Thomas Feuring

Fuzzy differential equations

We will concentrate on the second order, linear, constant coefficient ordinary differential equation for x in interval I. I can be [0, T], for T > 0 or I can be [0, ∞). The initial conditions are y(0) = γ 0, y′(0) = γ 1. We assume g is continuous on I.

Evolutionary algorithm solution to fuzzy problems: Fuzzy linear programming

In this paper we wish to find solutions to the fully fuzzified linear program where all the parameters and variables are fuzzy numbers. We first change the problem of maximizing a fuzzy number, the value of the objective function, into a multi-objective fuzzy linear programming problem. We prove that fuzzy flexible programming can be used to explore the whole undominated set to the multi-objective fuzzy linear program. An evolutionary algorithm is designed to solve the fuzzy flexible program and we apply this program to two applications to generate good solutions.

Fuzzy hierarchical analysis revisited

Abstract We present a new method of finding the fuzzy weights in fuzzy hierarchical analysis which is the direct fuzzification of the original method used by Saaty in the analytic hierarchy process. We test our new procedure in two cases where there are formulas for the crisp weights. An example is presented where there are five criteria and three alternatives.

Fuzzy mathematics in economics and engineering

Fuzzy Sets.- Solving Fuzzy Equations.- Fuzzy Mathematics in Finance.- Fuzzy Non-Linear Regression.- Operations Research.- Fuzzy Differential Equations.- Fuzzy Difference Equations.- Fuzzy Partial Differential Equations.- Fuzzy Eigenvalues.- Fuzzy Integral Equations.- Summary and Conclusions.- Evolutionary Algorithms.

Introduction to fuzzy partial differential equations

This paper considers solutions to elementary fuzzy partial differential equations. We follow the same strategy as in Buckley and Feuring (Fuzzy Sets and Systems, to appear) which is: (1) first check to see if the Buckley-Feuring method produces a solution; and (2) if the Buckley-Feuring method does not give a solution, then see if the Seikkala procedure generates a solution. Examples are presented showing the Buckley-Feuring solution and the Seikkala solution.

Fuzzy initial value problem for N th-order linear differential equations

We present two methods of solving nth-order linear differential equation that have fuzzy initial conditions. For the second order, constant coefficient, case we show that: (1) the first method always gives a solution; and (2) the second solution quite often does not exist, and when it does exist, it may be equal to the solution determined using the first method. We apply the first method to three example problems including an electrical circuit and a vibrating mass.

Fuzzy and neural : interactions and applications

Introduction.- Fuzzy Sets and Fuzzy Functions.- Neural Nets.- First Approximation Results.- Hybrid Neural Nets.- Neural Nets Solve Fuzzy Problems.- Fuzzy Neural Nets.- Second Approximation Results.- Hybrid Fuzzy Neural Nets.- Applications of Hybrid Fuzzy Neural Nets and Fuzzy Neural Nets.- Fuzzy Teaching Machine.- Summary, Future Research and Conclusions.

The fuzzy neural network approximation lemma

Abstract It is well known that artificial neural networks are universal approximators. But what about fuzzy neural networks? Only Buckley and Hayashi [1] presented a theoretical result for these networks: They showed that there are fuzzy functions which cannot be approximated by a certain fuzzy neural network. In this paper we answer the question for a special type of fuzzy neural networks — especially with regard to the fuzzy arithmetic that is used: It is simplified in order to minimize the computational expense as well as to simplify the theoretical examinations. We prove that the class of fuzzy functions which is identical to the class of all continuous real functions extended by means of the extension principle can be approximated by certain fuzzy neural networks.

Linear systems of first order ordinary differential equations: fuzzy initial conditions

Abstract We present two types of fuzzy solutions to linear systems of first order differential equations having fuzzy initial conditions. The first solution, called the extension principle solution, fuzzifies the crisp solution and then checks to see if its α-cuts satisfy the differential equations. The second solution, called the classical solution, solves the fuzzified differential equations and then checks to see if the solution always defines a fuzzy number. Three applications are presented: (1) predator–prey models; (2) the spread of infectious diseases; and (3) modeling an arms race.

Fuzzy Markov chains

We first review some of the basic results of finite Markov chains based on probability theory, then we present fuzzy finite Markov chains based on possibility theory, and compare the results of the two theories. Then we introduce finite horizon Markovian decision processes based on fuzzy Markov chains and study an example in detail showing our solution procedure.