On quasiperiodic solutions of semilinear systems of differential equations

We prove existence theorems for analytic quasi-periodic solutions for analytic systems of differential equations in a Banach space by the method of accelerated convergence. The results obtained are new even in the finite-dimensional case. Translated fromDinamicheskie Sistemy, Vol. 11, 1992.     

Analytic expansion of periodic solutions of integro-differential systems

We constructively obtain conditions for the existence of periodic solutions of systems of integro-differential equations in the degenerate case. By using integro-differential equations with a parameter, we construct an analytic representation of the solutions and obtain estimates for the convergence rate of iterative processes and for the exact solutions.     

On the computational efficiency index and some iterative methods for solving systems of nonlinear equations

In this paper two new iterative methods are built up and analyzed. A generalization of the efficiency index used in the scalar case to several variables in iterative methods for solving systems of nonlinear equations is revisited. Analytic proofs of the local order of convergence based on developments of multilineal functions and numerical concepts that will be used to illustrate the analytic results are given. An approximation of the computational order of convergence is computed independently of the knowledge of the root and the necessary time to get one correct decimal is studied in our examples.     

On the computational efficiency index and some iterative methods for solving systems of nonlinear equations

In this paper two new iterative methods are built up and analyzed. A generalization of the efficiency index used in the scalar case to several variables in iterative methods for solving systems of nonlinear equations is revisited. Analytic proofs of the local order of convergence based on developments of multilineal functions and numerical concepts that will be used to illustrate the analytic results are given. An approximation of the computational order of convergence is computed independently of the knowledge of the root and the necessary time to get one correct decimal is studied in our examples.     

Discrete least-squares global approximations to solutions of partial differential equations

Solutions of partial differential equations (PDEs) using globally nonvanishing approximating functions are discussed, and the particular case of global polynomial solutions is studied. Convergence and error bounds are examined. Examples are given and compared with analytic solutions. This method seems particularly well suited for elliptic PDEs with continuous boundary conditions and nonhomogeneous terms, even for irregular domains, offering geometric convergence rates. By providing the minimized residues, strong error indicators are obtained. This algorithm's implementation retains simplicity under a variety of applications.     

Convergence of solutions of von Karman evolution equations to equilibria

We consider von Karman evolution equations with nonlinear interior dissipation and with clamped boundary conditions. Under some conditions we prove that every energy solution converges to a stationary solution and establish a rate of convergence. Earlier this result was known in the case when the set of equilibria was finite and hyperbolic. In our argument we use the fact that the von Karman nonlinearity is analytic on an appropriate space and apply the Lojasiewicz–Simon method in the form suggested by A. Haraux and M. Jendoubi.     

Solution formulas for linear difference equations with applications to continued fractions

For any system of linear difference equations of arbitrary order, a family of solution formulas is constructed explicitly by way of relating the given system to simpler neighboring systems. These formulas are then used to investigate the asymptotic behavior of the solutions. When applying this difference equation method to second-order equations that belong to neighboring continued fractions, new results concerning convergence of continued fractions as well as meromorphic extension of analytic continued fractions beyond their convergence region are provided. This is demonstrated for analytic continued fractions whose elements tend to infinity. Finally, a recent result on the existence of limits of solutions to real difference equations having infinite order is extended to complex equations.     

On the rate of convergence of methods of projection-iterative type for fredholm equations with periodic analytic kernels

Optimal rates of convergence of projection-iterative methods and methods of Sokolov type are found for a certain class of Fredholm equations with analytic kernels that appear within the framework of the method of boundary integral equations.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 9, pp. 1155–1161, September, 1995.     

Optimal methods for specifying information in the solution of integral equations with analytic kernels

We determine the exact order of the minimum radius of information in the logarithmic scale for Fredholm integral equations of the second kind with periodic analytic kernels and free terms. We show that the information complexity of the solution of Fredholm equations with analytic kernels is greater in order than the complexity of the approximation of analytic functions. This distinguishes the analytic case from the case of finite smoothness.     

The th moment stability for the stochastic pantograph equation

In this paper, we investigate the αth moment asymptotical stability of the analytic solution and the numerical methods for the stochastic pantograph equation by using the Razumikhin technique. Especially the linear stochastic pantograph equations and the semi-implicit Euler method applying them are considered. The convergence result of the semi-implicit Euler method is obtained. The stability conditions of the analytic solution of those equations and the numerical method are given. Finally, some experiments are given.     

数值求解对流扩散方程有限分析方法的稳定性与收敛性

本文利用椭圆型偏微分方程所满足的最大最小值原理研究有限分析方法,证明了数值求解对流扩散方程有限分析方法的稳定性与收敛性,顺便指出了前人理论中的错误.     

Faedo–Galerkin Approximation of Solution for a Nonlocal Neutral Fractional Differential Equation with Deviating Argument

In this work, we study a class of nonlocal neutral fractional differential equations with deviated argument in the separable Hilbert space. We obtain an associated integral equation and then, consider a sequence of approximate integral equations. We investigate the existence and uniqueness of the mild solution for every approximate integral equation by virtue of the theory of analytic semigroup theory via the technique of Banach fixed point theorem. Next we demonstrate the convergence of the solutions of the approximate integral equations to the solution of the associated integral equation. The Faedo–Galerkin approximation of the solution is studied and demonstrated some convergence results. Finally, we give an example.

     

Convergence criterion of Newton's method for singular systems with constant rank derivatives

The present paper is concerned with the convergence problem of Newton's method to solve singular systems of equations with constant rank derivatives. Under the hypothesis that the derivatives satisfy a type of weak Lipschitz condition, a convergence criterion based on the information around the initial point is established for Newton's method for singular systems of equations with constant rank derivatives. Applications to two special and important cases: the classical Lipschitz condition and the Smale's assumption, are provided; the latter, in particular, extends and improves the corresponding result due to Dedieu and Kim in [J.P. Dedieu, M. Kim, Newton's method for analytic systems of equations with constant rank derivatives, J. Complexity 18 (2002) 187-209].     

Convergence of numerical solutions to stochastic age-dependent population equations with Markovian switching

In this paper, a class of stochastic age-dependent population equations with Markovian switching is considered. The main aim of this paper is to investigate the convergence of the numerical approximation of stochastic age-dependent population equations with Markovian switching. It is proved that the numerical approximation solutions converge to the analytic solutions of the equations under the given conditions. An example is given for illustration.     

Convergence of numerical solutions to stochastic pantograph equations with Markovian switching

In this paper, a class of stochastic pantograph equations with Markovian switching is considered. The main purpose is to investigate the convergence of the Euler method of the equations. It is proved that the Euler approximation solution converge to the analytic solution in probability under weaker conditions. An example is provided to illustrate our theory.     

Construction and analytic properties of integral manifolds of solutions of nonlinear systems of difference equations

We study properties of integral manifolds of a system of difference equations in the hyperbolic case. We prove the existence of analytic integral manifolds for a system of difference equations with analytic right sides. We examine an analytic dependence on the parameter.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 5, pp. 630–636, May, 1992.     

Newton's Method for Underdetermined Systems of Equations Under the γ-Condition

The convergence criterion of Newton's method for underdetermined system of equations under the γ-condition is established and the radius of the convergence ball is obtained. Applications to analytic operator are provided and some results due to Shub and Smale (SIAM J. Numer. Anal. 1996, 33:128–148) are extended and improved.     

A Padé-based algorithm for overcoming the Gibbs phenomenon

Truncated Fourier series and trigonometric interpolants converge slowly for functions with jumps in value or derivatives. The standard Fourier–Padé approximation, which is known to improve on the convergence of partial summation in the case of periodic, globally analytic functions, is here extended to functions with jumps. The resulting methods (given either expansion coefficients or function values) exhibit exponential convergence globally for piecewise analytic functions when the jump location(s) are known. Implementation requires just the solution of a linear system, as in standard Padé approximation. The new methods compare favorably in experiments with existing techniques.     

Analytic approximate solution of three‐dimensional Navier‐Stokes equations of flow between two stretchable disks

The similarity transform for the steady three‐dimensional Navier‐Stokes equations of flow between two stretchable disks gives a system of nonlinear ordinary differential equations which is analytically solved by applying a newly developed method, namely, the homotopy analysis method. The analytic solutions of the system of nonlinear ordinary differential equations are constructed in the series form. The convergence of the obtained series solutions is analyzed. The validity of our solutions is verified by the numerical results. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

对流扩散方程有限混合分析格式

介绍了对流扩散方程的混合有限分析法 ,得出了求解对流扩散方程隐式格式、离散算子 ,并且证明了这些格式解的存在性 ,分析了格式的截断误差