模糊运算和模糊有限元静力控制方程的求解

根据模糊数的区间形式表达和区间运算的性质,给出了模糊数和模糊变量的运算规则.据此并依据区间有限元理论,提出了结构模糊有限元静力控制方程的几种求解方法.方法可根据输入模糊数的隶属函数,给出结构响应量的可能性分布.且计算量小,易于实施.算例分析说明了方法是实用和可行的.     

Exact computer realization of certain computational algorithms

The question of organizing calculations in supermany-valued arithmetic on a computer is examined. A program is presented for a model computer based on a computer of type M-20, permitting the exact solution of a system of a system of linear algebraic equations to be obtained by the Gauss-Jordan method.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 58, pp. 116–121, 1976.     

Parametric continuation of the solitary traveling pulse solution in the reaction-diffusion system using the Newton-Krylov method

The matrix-free Newton-Krylov method that uses the GMRES algorithm (an iterative algorithm for solving systems of linear algebraic equations) is used for the parametric continuation of the solitary traveling pulse solution in a three-component reaction-diffusion system. Using the results of integration on a short time interval, we replace the original system of nonlinear algebraic equations by another system that has more convenient (from the viewpoint of the spectral properties of the GMRES algorithm) Jacobi matrix. The proposed parametric continuation proved to be efficient for large-scale problems, and it made it possible to thoroughly examine the dependence of localized solutions on a parameter of the model.     

Real intersection points of piecewise algebraic curves

The piecewise algebraic curve, as the set of zeros of a bivariate spline function, is a generalization of the classical algebraic curve. In this work, we present an algorithm for computing the real intersection points of piecewise algebraic curves. It is primarily based on the interval zeros of the univariate interval polynomial in Bernstein form. An illustrative example is provided to show that the proposed algorithm is flexible.     

Interval Newton Iteration in Multiple Precision for the Univariate Case

In this paper, interval arithmetic using an underlying multiple precision arithmetic is briefly presented. Then interval Newton iteration for solving nonlinear equations is introduced. A new Newton's algorithm based on multiple precision interval arithmetic is given, along with its properties: termination, arbitrary accuracy on the computed zeros, automatic and dynamic adaptation of the precision. Finally, some experiments illustrate the behaviour of this method.     

区间运算和静力区间有限元

用均值和离差两参数表征区间变量的不确定性,根据区间运算规则,论证了区间变量的运算特性.将区间分析和有限元方法相结合,提出了非概率不确定结构的一种区间有限元分析方法.将区间有限元静力控制方程中n自由度不确定位移场特征参数的求解归结为求解一2n阶线性方程组.实例分析表明文中方法是有效和可行的.     

The Sylvester-Ramanujan System of Equations and The Complex Power Moment Problem

A classical system of algebraic equations is treated as a finite power moment problem in C and investigated on this base. Being originated from the algebraic theory of binary forms, this system is closely related to an extraordinary number of different subjects in the classical and modern analysis. A survey of these relations is presented.     

Application of the modified operational matrices in multiterm variable‐order time‐fractional partial differential equations

In this paper, we present a novel discrete scheme based on Genocchi polynomials and fractional Laguerre functions to solve multiterm variable‐order time‐fractional partial differential equations (M‐V‐TFPDEs) in the large interval. In this purpose, the accurate modified operational matrices are constructed to reduce the problems into a system of algebraic equations. Also, the computational algorithm based on the method and modified operational matrices in the large interval is easily implemented. Furthermore, we discuss the error estimation of the proposed method. Ultimately, to confirm our theoretical analysis and accuracy of numerical approach, several examples are presented.     

Applications of interval arithmetic in solving polynomial equations by Wu''''s elimination method

Wu's elimination method is an important method for solving multivariate poly- nomial equations.In this paper,we apply interval arithmetic to Wu's method and convert the problem of solving polynomial equations into that of solving interval polynomial equa- tions.Parallel results such as zero-decomposition theorem are obtained for interval poly- nomial equations.The advantages of the new approach are two-folds:First,the problem of the numerical instability arisen from floating-point arithmetic is largely overcome.Second, the low efficiency of the algorithm caused by large intermediate coefficients introduced by exact compaction is dramatically improved.Some examples are provided to illustrate the effectiveness of the proposed algorithm.     

Legendre polynomial solutions of high-order linear Fredholm integro-differential equations

In this study, a Legendre collocation matrix method is presented to solve high-order Linear Fredholm integro-differential equations under the mixed conditions in terms of Legendre polynomials. The proposed method converts the equation and conditions to matrix equations, by means of collocation points on the interval [−1, 1], which corresponding to systems of linear algebraic equations with Legendre coefficients. Thus, by solving the matrix equation, Legendre coefficients and polynomial approach are obtained. Also examples that illustrate the pertinent features of the method are presented and by using the error analysis, the results are discussed.     

A method of solving a nonlinear boundary value problem with a parameter for a loaded differential equation

A nonlinear loaded differential equation with a parameter on a finite interval is studied. The interval is partitioned by the load points, at which the values of the solution to the equation are set as additional parameters. A nonlinear boundary value problem for the considered equation is reduced to a nonlinear multipoint boundary value problem for the system of nonlinear ordinary differential equations with parameters. For fixed parameters, we obtain the Cauchy problems for ordinary differential equations on the subintervals. Substituting the values of the solutions to these problems into the boundary condition and continuity conditions at the partition points, we compose a system of nonlinear algebraic equations in parameters. A method of solving the boundary value problem with a parameter is proposed. The method is based on finding the solution to the system of nonlinear algebraic equations composed.     

Applicability of the Interval Taylor Model to the Computational Proof of Existence of Periodic Trajectories in Systems of Ordinary Differential Equations

We consider the construction of the interval Taylor model used to prove the existence of periodic trajectories in systems of ordinary differential equations. Our model differs from the ones available in the literature in the method for describing the algorithms for the computation of arithmetic operations over Taylor models. In the framework of the current model, this permits reducing the computational expenditures for obtaining interval estimates on computers. We prove an assertion that permits establishing the existence of a periodic solution of a system of ordinary differential equations by verifying the convergence of the Picard iterations in the sense of embedding of the proposed Taylor models. An example illustrating how the resulting assertion can be used to prove the existence of a closed trajectory in the van der Pol system is given.     

Arithmetic of plane Cremona transformations and the dimensions of transfinite heterotic string space-time

It is shown that the two sequences of characteristic dimensions of transfinite heterotic string space-time found by El Naschie can be remarkably well accounted for in terms of the arithmetic of self-conjugate homaloidal nets of plane algebraic curves of orders 3–20. A firm algebraic geometrical justification is thus given not only for all the relevant dimensions of the classical theory, but also for the other two dimensions proposed by El Naschie, viz. the inverse of the quantum gravity coupling constant (≃42.36067977) and that of (one half of) the fine structure constant (≃68.54101967). A non-trivial coupling between the two El Naschie sequences is also revealed.     

Extensions of ValEncIA-IVP for reduction of overestimation,for simulation of differential algebraic systems,and for dynamical optimization

Simulation techniques are commonly used to analyze the influence of uncertainties of initial conditions and systemparameters on the trajectories of the state variables of dynamical systems. In this context, interval arithmetic approaches are of interest. They are capable of determining guaranteed bounds of all reachable states if worst-case bounds of the above-mentioned uncertainties are known. Furthermore, interval algorithms ensure the correctness of numerical results in spite of rounding errors which inevitably arise if floating point operations are carried out on a computer. However, naive implementations of interval algorithms often lead to overestimation, i.e., too conservative enclosures which can make the results meaningless. In this contribution, we summarize the basic routines of ValEncIA-IVP which computes interval enclosures of all reachable states of dynamical systems described by ordinary differential equations ODEs. ValEncIA-IVP , VAL idation of state ENC losures using I nterval A rithmetic for I nitial V alue P roblems, can be applied to the simulation of systems with both uncertain parameters and uncertain initial conditions. Advanced techniques for reduction of overestimation are demonstrated for a simplified catalytic reactor. Afirst approach to using VanEncIA-IVP for the simulation of sets of differential algebraic equations is outlined. Finally, an outlook on the integration of ValEncIA-IVP in an interval arithmetic framework for computation of optimal and robust control strategies for continuous-time processes is given. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A network model for incompressible two-fluid flow and its numerical solution

In this paper we consider an incompressible version of the two-fluid network model proposed by Porsching (Nu. Methods Part. Diff. Eq., 1 , 295–313 [1985]). The system of equations governing the model is a mixed system of differential and algebraic equations (DAEs). These DAEs are then recast, through proper transformation, into a system of ordinary differential equations on a submanifold of ?ⁿ, for which uniqueness, existence, and stability theorems are proved. Numerical simulations are presented.     

Criteria for the Trivial Solution of Differential Algebraic Equations with Small Nonlinearities to be Asymptotically Stable

Differential algebraic equations consisting of a constant coefficient linear part and a small nonlinearity are considered. Conditions that enable linearizations to work well are discussed. In particular, for index-2 differential algebraic equations, there results a kind of Perron Theorem that sounds as clear as its classical model.     

On an iterative method for simultaneous inclusion of polynomial complex zeros

Starting from disjoint discs which contain polynomial complex zeros, the iterative interval method of the third order for the simultaneous finding inclusive discs for complex zeros is formulated. The Lagrangean interpolation formula and complex circular arithmetic are used. The convergence theorem and the conditions for convergence are considered. The proposed method has been applied for solving an algebraic equation.     

Arithmetic partial differential equations, I

We develop an arithmetic analogue of linear partial differential equations in two independent “space-time” variables. The spatial derivative is a Fermat quotient operator, while the time derivative is the usual derivation. This allows us to “flow” integers or, more generally, points on algebraic groups with coordinates in rings with arithmetic flavor. In particular, we show that elliptic curves carry certain canonical “arithmetic flows” that are arithmetic analogues of the convection, heat, and wave equations, respectively. The same is true for the additive and the multiplicative group.     

Genetic algorithm for asymmetric traveling salesman problem with imprecise travel times

This paper presents a variant of the asymmetric traveling salesman problem (ATSP) in which the traveling time between each pair of cities is represented by an interval of values (wherein the actual travel time is expected to lie) instead of a fixed (deterministic) value as in the classical ATSP. Here the ATSP (with interval objective) is formulated using the usual interval arithmetic. To solve the interval ATSP (I-ATSP), a genetic algorithm with interval valued fitness function is proposed. For this purpose, the existing revised definition of order relations between interval numbers for the case of pessimistic decision making is used. The proposed algorithm is based on a previously published work and includes some new features of the basic genetic operators. To analyze the performance and effectiveness of the proposed algorithm and different genetic operators, computational studies of the proposed algorithm on some randomly generated test problems are reported.     

On the finiteness theorem of Siegel and Mahler concerning diophantine equations

In this paper we present a new proof, involving so-called nonstandard arguments, of Siegel's classical theorem on diophantine equations: Any irreducible algebraic equation f(x,y) = 0 of genus g > 0 admits only finitely many integral solutions. We also include Mahler's generalization of this theorem, namely the following: Instead of solutions in integers, we are considering solutions in rationals, but with the provision that their denominators should be divisible only by such primes which belong to a given finite set. Then again, the above equation admits only finitely many such solutions. From general nonstandard theory, we need the definition and the existence of enlargements of an algebraic number field. The idea of proof is to compare the natural arithmetic in such an enlargement, with the functional arithmetic in the function field defined by the above equation.