FUZZY RELATIONAL EQUATIONS: BRIDGING THEORY,METHODOLOGY AND PRACTICE

This study delivers a general overview of the theory and practice of fuzzy relational equations. We discuss various methods leading to the solutions of these equations starting from analytical approaches, moving through semi-analytic methods and finally elaborating on the neural-like style of finding solutions to relational constructs. The paper addresses important aspects of knowledge representation worked out by these equations and proposes a number of structural enhancements of the existing relational architectures.     

New travelling wave solutions to modified CH and DP equations

A new procedure for finding exact travelling wave solutions to the modified Camassa-Holm and Degasperis-Procesi equations is proposed. It turns out that many new solutions are obtained. Furthermore, these solutions are in general forms, and many known solutions to these two equations are only special cases of them.     

Equivalence of the probability measures generated by the solutions of nonlinear evolution differential equations in a Hilbert space, disturbed by Gaussian processes. Part II

The paper continues the studies in the equivalence of the measures generated by the solutions of nonlinear evolution differential equations with unbounded linear operators perturbed by random Gaussian processes in a Hilbert space, in particular í. Two different nonlinear evolution differential equations perturbed by the same random Gaussian process in the right-hand side are considered in the space í. The sufficient existence and uniqueness conditions are established for the solutions of these equations, the equivalence of the measures generated by the solutions is proved, and explicit formulas for the Radon–Nikodym density of the respective measures calculated in terms of the coefficients of the considered equations are written.     

Equivalence of the probability measures generated by the solutions of nonlinear evolution differential equations in a Hilbert space,disturbed by Gaussian processes. Part I

Nonlinear evolution differential equations with unbounded linear operators of disturbance by Gaussian random processes are considered in an abstract Hilbert space. For the Cauchy problem for the differential equations, the sufficient existence and uniqueness conditions for their solutions are proved and the sufficient conditions for the equivalence of the probability measures generated by these solutions are derived. Moreover, the corresponding Radon–Nikodym densities are calculated explicitly in terms of the coefficients or characteristics of the considered differential equations.     

Infinite fuzzy relation equations with continuous t-norms

This study develops a concept of infinite fuzzy relation equations with a continuous t-norm. It extends the work by Xiong and Wang [Q.Q. Xiong, X.P. Wang, Some properties of sup-min fuzzy relational equations on infinite domains, Fuzzy Sets and Systems 151 (2005) 393-402]. We describe attainable (respectively, unattainable) solutions, which are closely related to minimal solutions to the equations. It is shown that a solution set comprises both attainable and unattainable solutions. The study offers a characterization of these solutions. Under some assumptions, the solution set is presented and discussed. Two applications are also given.     

Nerve impulse equations: Computer solutions and displays

The differential equations for nerve impulse propagation (the Hodgkin-Huxley equations) are solved numerically by computer. Programs for performing these computations and displaying the results are described, examples of solutions are given, and the use of these programs in an undergraduate Neurobiology course is discussed.     

几个非线性偏微分方程的非古典对称及相似解

研究了一些非线性偏微分方程的非古典势对称和非古典对称,得到了某些方程的新的势对称和新的对称,同时也得到了其伴随系统的新的对称,并求出了一些相似解．这些解对进一步研究这些非线性偏微分方程所描述的物理现象具有广泛的应用价值．     

Stability of Discontinuous Retarded Functional Differential Equations With Applications

We present Lyapunov stability results, including converse theorems, for a class of discontinuous dynamical systems (DDS) determined by the solutions of retarded functional differential equations. We demonstrate the applicability of these results in the analysis of several specific important classes of DDS determined by functional differential equations and differential-difference equations.     

Numerical Solution of Fuzzy Equations with Z-numbers Using Neural Networks

In this paper, the uncertainty property is represented by the Z-number as the coefficients of the fuzzy equation. This modification for the fuzzy equation is suitable for nonlinear system modeling with uncertain parameters. We also extend the fuzzy equation into dual type, which is natural for linearin-parameter nonlinear systems. The solutions of these fuzzy equations are the controllers when the desired references are regarded as the outputs. The existence conditions of the solutions (controllability) are proposed. Two types of neural networks are implemented to approximate solutions of the fuzzy equations with Z-number coefficients.     

First order linear fuzzy differential equations under generalized differentiability

First order linear fuzzy differential equations are investigated. We interpret a fuzzy differential equation by using the strongly generalized differentiability concept, because under this interpretation, we may obtain solutions which have a decreasing length of their support (which means a decreasing uncertainty). In several applications the behaviour of these solutions better reflects the behaviour of some real-world systems. Derivatives of the H-difference and the product of two functions are obtained and we provide solutions of first order linear fuzzy differential equations, using different versions of the variation of constants formula. Some examples show the rich behaviour of the solutions obtained.     

Comparison of Legendre polynomial approximation and variational iteration method for the solutions of general linear Fredholm integro-differential equations

In this study it is shown that the numerical solutions of linear Fredholm integro-differential equations obtained by using Legendre polynomials can also be found by using the variational iteration method. Furthermore the numerical solutions of the given problems which are solved by the variational iteration method obviously converge rapidly to exact solutions better than the Legendre polynomial technique. Additionally, although the powerful effect of the applied processes in Legendre polynomial approach arises in the situations where the initial approximation value is unknown, it is shown by the examples that the variational iteration method produces more certain solutions where the first initial function approximation value is estimated. In this paper, the Legendre polynomial approximation (LPA) and the variational iteration method (VIM) are implemented to obtain the solutions of the linear Fredholm integro-differential equations and the numerical solutions with respect to these methods are compared.     

Global stability analysis of the Runge-Kutta methods for volterra integral and integro-differential equations with degenerate kernels

We investigate the stability of the numerical solutions resulting from applying very general classes of Runge-Kutta methods to Volterra integral and integro-differential equations with degenerate kernels. The results are generalizations of previous results obtained by the authors for exact collocation methods for these equations.     

Extended Jacobian elliptic function algorithm with symbolic computation to construct new doubly-periodic solutions of nonlinear differential equations

With the aid of computerized symbolic computation, the extended Jacobian elliptic function expansion method and its algorithm are presented by using some relations among ten Jacobian elliptic functions and are very powerful to construct more new exact doubly-periodic solutions of nonlinear differential equations in mathematical physics. The new (2+1)-dimensional complex nonlinear evolution equations is chosen to illustrate our algorithm such that sixteen families of new doubly-periodic solutions are obtained. When the modulus m→1 or 0, these doubly-periodic solutions degenerate as solitonic solutions including bright solitons, dark solitons, new solitons as well as trigonometric function solutions.     

Certain operators of fractional calculus and their applications to differential equations

In many recent works, one can find remarkable demonstrations of the usefulness of certain fractional calculus operators in the derivation of (explicit) particular solutions of a number of linear ordinary and partial differential equations of the second and higher orders. Our main objective in the present sequel to these earlier works, is to show how readily and systematically some recent contributions on this subject, involving linear ordinary and partial differential equations of order , can be obtained (in a unified manner) by suitably appealing to some general theorems on (explicit) particular solutions of a certain family of linear ordinary fractional differintegral equations.     

On the numerical solution of some algebraic integral equations

We discuss the numerical solution of some algebraic integral equations and integral equations of Lichtenstein type by means of a variational method and by using the subspace ofL-splines. For these approximate solutions the proofs of existence, uniqueness, and convergence as well as error estimates are given.     

Relational heuristics for the design of deterministic programs

Summary It has been shown that the design of deterministic programs can be formulated as the resolution of relational equations. Because relational calculi are not sufficiently structured, there are no algorithmic solutions to relational equations. In this paper, we formulate some heuristic solutions to these equations.Part of this work was carried out while the first and second author were at Laval University in Québec. Canada: it was supported by the National Research Council of Canada through a research grant to the first author and a scholarship to the second author. Presently, this research is supported by a grant to the first and third authors from the Tunisian Council on Scientific and Technical Research     

A Level Set Framework for Capturing Multi-Valued Solutions of Nonlinear First-Order Equations

We introduce a level set method for the computation of multi-valued solutions of a general class of nonlinear first-order equations in arbitrary space dimensions. The idea is to realize the solution as well as its gradient as the common zero level set of several level set functions in the jet space. A very generic level set equation for the underlying PDEs is thus derived. Specific forms of the level set equation for both first-order transport equations and first-order Hamilton-Jacobi equations are presented. Using a local level set approach, the multi-valued solutions can be realized numerically as the projection of single-valued solutions of a linear equation in the augmented phase space. The level set approach we use automatically handles these solutions as they appear     

An application of rationalized Haar functions to solution of linear partial differential equations

Rationalized Haar functions (RHFs, for short) are applied for solving linear first- and second-order partial differential equations (PDEs, for short). For this purpose, new operational matrices of integration and differentiation based on a double rationalized Haar series are derived. By using these operational matrices for their solution, the PDEs are transformed into matrix equations quite easily. Coefficients of double rationalized Haar series related to their solutions can be obtained by solving these matrix equations. Some numerical examples are also included.     

薄膜体声波谐振器的FDTD数值分析

提出了基于时域有限差分方法对薄膜体声波谐振器进行数值分析的新方法。利用时域有限差分法理论对压电材料的控制方程,牛顿方程和电学方程在空间和时间进行了离散化,通过得到的差分方程直接得出了声场传播的时域数值解。使用该数值方法对薄膜体声波谐振器的电学特性阻抗进行了分析,并将结果与一维Mason模型的解析解进行了比较验证。     

Condition for existence of solutiuons of linear equations with arbitrary vectors and its relation to controllability

Necessary and sufficient conditions are given for the existence of solutions for a class of linear equations with arbitrary constant vectors. It is shown that these conditions are useful for studying and unifying various concepts related to controllability of linear discrete-time systems. The result can be regarded as the dual to that of the partial uniqueness of solutions of linear equations.