Numerical solution of fuzzy delay differential equations under generalized differentiability by Euler's method

In this paper, we interpret a fuzzy delay differential equations using the concept of generalized differentiability. Using the Generalized Characterization Theorem, we investigate the problem of finding a numerical approximation of solutions. The Euler approximation method is implemented and its error analysis is discussed. The applicability of the theoretical results is illustrated with some examples

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 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

Fuzzy Stochastic Differential Equations Driven by Semimartingales-Different Approaches

The first aim of the paper is to present a survey of possible approaches for the study of fuzzy stochastic differential or integral equations. They are stochastic counterparts of classical approaches known from the theory of deterministic fuzzy differential equations. For our aims we present first a notion of fuzzy stochastic integral with a semimartingale integrator and its main properties. Next we focus on different approaches for fuzzy stochastic differential equations. We present the existence of fuzzy solutions to such equations as well as their main properties. In the first approach we treat the fuzzy equation as an abstract relation in the metric space of fuzzy sets over the space of square integrable random vectors. In the second one the equation is interpreted as a system of stochastic inclusions. Finally, in the last section we discuss fuzzy stochastic integral equations with solutions being fuzzy stochastic processes. In this case the notion of the stochastic Itô’s integral in the equation is crisp; that is, it has single-valued level sets. The second aim of this paper is to show that there is no extension to more general diffusion terms

List of contents

Rev. iberoam. bioecon. cambio clim. Vol.1(1) 2015; 95-114Los cambios medioambientales globales hacen pensar en un aumento futuro de la aridez, por ello es necesario buscar alternativas que permitan un uso más eficiente del agua y reducir su consumo, teniendo en cuenta que es un recurso limitado. En la actualidad, aproximadamente el 59,7% del total de agua planificada para todos los usos en Cuba se utiliza en la agricultura, pero no más del 50% de esa agua se convierte directamente en productos agrícolas. El estudio de las funciones agua-rendimiento y su uso dentro de la planificación del agua para riego es una vía importante para trazar estrategias de manejo que contribuyan al incremento en la producción agrícola. Utilizando los datos de agua aplicada por riego y los rendimientos obtenidos en más de 100 experimentos de campo realizados fundamentalmente en suelo Ferralítico Rojo de la zona sur de La Habana y con ayuda de herramientas de análisis de regresión en este trabajo se estiman las funciones agua aplicada-rendimientos para algunos cultivos agrícolas y se analizan las posibles estrategias de optimización del riego a seguir en función de la disponibilidad de agua. Seleccionar una estrategia de máxima eficiencia del riego puede conducir a reducciones de agua a aplicar entre un 21,6 y 46,8%, incrementos de la productividad del agua entre 17 y 32% y de la relación beneficios/costo estimada de hasta un 3,4%. Lo anterior indica la importancia desde el punto de vista económico que puede llegar a alcanzar el uso de esta estrategia en condiciones de déficit hídrico. El conocimiento de las funciones agua aplicada por riego-rendimiento y el uso de la productividad del agua, resultan parámetros factibles de introducir como indicadores de eficiencia en el planeamiento del uso del agua en la agricultura, con lo cual es posible reducir los volúmenes de agua a aplicar y elevar la relación beneficio-costo actual.Rev. iberoam. bioecon. cambio clim. Vol.1(1) 2015; 95-11

Detectability of non-differentiable generalized synchrony

Generalized synchronization of chaos is a type of cooperative behavior in directionally-coupled oscillators that is characterized by existence of stable and persistent functional dependence of response trajectories from the chaotic trajectory of driving oscillator. In many practical cases this function is non-differentiable and has a very complex shape. The generalized synchrony in such cases seems to be undetectable, and only the cases, in which a differentiable synchronization function exists, are considered to make sense in practice. We show that this viewpoint is not always correct and the non-differentiable generalized synchrony can be revealed in many practical cases. Conditions for detection of generalized synchrony are derived analytically, and illustrated numerically with a simple example of non-differentiable generalized synchronization.Comment: 8 pages, 8 figures, submitted to PR

An impulsive approach for numerical investigation of hybrid fuzzy differential equations and intuitionistic treatment for fuzzy ordinary and partial differential equations

Many evolution processes are characterized by the fact that at certain moments of time, they experience a change of state abruptly. It is assume naturally, that those perturbations act instantaneously, in the form of impulses. The impulsive differential equations, by means differential equations involving impulse effects, are seen as a natural description of observed evolution phenomenon of several real world problems. For example, systems with impulse effect have applications in physics, biotechnolagy, industrial robotics, pharmacokinetics, population dynamics, ecology, optimal control production theory and many others. Therefore, it is beneficial to study the theory of impulsive differential equations as a well deserved discipline, due to the increase applications of impulsive differential equations in various fields in the future. However, in many mathematical modelling of the real world problems, fuzziness and impulsiveness occurs simultaneously. This problem would be better modelled by impulsive fuzzy differential equations. Therefore, this research applies the theory of impulsive fuzzy differential equations by combining the theories of impulsive differential equations and fuzzy differential equations. The numerical algorithms are developed and the solutions are verified by comparing the results with the analytical solutions. The novel method for the first order linear impulsive hzzy differential equations under generalized differentiability is also proposed analytically and numerically, The convergence theor~m for the impulsive fuzzy differential equations (FDE) under generalized differentiability is defined. In this study, Ant Colony Programming (ACP) was used to find the optimal solution of FDE. Results obtained show that the method is effective in solving fuzzy differential equation. The solution in this method is equivaIent to the exact solution of the problem. Modified Romberg's method and Modified Two-step Simpson's 318 method are used to solve FDE with hzzy IVP has been successfully derived. The result has been shown that Modified Rornberg's method gave smaller error than the Standard Euler's method. Therefore Modified Romberg's method can estimate the solution of fizzy differential equation more effectively than the Euler's method in solving fuzzy differential equation. Meanwhile, by using the modified wo-step Simpson's 318 methods, it has been shown that the solution of FDE provide more accurate approximation to the exact solution and it also gives better results than the Runge-Kutta method. In other words, Modified Twostep Simpson's 318 method is an effective method to solve fuzzy differential equation compared to the Runge-Kutta method

FUZZY DELAY DIFFERENTIAL EQUATIONS WITH HYBRID SECOND AND THIRD ORDERS RUNGE-KUTTA METHOD

This paper considers fuzzy delay differential equations with known statedelays. A dynamic problem is formulated by time-delay differential equations and an efficient scheme using a hybrid second and third orders Runge-Kutta method is developed and applied. Runge-Kutta is well-established methods and can be easily modified to overcome the discontinuities, which occur in delay differential equations. Our objective is to develop a scheme for solving fuzzy delay differential equations. A numerical example was run, and the solutions were validated with the exact solution. The numerical results from C program will show that the hybrid Runge-Kutta scheme able to calculate the fuzzy solutions successfully

Numerical Solution of Fuzzy Arbitrary Order Predator-Prey Equations

This paper seeks to investigate the numerical solution of fuzzy arbitrary order predator-prey equations using the Homotopy Perturbation Method (HPM). Fuzziness in the initial conditions is taken to mean convex normalised fuzzy sets viz. triangular fuzzy number. Comparisons are made between crisp solution given by others and fuzzy solution in special cases. The results obtained are depicted in plots and tables to demonstrate the efficacy and powerfulness of the methodology