Reachable solution set of a fuzzy relation equation

The largest element of the solution set of a fuzzy relation equation has been found by E. Sanchez (Inform. and Control30 (1976), 38–48) but the smallest element does not exist. It is difficult to expose the solution of the fuzzy relation equation. In the case of the determinate relation equations, complete consequences have been found by Wang Peizhuang and Yuan Meng (“Relation Equation and Relation Inequalities,” Selected papers on fuzzy subsets, Beijing Normal University, March 1980). In the case of the fuzzy relation equations, Wang and Yuan have given a class of special solutions which probably possesses some minimality characterizations. In this paper, the reachable solution set of the fuzzy relation equation is given. For the fuzzy relation equation on the finite sets, a neat and efficient method for solving it is given.     

Basic Theory of Fuzzy Difference Equations

The notion of fuzzy difference equation is introduced. Using Lyapunov type of function a comparison theorem for the fuzzy difference equation is obtained in terms of ordinary difference equations, which is used as a tool to study the stability results of the fuzzy difference equations.     

On the “Destruction” of Solutions of Nonlinear Wave Equations of Sobolev Type with Cubic Sources

We consider model three-dimensional wave nonlinear equations of Sobolev type with cubic sources, and foremost, model three-dimensional equations of Benjamin-Bona-Mahony and Rosenau types with model cubic sources. An essentially three-dimensional nonlinear equation of spin waves with cubic source is also studied. For these equations, we investigate the first initial boundary-value problem in a bounded domain with smooth boundary. We prove local solvability in the strong generalized sense and, for an equation of Benjamin-Bona-Mahony type with source, we prove the unique solvability of a “weakened” solution. We obtain sufficient conditions for the “destruction” of the solutions of the problems under consideration. These conditions have the sense of a “large” value of the initial perturbation in the norms of certain Banach spaces. Finally, for an equation of Benjamin-Bona-Mahony type, we prove the “failure” of a “weakened” solution in finite time.     

Li–Yorke chaos in higher dimensions: a review

Li and Yorke not only introduced the term “chaos” along with a mathematically rigorous definition of what they meant by it, but also gave a condition for chaos in scalar difference equations, their equally famous “period three implies chaos” result. Generalizations of the Li and Yorke definition of chaos to difference equations in ? ⁿ are reviewed here as well as higher dimensional conditions ensuring its existence, specifically the “snap-back repeller” condition of Marotto and its counterpart for saddle points. In addition, further generalizations to mappings in Banach spaces and complete metric spaces are considered. These will be illustrated with various simple examples including an application to chaotic dynamics on the metric space (?  ⁿ , D) of fuzzy sets on the base space ? ⁿ .     

Construction and Study of Exact Solutions to A Nonlinear Heat Equation

We construct and study exact solutions to a nonlinear second order parabolic equation which is usually called the “nonlinear heat equation” or “nonlinear filtration equation” in the Russian literature and the “porous medium equation” in other countries. Under examination is the special class of solutions having the form of a heat wave that propagates through cold (zero) background with finite velocity. The equation degenerates on the boundary of a heat wave (called the heat front) and its order decreases. The construction of these solutions by passing to an overdetermined system and analyzing its solvability reduces to integration of nonlinear ordinary differential equations of the second order with an initial condition such that the equations are not solvable with respect to the higher derivative. Some admissible families of heat fronts and the corresponding exact solutions to the problems in question are obtained. A detailed study of the global properties of solutions is carried out by the methods of the qualitative theory of differential equations and power geometry which are adapted for degenerate equations. The results are interpreted from the point of view of the behavior and properties of heat waves with a logarithmic front.     

On two conjectures about Dennis-Moré conditions

Numerical Algorithms - Quasi-Newton methods for solving nonlinear systems of equations are generally defined in order to satisfy a “direct secant equation” or an “inverse secant...     

The impact of finite precision arithmetic and sensitivity on the numerical solution of partial differential equations

In this paper, we address some fundamental issues concerning “time marching” numerical schemes for computing steady state solutions of boundary value problems for nonlinear partial differential equations. Simple examples are used to illustrate that even theoretically convergent schemes can produce numerical steady state solutions that do not correspond to steady state solutions of the boundary value problem. This phenomenon must be considered in any computational study of nonunique solutions to partial differential equations that govern physical systems such as fluid flows. In particular, numerical calculations have been used to “suggest” that certain Euler equations do not have a unique solution. For Burgers' equation on a finite spatial interval with Neumann boundary conditions the only steady state solutions are constant (in space) functions. Moreover, according to recent theoretical results, for any initial condition the corresponding solution to Burgers' equation must converge to a constant as t → ∞. However, we present a convergent finite difference scheme that produces false nonconstant numerical steady state “solutions.” These erroneous solutions arise out of the necessary finite floating point arithmetic inherent in every digital computer. We suggest the resulting numerical steady state solution may be viewed as a solution to a “nearby” boundary value problem with high sensitivity to changes in the boundary conditions. Finally, we close with some comments on the relevance of this paper to some recent “numerical based proofs” of the existence of nonunique solutions to Euler equations and to aerodynamic design.     

Interpolation Problems for Entire Functions Induced by Regular Hexagons

We consider linear equations for analytic functions in the plane with cuts along a “half” of the boundary of a hexagon. We propose a regularization method, reducing them to an equation with difference kernel. Applications are given to the moment problem for entire functions of exponential type.     

Numerical solutions of diffusive logistic equation

In this paper we investigate numerically positive solutions of a superlinear Elliptic equation on bounded domains.The study of Diffusive logistic equation continues to be an active field of research. The subject has important applications to population migration as well as many other branches of science and engineering. In this paper the “finite difference scheme” will be developed and compared for solving the one- and three-dimensional Diffusive logistic equation. The basis of the analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach, developed from many authors these years.     

Determinants of elliptic hypergeometric integrals

We start from an interpretation of the BC ₂-symmetric “Type I” (elliptic Dixon) elliptic hypergeometric integral evaluation as a formula for a Casoratian of the elliptic hypergeometric equation and then generalize this construction to higher-dimensional integrals and higher-order hypergeometric functions. This allows us to prove the corresponding formulas for the elliptic beta integral and symmetry transformation in a new way, by proving that both sides satisfy the same difference equations and that these difference equations satisfy a needed Galois-theoretic condition ensuring the uniqueness of the simultaneous solution.     

Fuzzy Volterra Integral Equations: A Stacking Theorem Approach

New existence results are established for fuzzy Volterra equations with a “local” integrably boundedness condition using: (i) the Banach Alaoglu theorem; and (ii) the stacking theorem.     

求某些非线性偏微分方程特解的一个简洁方法

简单介绍了应用一个简洁的“试探函数法”求解非线性偏微分方程的基本步骤,主要研究了两大类方程,一类是Burgers方程或KdV方程的推广,另一类是具有特殊非线性反应率的Fisher方程.不难看出,这个方法是简洁的,并且可望进一步扩展.     

The Global Random Attractor for a Class of Stochastic Porous Media Equations

We prove new L ²-estimates and regularity results for generalized porous media equations “shifted by” a function-valued Wiener path. To include Wiener paths with merely first spatial (weak) derivates we introduce the notion of “ζ-monotonicity” for the non-linear function in the equation. As a consequence we prove that stochastic porous media equations have global random attractors. In addition, we show that (in particular for the classical stochastic porous media equation) this attractor consists of a random point.     

Geometric programming problem with single-term exponents subject to max-product fuzzy relational equations

In this paper, an optimization model for minimizing an objective function with single-term exponents subject to fuzzy relational equations specified in max-product composition is presented. The solution set of such a fuzzy relational equation is a non-convex set. First, we present some properties for the optimization problem under the assumptions of both negative and nonnegative exponents in the objective function. Second, an efficient procedure is developed to find an optimal solution without looking for all the potential minimal solutions and without using the value matrix. An example is provided to illustrate the procedure.     

Algorithms for the solution of systems of coupled second-order ordinary differential equations

Several step-by-step methods for the computer solution systems of coupled second-order ordinary differential equations, are examined from the point of view of efficiency “time-wise” and “storage-wise”. Particular reference is made to a system arising in the close-coupling approximation of the Schroedinger equation. The stability of the solution is also considered.     

Spectral problem of the radial Schrödinger equation with confining power potentials

We suggest an approach in which the Schrödinger equation for several widely used potentials is reduced to the eigenvalue problem for an infinite system of algebraic equations. The method is convenient for both analytical and numerical calculations. With the help of this approach, the mass spectra of “charmonium” and “bottomonium” are calculated for the “Cornell” potential, and for the sum of the Coulomb and oscillator potentials. The method proposed allows one to determine the mass spectra of relativistic Schrödinger-type equations. Good agreement with experimental data is achieved.     

The Structure of Internal Layers for Unstable Nonlinear Diffusion Equations

We study the structure of diffusive layers in solutions of unstable nonlinear diffusion equations. These equations are regularizations of the forward-backward heat equation and have diffusion coefficients that become negative. Such models include the Cahn-Hilliard equation and the pseudoparabolic viscous diffusion equation. Using singular perturbation methods we show that the balance between diffusion and higher-order regularization terms uniquely determines the interface structure in these equations. It is shown that the well-known “equal area” rule for the Cahn-Hilliard equation is a special case of a more general rule for shock construction in the viscous Cahn-Hilliard equation.     

Exact solutions of one-dimensional nonlinear shallow water equations over even and sloping bottoms

We establish an equivalence of two systems of equations of one-dimensional shallow water models describing the propagation of surface waves over even and sloping bottoms. For each of these systems, we obtain formulas for the general form of their nondegenerate solutions, which are expressible in terms of solutions of the Darboux equation. The invariant solutions of the Darboux equation that we find are simplest representatives of its essentially different exact solutions (those not related by invertible point transformations). They depend on 21 arbitrary real constants; after “proliferation” formulas derived by methods of group theory analysis are applied, they generate a 27-parameter family of essentially different exact solutions. Subsequently using the derived infinitesimal “proliferation” formulas for the solutions in this family generates a denumerable set of exact solutions, whose linear span constitutes an infinite-dimensional vector space of solutions of the Darboux equation. This vector space of solutions of the Darboux equation and the general formulas for nondegenerate solutions of systems of shallow water equations with even and sloping bottoms give an infinite set of their solutions. The “proliferation” formulas for these systems determine their additional nondegenerate solutions. We also find all degenerate solutions of these systems and thus construct a database of an infinite set of exact solutions of systems of equations of the one-dimensional nonlinear shallow water model with even and sloping bottoms.     

Diffraction of water waves due to a presence of two headlands

This paper is concerned with a boundary value problem for the Helmholtz equation on a horizontal infinite strip with obstacles. The derivation of Helmholtz equation from shallow water equations is given and the boundary value problem with an arbitrary shap of headland is stated. The boundary conditions are of the general Neumann type, and thus we use the finite difference method in numerical solution. Helholtz equation is replaced by the five-points formula and for the points close to the boundary, Taylors expansions are made useful with non-uniform spacing. For solving the resulting system of linear equations, the “Mathematica” package is used. The graphs show the velocity potential contours in the cases, of semielliptic, semicircular and narrow headland. Also, we discuss the problem in the presence of two headlands.     

On the dynamic programming principle for uniformly nondegenerate stochastic differential games in domains and the Isaacs equations

We prove the dynamic programming principle for uniformly nondegenerate stochastic differential games in the framework of time-homogeneous diffusion processes considered up to the first exit time from a domain. In contrast with previous results established for constant stopping times we allow arbitrary stopping times and randomized ones as well. There is no assumption about solvability of the the Isaacs equation in any sense (classical or viscosity). The zeroth-order “coefficient” and the “free” term are only assumed to be measurable in the space variable. We also prove that value functions are uniquely determined by the functions defining the corresponding Isaacs equations and thus stochastic games with the same Isaacs equation have the same value functions.