M-fractional derivative under interval uncertainty: theory, properties and applications

In the recent years some efforts were made to propose simple and well-behaved fractional derivatives that inherit the classical properties from the first order derivative. In this regards, the truncated M-fractional derivative for α-differentiable function was recently introduced that is a generalization of four fractional derivatives presented in the literature and has their important features. In this research, we aim to generalize this novel and effective derivative under interval uncertainty. The concept of interval truncated M-fractional derivative is introduced and some of the distinguished properties of this interesting fractional derivative such as Rolle’s and mean value theorems, are developed for the interval functions. In addition, the existence and uniqueness conditions of the solution for the interval fractional differential equations (IFDEs) based on this new derivative are also investigated. Finally, we present the applicability of this novel interval fractional derivative for IFDEs based on the notion of w-increasing (w-decreasing) by solving a number of test problems

An eigenvalue-eigenvector method for solving a system of fractional differential equations with uncertainty

A new method is proposed for solving systems of fuzzy fractional differential equations (SFFDEs) with fuzzy initial conditions involving fuzzy Caputo differentiability. For this purpose, three cases are introduced based on the eigenvalue-eigenvector approach; then it is shown that the solution of system of fuzzy fractional differential equations is vector of fuzzy-valued functions. Then the method is validated by solving several examples

Fractional and time-scales differential equations

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Combined effect of the magnetic field, orientation, and filling ratio on cylindrical pulsating heat pipe using distilled water and distilled water/Fe3O4 nanofluid

To investigate the effect of the magnetic field, a pulsating heat pipe was made in the shape of a cylinder and Fe3O4 nanoparticles (%0.1 wt) were used with the base fluid of distilled water as the working fluid. (Tetramethyl ammonium hydroxide) TMAH surfactant was used as a stabilizer. To investigate the effect of gravity on the performance of the pipe, the device was tested at different angles from zero to 90 degrees. In this research, the effect of different variables, including the type of working fluid (distilled water vs. nanofluid), filling ratio, slope, and amount of heat input to the evaporator (30–300 W), in two different states, once without the influence of the magnetic field and once again with the application of a magnetic field was investigated. The results of the tests showed that the performance of the device at 50 % filling ratio is better than 60 % filling ratio. The use of nanoparticles improved the performance of the device. Inclining the device increases the thermal resistance so that the device performs poorly in the horizontal mode in all modes except when it is under the influence of a magnetic field. The use of nanofluid, as well as the application of a magnetic field, makes the start-up time of the device decrease by 37 % and 30 %, respectively, compared to distilled water. The temperature of the start of fluctuations also decreases by 24 % and 32 %, respectively

Some kinds of the controllable problems for fuzzy control dynamic systems

In this work, we have discussed the fuzzy solutions for fuzzy controllable problem, fuzzy feedback problem, and fuzzy global controllable (GC) problems. We use the method of successive approximations under the generalized Lipschitz condition for the local existence and furthermore, we have described the contraction principle under suitable conditions for global existence and uniqueness of fuzzy solutions. We have too the GC results for fuzzy systems. Some examples and computer simulation illustrating our approach are also given for these controllable problems

Developing a Local Neurofuzzy Model for Short-Term Wind Power Forecasting

Large scale integration of wind generation capacity into power systems introduces operational challenges due to wind power uncertainty and variability. Therefore, accurate wind power forecast is important for reliable and economic operation of the power systems. Complexities and nonlinearities exhibited by wind power time series necessitate use of elaborative and sophisticated approaches for wind power forecasting. In this paper, a local neurofuzzy (LNF) approach, trained by the polynomial model tree (POLYMOT) learning algorithm, is proposed for short-term wind power forecasting. The LNF approach is constructed based on the contribution of local polynomial models which can efficiently model wind power generation. Data from Sotavento wind farm in Spain was used to validate the proposed LNF approach. Comparison between performance of the proposed approach and several recently published approaches illustrates capability of the LNF model for accurate wind power forecasting

Uncertain viscoelastic models with fractional order: a new spectral tau method to study the numerical simulations of the solution

The analysis of the behaviors of physical phenomena is important to discover significant features of the character and the structure of mathematical models. Frequently the unknown parameters involve in the models are assumed to be unvarying over time. In reality, some of them are uncertain and implicitly depend on several factors. In this study, to consider such uncertainty in variables of the models, they are characterized based on the fuzzy notion. We propose here a new model based on fractional calculus to deal with the Kelvin–Voigt (KV) equation and non-Newtonian fluid behavior model with fuzzy parameters. A new and accurate numerical algorithm using a spectral tau technique based on the generalized fractional Legendre polynomials (GFLPs) is developed to solve those problems under uncertainty. Numerical simulations are carried out and the analysis of the results highlights the significant features of the new technique in comparison with the previous findings. A detailed error analysis is also carried out and discussed

Tau method for the numerical solution of a fuzzy fractional kinetic model and its application to the oil palm frond as a promising source of xylose

The Oil Palm Frond (a lignocellulosic material) is a high-yielding energy crop that can be utilized as a promising source of xylose. It holds the potential as a feedstock for bioethanol production due to being free and inexpensive in terms of collection, storage and cropping practices. The aim of the paper is to calculate the concentration and yield of xylose from the acid hydrolysis of the Oil Palm Frond through a fuzzy fractional kinetic model. The approximate solution of the derived fuzzy fractional model is achieved by using a tau method based on the fuzzy operational matrix of the generalized Laguerre polynomials. The results validate the effectiveness and applicability of the proposed solution method for solving this type of fuzzy kinetic model

On new solutions of linear system of first -order fuzzy differential equations with fuzzy coefficient

In this paper, we firstly introduce system of first order fuzzy differential equations. Then, we convert the problem to two crisp systems of first order differential equations. For numerical aspects, we apply exponentially fitted Runge Kutta method to solve the fuzzy problems. We solve some well-known examples in order to demonstrate the applicability and accuracy of results