valued domain, namely fractional -differential equations. For this aim, we firstly introduce the concepts of Caputo fractional derivative and Riemann-Liouville fractional integral of -number-valued-functions and prove the completeness of the metric space of -number-valued-functions with some useful properties. Next, the fixed point approach is applied to investigate the existence and uniqueness of integral solution of the initial value problem to fractional -differential equations. At last, some stability problems for the proposed fractional -differential equations are discussed, that is the stability in Hyers-Ulam sense of the initial value problem and the Lyapunov stability of the equilibrium point to the fractional -differential equations. Some illustrating examples are given to demonstrate the correctness of theoretical results.

On implicit coupled systems of fuzzy fractional delay differential equations with triangular fuzzy functions

<abstract> In this paper, we introduce and study an implicit coupled system of fuzzy fractional delay differential equations involving fuzzy initial values and fuzzy source functions of triangular type. We assume that these initial values and source functions are triangular fuzzy functions and define the solution of the implicit coupled system as a triangular fuzzy function matrix consisting of real functional matrices. The method of triangular fuzzy function, fractional steps and fuzzy terms separation are used to solve the implicit coupled systems. Further, we prove existence and uniqueness of solution for the considered systems, and also construct a solution algorithm. Finally, an example is given to illustrate our main results and some further work are presented. </abstract>

A Fuzzy Method for Solving Fuzzy Fractional Differential Equations Based on the Generalized Fuzzy Taylor Expansion

In this field of research, in order to solve fuzzy fractional differential equations, they are normally transformed to their corresponding crisp problems. This transformation is called the embedding method. The aim of this paper is to present a new direct method to solve the fuzzy fractional differential equations using fuzzy calculations and without this transformation. In this work, the fuzzy generalized Taylor expansion by using the sense of fuzzy Caputo fractional derivative for fuzzy-valued functions is presented. For solving fuzzy fractional differential equations, the fuzzy generalized Euler’s method is introduced and applied. In order to show the accuracy and efficiency of the presented method, the local and global truncation errors are determined. Moreover, the consistency, convergence, and stability of the generalized Euler’s method are proved in detail. Eventually, the numerical examples, especially in the switching point case, show the flexibility and the capability of the presented method.