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

Fuzzy Solutions to Second Order Three Point Boundary Value Problem

In this manuscript, the proposed work is to study the existence of second-order differential equations with three point boundary conditions. Existence is proved using fuzzy set valued mappings of a real variable whose values are normal, convex, upper semi continuous and compactly supported fuzzy sets. The sufficient conditions are also provided to establish the existence results of fuzzy solutions of second order differential equations for three point boundary value problem. By using Banach fixed point principle, a new existence theorem of solutions for these equations in the metric space of normal fuzzy convex sets with distance given by the maximum of the Hausdorff distance between level sets is obtained. Then to further establish the existence, fixed point theorem for absolute retracts is used by taking consideration that space of fuzzy sets can be embedded isometrically as a cone in Banach space. Finally, an example is provided to illustrate the result

Global existence of solutions for fuzzy second-order differential equations under generalized H-differentiability

AbstractIn this paper, we study the global existence of solutions for second-order fuzzy differential equations with initial conditions under generalized H-differentiability. Second derivative of the H-difference of two functions under generalized H-differentiability is obtained. Two theorems which assure global existence of solutions for second-order fuzzy differential equations are given and proved. Some examples are given to illustrate these results

Status of the differential transformation method

Further to a recent controversy on whether the differential transformation method (DTM) for solving a differential equation is purely and solely the traditional Taylor series method, it is emphasized that the DTM is currently used, often only, as a technique for (analytically) calculating the power series of the solution (in terms of the initial value parameters). Sometimes, a piecewise analytic continuation process is implemented either in a numerical routine (e.g., within a shooting method) or in a semi-analytical procedure (e.g., to solve a boundary value problem). Emphasized also is the fact that, at the time of its invention, the currently-used basic ingredients of the DTM (that transform a differential equation into a difference equation of same order that is iteratively solvable) were already known for a long time by the "traditional"-Taylor-method users (notably in the elaboration of software packages --numerical routines-- for automatically solving ordinary differential equations). At now, the defenders of the DTM still ignore the, though much better developed, studies of the "traditional"-Taylor-method users who, in turn, seem to ignore similarly the existence of the DTM. The DTM has been given an apparent strong formalization (set on the same footing as the Fourier, Laplace or Mellin transformations). Though often used trivially, it is easily attainable and easily adaptable to different kinds of differentiation procedures. That has made it very attractive. Hence applications to various problems of the Taylor method, and more generally of the power series method (including noninteger powers) has been sketched. It seems that its potential has not been exploited as it could be. After a discussion on the reasons of the "misunderstandings" which have caused the controversy, the preceding topics are concretely illustrated.Comment: To appear in Applied Mathematics and Computation, 29 pages, references and further considerations adde

Application of Adomian decomposition method to solve hybrid fuzzy differential equations

AbstractIn this paper, we study the numerical solution of hybrid fuzzy differential equations by using Adomian decomposition method (ADM). This is powerful method which consider the approximate solution of a nonlinear equation as an infinite series usually converging to the accurate solution. Several numerical examples are given and by comparing the numerical results obtained from ADM and predictor corrector method (PCM), we have studied their accuracy

On exact solutions of a class of interval boundary value problems

summary:In this article, we deal with the Boundary Value Problem (BVP) for linear ordinary differential equations, the coefficients and the boundary values of which are constant intervals. To solve this kind of interval BVP, we implement an approach that differs from commonly used ones. With this approach, the interval BVP is interpreted as a family of classical (real) BVPs. The set (bunch) of solutions of all these real BVPs we define to be the solution of the interval BVP. Therefore, the novelty of the proposed approach is that the solution is treated as a set of real functions, not as an interval-valued function, as usual. It is well-known that the existence and uniqueness of the solution is a critical issue, especially in studying BVPs. We provide an existence and uniqueness result for interval BVPs under consideration. We also present a numerical method to compute the lower and upper bounds of the solution bunch. Moreover, we express the solution by an analytical formula under certain conditions. We provide numerical examples to illustrate the effectiveness of the introduced approach and the proposed method. We also demonstrate that the approach is applicable to non-linear interval BVPs

On solving fuzzy delay differential equation using bezier curves

In this article, we plan to use Bezier curves method to solve linear fuzzy delay differential equations. A Bezier curves method is presented and modified to solve fuzzy delay problems taking the advantages of the fuzzy set theory properties. The approximate solution with different degrees is compared to the exact solution to confirm that the linear fuzzy delay differential equations process is accurate and efficient. Numerical example is explained and analyzed involved first order linear fuzzy delay differential equations to demonstrate these proper features of this proposed problem

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