Fractional Fuzzy Research Articles

On the Existence and Uniqueness for High Order Fuzzy Fractional Differential Equations with Uncertainty

A class fuzzy fractional differential equation (FFDE) involving Riemann-LiouvilleH-differentiability of arbitrary orderq>1is considered. Using Krasnoselskii-Krein type conditions, Kooi type conditions, and Rogers conditions we establish the uniqueness and existence of the solution after determining the equivalent integral form of the solution.

A Systematic Spectral-Tau Method for the Solution of Fuzzy Fractional Diffusion and Fuzzy Fractional Wave Equations

A Systematic Spectral-Tau Method for the Solution of Fuzzy Fractional Diffusion and Fuzzy Fractional Wave Equations

Fuzzy Fractional Order Sliding Mode Control for Automatic Clutch of Vehicle AMT

The automatic clutch driving system of vehicle automated manual transmission (AMT) is a typical nonlinear system. A fast and accurate position control is difficult to achieve because of the nonlinearity of the load changes caused by the clutch diaphragm spring, the cushion spring, and other nonlinear components and because of the uncertainty of the load changes caused by the cushion spring deformation in the clutch engagement process at varying temperatures on the friction surface. In this paper, based on the theory of fractional calculus and traditional sliding mode control, the method of fuzzy fractional order sliding mode control (FFOSMC) is proposed for clutch control. By selecting the fractional order sliding surface, the laws of control are designed; and self-tuning of the switch gain is realized with the fuzzy controller. Theoretical analysis and numerical simulation revealed that compared with the traditional fuzzy integer order sliding mode controller, FFOSMC achieves better dynamic performance in position control and better robustness against load disturbance and other uncertain factors.

Numerical Simulations for Solving Fuzzy Fractional Differential Equations by Max-Min Improved Euler Methods

Abstract In this paper, an extension is introduced into Max-Min Improved Euler methods for solving initial value problems of fuzzy fractional differential equations (FFDEs). Two modified fractional Euler type methods have been proposed and investigated to obtain numerical solutions of linear and nonlinear FFDEs. The proposed algorithms are tested on various illustrative examples. Exact values are also simulated to compare and discuss the closeness and accuracy of approximations so obtained. Comparatively, tables and graphs results reveal the complete reliability, efficiency and accuracy of the proposed methods.

Fuzzy Fractional Order Proportional Integral Controllers for UAV Lateral Attitude Control System

This article is based on the theory of fractional calculus control, and put forward a kind of fractional order PI controller design method for lateral attitude control system of unmanned aerial vehicle (UAV) model. And the unit step response of the control system is analyzed in the simulation to improve the UAV flight control system stability and robustness. Use the controller parameter tuning method and combined with fuzzy reasoning to design IOPID controller, FOPI controller and fuzzy fractional order PI controller. Then, by exploiting Matlab, the frequency domain response and unit step response characteristics of the different control systems can be plotted. The results verified that the designed fuzzy fractional order controller for the attitude control system is effective.

Design of Fuzzy Fractional PD + I Controllers Tuned by a Genetic Algorithm

The fractional-order concepts are a useful tool to describe several physical phenomena, and nowadays they are widely used in the field of automatic control. A genetic algorithm (GA) is a search process for finding approximate solutions in optimization problems. The GA provides further flexibility and robustness that are unique for signal process. In this paper we consider the development of an optimal fuzzy fractional PD + I controller in which the parameters are tuned by a GA. The performance of the proposed fuzzy fractional control is illustrated through some application examples.

On Existence and Uniqueness of Solution of Fuzzy Fractional Differential Equations

The purpose of this paper is to study the fuzzy fractional dieren- tial equations. We prove that fuzzy fractional dierential equation is equivalent to the fuzzy integral equation and then using this equivalence existence and uniqueness result is establish. Fuzzy derivative is consider in the Goetschel- Voxman sense and fractional derivative is consider in the Riemann Liouville sense. At the end, we give the applications of the main result.

Solving Fuzzy Fractional Boundary Value Problems Using Fractional Differential Transform Method

In this paper we shall present an approximate solution for fuzzy fractional boundary value problems (FFBVP's) based on the Fractional Differential Transform Method (FDTM) which is proposed to solve linear and nonlinear FFBVP ,s . The fuzziness will appear in the boundary conditions, to be fuzzy numbers. The solution of our model equations are calculated in the form of convergent series with easily computable components. Some examples are solved as an illustrations, the numerical results shows that the followed approach is easy to implement and accurate when applied to FFBVP ,s .

Fuzzy Fractional Neural Network Approximation by Fuzzy Quasi-interpolation Operators

Fuzzy Fractional Neural Network Approximation by Fuzzy Quasi-interpolation Operators

Application of Fuzzy Fractional Kinetic Equations to Modelling of the Acid Hydrolysis Reaction

In view of the usefulness and a great importance of the kinetic equation in specific chemical engineering problems, we discuss the numerical solution of a simple fuzzy fractional kinetic equation applied for the hemicelluloses hydrolysis reaction. The fuzzy approximate solution is derived based on the Legendre polynomials to the fuzzy fractional equation calculus. Moreover, the complete error analysis is explained based on the application of fuzzy Caputo fractional derivative. The main advantage of the present method is its superior accuracy which is obtained by using a limited number of Legendre polynomials. The method is computationally interesting, and the numerical results demonstrate the effectiveness and validity of the method for solving fuzzy fractional differential equations.

Solving Fuzzy Fractional Differential Equations Using Zadeh's Extension Principle

We study a fuzzy fractional differential equation (FFDE) and present its solution using Zadeh's extension principle. The proposed study extends the case of fuzzy differential equations of integer order. We also propose a numerical method to approximate the solution of FFDEs. To solve nonlinear problems, the proposed numerical method is then incorporated into an unconstrained optimisation technique. Several numerical examples are provided.

Numerical Solution of Fuzzy Fractional Pharmacokinetics Model Arising from Drug Assimilation into the Bloodstream

We propose a Jacobi tau method for solving a fuzzy fractional pharmacokinetics. This problem can model the concentration of the drug in the blood as time increases. The proposed approach is based on the Jacobi tau (JT) method. To illustrate the reliability of the method, some special cases of the equations are solved as test examples. The method reduces the solution of the problem to the solution of a system of algebraic equations. Error analysis included the fractional derivative error estimation, and the upper bound of the absolute errors is introduced for this method.

Modified fractional Euler method for solving Fuzzy Fractional Initial Value Problem

In this paper, the solution to Fuzzy Fractional Initial Value Problem [FFIVP] under Caputo-type fuzzy fractional derivatives by a modified fractional Euler method is presented. The Caputo-type fuzzy fractional derivatives are defined based on Hukuhara difference and strongly generalized fuzzy differentiability. The modified fractional Euler method based on a generalized Taylor’s formula and a modified trapezoidal rule is used for solving initial value problem under fuzzy fractional differential equation of order β∈0,1. Solving two examples of linear and nonlinear FFIVP illustrates the method.

Fuzzy acceptance sampling plans for inspection of geospatial data with ambiguity in quality characteristics

This paper presents a method for designing a fuzzy acceptance sampling plan (FASP) for the case where sampling parameters are defined as fuzzy numbers, in particular for quality inspection of geospatial data with ambiguous quality characteristics. In contrast to existing fuzzy sampling plans that concentrate solely on ambiguity in the fraction of nonconforming items, the proposed method includes three cases with different fuzzy sampling parameters. These are the fuzzy fraction of nonconforming items, fuzzy sample rate, and fuzzy fraction of nonconforming items and fuzzy sample rate together. The design of the fuzzy sampling plan is modeled as a fuzzy optimization problem dealing with two cases in terms of lot size based on the fuzzy Hypergeometric and Poisson distributions.The proposed method is implemented to design fuzzy sampling plans for quality inspection of geospatial mineral products in Qinghai Province, China. The results show that (1) the proposed method has the advantage of performing quality inspection for geospatial data products with uncertain parameters; (2) in contrast to a traditional sampling plan having a single OC-curve, the OC-band of a fuzzy sampling plan has the lower and upper bounds; and (3) in contrast to existing fuzzy sampling plans which account primarily for uncertainty in the fraction of nonconforming items, the proposed method completely covers fuzzy sampling parameters.

Fuzzy Fractional Monodromy and the Section–Hyperboloid

A theorem about the matrix of fractional monodromy will be formulated. The monodromy corresponds to going around a fiber with a singular point of oscillator type with arbitrary resonance. The reason of fractional monodromy and fuzziness of such a monodromy is explained. Some ideas for the proof of the theorem are given. A few remarks about the semi-global structure of singular lagrangian fibration are made.

Characterization of trabecular bone structure from high-resolution magnetic resonance images using fuzzy logic

The purpose of this work was to apply fuzzy logic image processing techniques to characterize the trabecular bone structure with high-resolution magnetic resonance images. Fifteen ex vivo high-resolution magnetic resonance images of specimens of human radii at 1.5 T and 12 in vivo high-resolution magnetic resonance images of the calcanei of peri- and postmenopausal women at 3 T were obtained. Soft segmentation using fuzzy clustering was applied to MR data to obtain fuzzy bone volume fraction maps, which were then analyzed with three-dimensional (3D) fuzzy geometrical parameters and measures of fuzziness. Geometrical parameters included fuzzy perimeter and fuzzy compactness, while measures of fuzziness included linear index of fuzziness, quadratic index of fuzziness, logarithmic fuzzy entropy, and exponential fuzzy entropy. Fuzzy parameters were validated at 1.5 T with 3D structural parameters computed from microcomputed tomography images, which allow the observation of true trabecular bone structure and with apparent MR structural indexes at 1.5 T and 3 T. The validation was statistically performed with the Pearson correlation coefficient as well as with the Bland-Altman method. Bone volume fraction correlation values (r) were up to .99 (P<.001) with good agreements based on Bland-Altman analysis showing that fuzzy clustering is a valid technique to quantify this parameter. Measures of fuzziness also showed consistent correlations to trabecular number parameters (r>.85; P<.001) and good agreements based on Bland-Altman analysis, suggesting that the level of fuzziness in high-resolution magnetic resonance images could be related to the trabecular bone structure.