A Taylor method for numerical solution of generalized pantograph equations with linear functional argument

This paper is concerned with a generalization of a functional differential equation known as the pantograph equation which contains a linear functional argument. In this paper, we introduce a numerical method based on the Taylor polynomials for the approximate solution of the pantograph equation with retarded case or advanced case. The method is illustrated by studying the initial value problems. The results obtained are compared by the known results.     

Holomorphic solutions to functional differential equations

The pantograph equation is perhaps one of the most heavily studied class of functional differential equations owing to its numerous applications in mathematical physics, biology, and problems arising in industry. This equation is characterized by a linear functional argument. Heard (1973) [10] considered a generalization of this equation that included a nonlinear functional argument. His work focussed on the asymptotic behaviour of solutions for a real variable x as x→∞. In this paper, we revisit Heard's equation, but study it in the complex plane. Using results from complex dynamics we show that any nonconstant solution that is holomorphic at the origin must have the unit circle as a natural boundary. We consider solutions that are holomorphic on the Julia set of the nonlinear argument. We show that the solutions are either constant or have a singularity at the origin. There is a special case of Heard's equation that includes only the derivative and the functional term. For this case we construct solutions to the equation and illustrate the general results using classical complex analysis.     

A natural boundary for solutions to the second order pantograph equation

Pantograph equations are characterized by the presence of a linear functional argument. These equations arise in several applications and often the argument has a repelling fixed point at the origin. Recently, Marshall et al. [J. Math. Anal. Appl. 268 (2002) 157-170] studied a related class of functional differential equations with nonlinear functional arguments and showed that, generically, solutions to such equations have a natural boundary. Their approach uses some well-known properties of the Julia set and relies heavily on the nonlinearity of the functional argument. The method is not directly applicable to pantograph type equations though some of the techniques can be exploited. In this paper we show that solutions to pantograph equations generally have natural boundaries. We focus on a special set of solutions that have the imaginary axis as a natural boundary.     

A new Jacobi rational–Gauss collocation method for numerical solution of generalized pantograph equations

This paper is concerned with a generalization of a functional differential equation known as the pantograph equation which contains a linear functional argument. In this article, a new spectral collocation method is applied to solve the generalized pantograph equation with variable coefficients on a semi-infinite domain. This method is based on Jacobi rational functions and Gauss quadrature integration. The Jacobi rational-Gauss method reduces solving the generalized pantograph equation to a system of algebraic equations. Reasonable numerical results are obtained by selecting few Jacobi rational–Gauss collocation points. The proposed Jacobi rational–Gauss method is favorably compared with other methods. Numerical results demonstrate its accuracy, efficiency, and versatility on the half-line.     

A Bessel collocation method for numerical solution of generalized pantograph equations

This article is concerned with a generalization of a functional differential equation known as the pantograph equation which contains a linear functional argument. In this article, we introduce a collocation method based on the Bessel polynomials for the approximate solution of the pantograph equations. The method is illustrated by studying the initial value problems. The results obtained are compared by the known results. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Holomorphic solutions to pantograph type equations with neutral fixed points

Pantograph type equations have been studied extensively owing to the numerous applications in which these equations arise. These studies focused primarily on the case when the functional argument is linear, and the origin is either a repelling or attracting fixed point. The nonlinear case has been studied by Oberg [Trans. Amer. Math. Soc. 161 (1971) 302-327] and Marshall et al. [J. Math. Anal. Appl. 268 (2002) 157-170], but the focus again was on repelling or attracting fixed points. Oberg (op. cit.), however, did consider briefly the neutral fixed point case and found a connexion with Siegel discs. In this paper we build on Oberg's work and study the neutral fixed point case. We show that, for nonlinear functional arguments with neutral fixed points, pantograph type equations have nonconstant holomorphic solutions only if the functional argument has a Siegel disc centered at the fixed point. We then show that the boundary of the Siegel disc forms a natural boundary for the nonconstant holomorphic solutions.     

Local Controllability of Functional Integrodifferential Equations in Banach Space

In this paper, we establish a set of sufficient conditions for the local controllability of functional integrodifferential equations in Banach space. The results are obtained by using the methods of analytic semigroups, fractional powers of operators, and a fixed-point argument. These results generalize previous results on bounded linear operators to unbounded linear operators in which the equation involves a nonlinear delay term. An application to a partial integrodifferential equation is given.     

Well-posedness of fully nonlinear and nonlocal critical parabolic equations

In this paper, we prove the existence of smooth solutions in Sobolev spaces to fully nonlinear and nonlocal parabolic equations with critical index. Our argument is to transform the fully nonlinear equation into a quasi-linear nonlocal parabolic equation.     

Asymptotic behavior of solutions to a nonlinear Stefan problem with different moving parameters

This paper deals with a nonlinear diffusion equation with double free boundaries possessing different moving parameters. We present the spreading–vanishing dichotomy and threshold between spreading and vanishing. Moreover, when spreading happens, using the zero number argument we provide sharp estimates of spreading speeds of expanding fronts, and describe how the solution approaches the semi-wave.     

Stabilization for a Fourth Order Nonlinear Schrödinger Equation in Domains with Moving Boundaries

In the present paper we study the well-posedness using the Galerkin method and the stabilization considering multiplier techniques for a fourth-order nonlinear Schrödinger equation in domains with moving boundaries. We consider two situations for the stabilization: the conservative case and the dissipative case.     

EXPONENTIAL STABILITY FOR NONLINEAR HYBRID STOCHASTIC PANTOGRAPH EQUATIONS AND NUMERICAL APPROXIMATION

The paper develops exponential stability of the analytic solution and convergence in probability of the numerical method for highly nonlinear hybrid stochastic pantograph equation. The classical linear growth condition is replaced by polynomial growth conditions, under which there exists a unique global solution and the solution is almost surely exponentially stable. On the basis of a series of lemmas, the paper establishes a new criterion on convergence in probability of the Euler-Maruyama approximate solution. The criterion is very general so that many highly nonlinear stochastic pantograph equations can obey these conditions. A highly nonlinear example is provided to illustrate the main theory.     

Pseudo Almost Periodic Solutions of Nonlinear Hyperbolic Equations with Piecewise Constant Argument

In this paper, we investigate the pseudo almost periodicity of the unique bounded solution for a nonlinear hyperbolic equation with piecewise constant argument. The equation under consideration is a mathematical model for the dynamics of gas absorption,     

带逐段常变量的非线性双曲微分方程的伪概周期解

In this paper,we investigate the pseudo almost periodicity of the unique bounded solution for a nonlinear hyperbolic equation with piecewise constant argument.The equation under consideration is a mathematical model for the dynamics of gas absorption.     

Stochastic Evolution Equations Driven by a Fractional White Noise

Abstract

In this paper we study stochastic evolution equations driven by a fractional white noise with arbitrary Hurst parameter in infinite dimension. We establish the existence and uniqueness of a mild solution for a nonlinear equation with multiplicative noise under Lipschitz condition by using a fixed point argument in an appropriate inductive limit space. In the linear case with additive noise, a strong solution is obtained. Those results are applied to stochastic parabolic partial differential equations perturbed by a fractional white noise.     

带不定权非线性边界的p-Laplacian问题解的存在性

研究了一类带不定权非线性边界的p-Laplacian椭圆方程.获得了当非线性边界的特征值参数小于第二特征值时,该方程存在非平凡解.主要工具为环绕定理.     

Nagumo type existence results for second-order nonlinear dynamic BVPS

In this paper, we present a Nagumo condition for time scales equations that includes as a particular case a type of the classical one for differential equations involving a certain restriction. The established Nagumo condition further allows us to prove the existence of at least one solution lying between a pair of lower and upper solutions of a nonlinear second-order dynamic equation with nonlinear functional boundary value conditions.     

Topological linearization of DEPCAGs with unbounded nonlinear terms

In this paper, we study the global topological linearization of a differential equation with piecewise constant argument of generalized type (DEPCAG) when the nonlinear term is unbounded. Some sufficient conditions are established for the topological conjugacy between a nonlinear system and its linear system. Our work generalizes the main result of Pinto and Robledo in [25].     

Generalized Jacobi spectral Galerkin method for fractional pantograph differential equation

This work is concerned with the extension of the Jacobi spectral Galerkin method to a class of nonlinear fractional pantograph differential equations. First, the fractional differential equation is converted to a nonlinear Volterra integral equation with weakly singular kernel. Second, we analyze the existence and uniqueness of solutions for the obtained integral equation. Then, the Galerkin method is used for solving the equivalent integral equation. The error estimates for the proposed method are also investigated. Finally, illustrative examples are presented to confirm our theoretical analysis.     

中立型比例延迟微分系统线性θ-方法的渐近估计

本文研究了一类线性非自治中立型比例延迟微分系统线性θ-方法的渐近稳定性,并借助于泛函不等式得到了数值解的渐近估计.此渐近估计不仅比数值渐近稳定性描述得更加精确,而且还能给出非稳定情形数值解的上界估计式.数值算例验证了相关理论结果.     

Operational matrices of Chebyshev cardinal functions and their application for solving delay differential equations arising in electrodynamics with error estimation

In this paper, a new and effective direct method to determine the numerical solution of pantograph equation, pantograph equation with neutral term and Multiple-delay Volterra integral equation with large domain is proposed. The pantograph equation is a delay differential equation which arises in quite different fields of pure and applied mathematics, such as number theory, dynamical systems, probability, mechanics and electrodynamics. The method consists of expanding the required approximate solution as the elements of Chebyshev cardinal functions. The operational matrices for the integration, product and delay of the Chebyshev cardinal functions are presented. A general procedure for forming these matrices is given. These matrices play an important role in modelling of problems. By using these operational matrices together, a pantograph equation can be transformed to a system of algebraic equations. An efficient error estimation for the Chebyshev cardinal method is also introduced. Some examples are given to demonstrate the validity and applicability of the method and a comparison is made with existing results.