A survey on fuzzy fractional differential and optimal control nonlocal evolution equations

We survey some representative results on fuzzy fractional differential equations, controllability, approximate controllability, optimal control, and optimal feedback control for several different kinds of fractional evolution equations. Optimality and relaxation of multiple control problems, described by nonlinear fractional differential equations with nonlocal control conditions in Banach spaces, are considered.Comment: This is a preprint of a paper whose final and definite form is with 'Journal of Computational and Applied Mathematics', ISSN: 0377-0427. Submitted 17-July-2017; Revised 18-Sept-2017; Accepted for publication 20-Sept-2017. arXiv admin note: text overlap with arXiv:1504.0515

A NUMERICAL METHOD FOR SOLVING FUZZY INITIAL VALUE PROBLEMS

In this thesis, the optimized one-step methods based on the hybrid block method (HBM) are derived for solving first and second-order fuzzy initial value problems. The off-step points are chosen to minimize the local truncation error of the proposed methods. Several theoretical properties of the proposed methods, such as stability, convergence, and consistency are investigated. Moreover, the regions of absolute stability of the proposed methods are plotted. Numerical results indicate that the proposed methods have order three and they are stable and convergent. In addition, several numerical examples are presented to show the efficiency and accuracy of the proposed methods. Results are compared with the existing ones in the literature. Even though the one off-step point is used, the results of the proposed methods are better than the ones obtained by other methods with a less computational cost

Regularization scheme for uncertain fuzzy differential equations: Analysis of solutions

In this paper a regularization scheme for a family of uncertain fuzzy systems of differential equations with respect to the uncertain parameters is introduced. Important fundamental properties of the solutions are discussed on the basis of the established technique and new results are proposed. More precisely, existence and uniqueness criteria of solutions of the regularized equations are established. In addition, an estimation is proposed for the distance between two solutions. Our results contribute to the progress in the area of the theory of fuzzy systems of differential equations and extend the existing results to the uncertain case. The proposed results also open the horizon for generalizations including equations with delays and some modifications of the Lyapunov theory. In addition, the opportunities for applications of such results to the design of efficient fuzzy controllers are numerous

Numerical And Approximate- Analytical Solution Of Fuzzy Initial Value Problems

Persamaan pembezaan kabur ( FDEs ) digunakan untuk memodel masalah tertentu dalam bidang sains dan kejuruteraan dan telah dikaji oleh ramai penyelidik . Masalah tertentu memerlukan penyelesaian FDEs yang memenuhi keadaan awal kabur menimbulkan masalah awal kabur ( FIVPs ). Contoh masalah seperti ini boleh didapati dalam fizik, kejuruteraan, model penduduk, dinamik reaktor nuklear, masalah perubatan, rangkaian neural dan teori kawalan. Walau bagaimanapun, kebanyakan masalah nilai awal kabur tidak boleh diselesaikan dengan tepat. Tambahan pula, penyelesaian analisis tepat yang diperoleh juga mungkin begitu sukar untuk dinilai dan oleh itu kaedah berangka dan analisis hampiran perlu untuk memperoleh penyelesaian. Fuzzy differential equations (FDEs) are used for the modeling of some problems in science and engineering and have been studied by many researchers. Certain problems require the solution of FDEs which satisfy fuzzy initial conditions giving rise to fuzzy initial problems (FIVPs). Examples of such problems can be found in physics, engineering, population models, nuclear reactor dynamics, medical problems, neural networks and control theory. However, most fuzzy initial value problems cannot be solved exactly. Furthermore, exact analytical solutions obtained may also be so difficult to evaluate and therefore numerical and approximate- analytical methods may be necessary to evaluate the solution

Existence of solutions to uncertain differential equations of nonlocal type via an extended Krasnosel’skii fixed point theorem

In the present study, we investigate the existence of the solutions to a type of uncertain differential equations subject to nonlocal derivatives. The approach is based on the application of an extended Krasnosel’skii fixed point theorem valid on fuzzy metric spaces. With this theorem, we deduce that the problem of interest has a fuzzy solution, which is defined on a certain interval. Our approach includes the consideration of a related integral problem, to which the above-mentioned tools are applicable. We finish with some physical motivationsWe are grateful to the Editor and the anonymous Referees for their comments and suggestions that helped to improve the paper. The research of J.J.N. and R.R.L. is supported by grant numbers PID2020-113275GB-I00 (AEI/FEDER, UE), MTM2016-75140-P (AEI/FEDER, UE), and ED431C 2019/02 (GRC Xunta de Galicia). The visit of A.K. to the University of Santiago de Compostela has been partially supported by the Agencia Estatal de Investigación (AEI) of Spain, co-financed by the European Fund for Regional Development (FEDER) corresponding to the 2014-2020 multiyear financial framework, project MTM2016-75140-P; and by Xunta de Galicia under grant ED431C 2019/02. Open Access funding provided thanks to the CRUE-CSIC agreement with Springer NatureS

A kernel least mean square algorithm for fuzzy differential equations and its application in earth's energy balance model and climate

Abstract This paper concentrates on solving fuzzy dynamical differential equations (FDDEs) by use of unsupervised kernel least mean square (UKLMS). UKLMS is a nonlinear adaptive filter which works by applying kernel trick to LMS adaptive filter. UKLMS estimates multivariate function which is embedded to estimate the solution of FDDE. Adaptation mechanism of UKLMS helps for finding solution of FDDE in a recursive scenario. Without any desired response, UKLMS finds nonlinear functions. For this purpose, an approximate solution of FDDE is constructed based on adaptable parameters of UKLMS. An optimization algorithm, optimizes the values of adaptable parameters of UKLMS. The proposed algorithm is applied for solving Earth energy balance model (EBM) which is considered as a fuzzy differential equation for the first time. The method in comparison with the other existing approaches (such as numerical methods) has some advantages such as more accurate solution and also that the obtained solution has a functional form, thus the solution can be obtained at each time in training interval. Low error and applicability of developed algorithm are examined by applying it for solving several problems. After comparing the numerical results, with relative previous works, the superiority of the proposed method will be illustrated

Numerical Solution of First-Order Linear Differential Equations in Fuzzy Environment by Runge-Kutta-Fehlberg Method and Its Application

The numerical algorithm for solving “first-order linear differential equation in fuzzy environment” is discussed. A scheme, namely, “Runge-Kutta-Fehlberg method,” is described in detail for solving the said differential equation. The numerical solutions are compared with (i)-gH and (ii)-gH differential (exact solutions concepts) system. The method is also followed by complete error analysis. The method is illustrated by solving an example and an application