Higher order numerical approximation for time dependent singularly perturbed differential‐difference convection‐diffusion equations

In this article, we develop a higher order numerical approximation for time dependent singularly perturbed differential‐difference convection‐diffusion equations. A priori bounds on the exact solution and its derivatives, which are useful for the error analysis of the numerical method are given. We approximate the retarded terms of the model problem using Taylor's series expansion and the resulting time‐dependent singularly perturbed problem is discretized by the implicit Euler scheme on uniform mesh in time direction and a special hybrid finite difference scheme on piecewise uniform Shishkin mesh in spatial direction. We first prove that the proposed numerical discretization is uniformly convergent of , where and denote the time step and number of mesh‐intervals in space, respectively. After that we design a Richardson extrapolation scheme to increase the order of convergence in time direction and then the new scheme is proved to be uniformly convergent of . Some numerical tests are performed to illustrate the high‐order accuracy and parameter uniform convergence obtained with the proposed numerical methods.     

Admissible meromorphic solutions of algebraic differential‐difference equations

By using Nevanlinna theory, we generalize a result given by Wittich to complex differential‐difference equations. The result obtained is that the differential‐difference equation in f which is of only one dominant term, has no admissible meromorphic solution f with hyper‐order less than 1 provided N(r,f) = S(r,f).     

On existence of solutions of differential‐difference equations

This paper applies Nevanlinna theory of value distribution to discuss existence of solutions of certain types of non‐linear differential‐difference equations such as (5) and (8) given in the succeeding paragraphs. Existence of solutions of differential equations and difference equations can be said to have been well studied, that of differential‐difference equations, on the other hand, have been paid little attention. Such mixed type equations have great significance in applications. This paper, in particular, generalizes the Rellich–Wittich‐type theorem and Malmquist‐type theorem about differential equations to the case of differential‐difference equations. Copyright © 2015 John Wiley & Sons, Ltd.     

A Taylor polynomial approach for solving the most general linear Fredholm integro‐differential‐difference equations

In this study, a matrix method is developed to solve approximately the most general higher order linear Fredholm integro‐differential‐difference equations with variable coefficients under the mixed conditions in terms of Taylor polynomials. This technique reduces the problem into the linear algebraic system. The method is valid for any combination of differential, difference and integral equations. An initial value problem and a boundary value problem are also presented to illustrate the accuracy and efficiency of the method. Copyright © 2012 John Wiley & Sons, Ltd.     

Complete flux scheme for parabolic singularly perturbed differential‐difference equations

In this study, we investigate the concept of the complete flux (CF) obtained as a solution to a local boundary value problem (BVP) for a given parabolic singularly perturbed differential‐difference equation (SPDDE) with modified source term to propose an efficient complete flux‐finite volume method (CF‐FVM) for parabolic SPDDE which is μ‐ and ?‐uniform method where μ, ? are shift and perturbation parameters, respectively. The proposed numerical method is shown to be consistent, stable, and convergent and has been successfully implemented on three test problems.     

Exact solutions of some systems of fractional differential‐difference equations

In this paper, the ‐expansion method is proposed to establish hyperbolic and trigonometric function solutions for fractional differential‐difference equations with the modified Riemann–Liouville derivative. The fractional complex transform is proposed to convert a fractional partial differential‐difference equation into its differential‐difference equation of integer order. We obtain the hyperbolic and periodic function solutions of the nonlinear time‐fractional Toda lattice equations and relativistic Toda lattice system. The proposed method is more effective and powerful for obtaining exact solutions for nonlinear fractional differential–difference equations and systems. Copyright © 2014 John Wiley & Sons, Ltd.     

Discrete Jacobi sub‐equation method for nonlinear differential–difference equations

We will propose a unified algebraic method to construct Jacobi elliptic function solutions to differential–difference equations (DDEs). The solutions to DDEs in terms of Jacobi elliptic functions sn, cn and dn have a unified form and can be presented through solving the associated algebraic equations. To illustrate the effectiveness of this method, we apply the algorithm to some physically significant DDEs, including the discrete hybrid equation, semi‐discrete coupled modified Korteweg–de Vries and the discrete Klein–Gordon equation, thereby generating some new exact travelling periodic solutions to the discrete Klein–Gordon equation. A procedure is also given to determine the polynomial expansion order of Jacobi elliptic function solutions to DDEs. Copyright © 2010 John Wiley & Sons, Ltd.     

Superconvergence analysis for time‐dependent Maxwell's equations in metamaterials

In this article, we consider the time‐dependent Maxwell's equations modeling wave propagation in metamaterials. One‐order higher global superclose results in the L² norm are proved for several semidiscrete and fully discrete schemes developed for solving this model using nonuniform cubic and rectangular edge elements. Furthermore, L^∞ superconvergence at element centers is proved for the lowest order rectangular edge element. To our best knowledge, such pointwise superconvergence result and its proof are original, and we are unaware of any other publications on this issue. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential 2011     

Vortex dynamics of the full time‐dependent Ginzburg‐Landau equations

In the Ginzburg‐Landau model for superconductivity a large Ginzburg‐Landau parameter κ corresponds to the formation of tight, stable vortices. These vortices are located exactly where an applied magnetic field pierces the superconducting bulk, and each vortex induces a quantized supercurrent about the vortex. The energy of large‐κ solutions blows up near each vortex which brings about difficulties in analysis. Rigorous asymptotic static theory has previously established the existence of a finite number of the vortices, and these vortices are located precisely at the critical points of the renormalized energy (the free energy less the vortex self‐induction energy). A rigorous study of the full time‐dependent Ginzburg‐Landau equations under the classical Lorentz gauge is done under the asymptotic limit κ → ∞. Under slow times the vortices remain pinned to their initial configuration. Under a fast time of order κ the vortices move according to a steepest descent of the renormalized energy. © 2002 John Wiley & Sons, Inc.     

A highly accurate Jacobi collocation algorithm for systems of high‐order linear differential–difference equations with mixed initial conditions

In this paper, a shifted Jacobi–Gauss collocation spectral algorithm is developed for solving numerically systems of high‐order linear retarded and advanced differential–difference equations with variable coefficients subject to mixed initial conditions. The spatial collocation approximation is based upon the use of shifted Jacobi–Gauss interpolation nodes as collocation nodes. The system of differential–difference equations is reduced to a system of algebraic equations in the unknown expansion coefficients of the sought‐for spectral approximations. The convergence is discussed graphically. The proposed method has an exponential convergence rate. The validity and effectiveness of the method are demonstrated by solving several numerical examples. Numerical examples are presented in the form of tables and graphs to make comparisons with the results obtained by other methods and with the exact solutions more easier. Copyright © 2015 John Wiley & Sons, Ltd.     

On contraction of nonlinear difference equations with time‐varying delays

General nonlinear difference equations with time‐varying delays are considered. Explicit criteria for contraction of such equations are presented. Then some simple sufficient conditions for global exponential stability of equilibria and for stability of invariant sets are derived. Furthermore, explicit criteria for existence, uniqueness and global exponential stability of periodic solutions are derived. Finally, the obtained results are applied to time‐varying discrete‐time neural networks with delay.     

Nonintrusive reduced‐order modeling of parametrized time‐dependent partial differential equations

We propose a nonintrusive reduced‐order modeling method based on the notion of space‐time‐parameter proper orthogonal decomposition (POD) for approximating the solution of nonlinear parametrized time‐dependent partial differential equations. A two‐level POD method is introduced for constructing spatial and temporal basis functions with special properties such that the reduced‐order model satisfies the boundary and initial conditions by construction. A radial basis function approximation method is used to estimate the undetermined coefficients in the reduced‐order model without resorting to Galerkin projection. This nonintrusive approach enables the application of our approach to general problems with complicated nonlinearity terms. Numerical studies are presented for the parametrized Burgers' equation and a parametrized convection‐reaction‐diffusion problem. We demonstrate that our approach leads to reduced‐order models that accurately capture the behavior of the field variables as a function of the spatial coordinates, the parameter vector and time. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

A defect correction method for the time‐dependent Navier‐Stokes equations

A method for solving the time dependent Navier‐Stokes equations, aiming at higher Reynolds' number, is presented. The direct numerical simulation of flows with high Reynolds' number is computationally expensive. The method presented is unconditionally stable, computationally cheap, and gives an accurate approximation to the quantities sought. In the defect step, the artificial viscosity parameter is added to the inverse Reynolds number as a stability factor, and the system is antidiffused in the correction step. Stability of the method is proven, and the error estimations for velocity and pressure are derived for the one‐ and two‐step defect‐correction methods. The spacial error is O(h) for the one‐step defect‐correction method, and O(h²) for the two‐step method, where h is the diameter of the mesh. The method is compared to an alternative approach, and both methods are applied to a singularly perturbed convection–diffusion problem. The numerical results are given, which demonstrate the advantage (stability, no oscillations) of the method presented. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Asymptotic behaviour of time‐dependent Ginzburg–Landau equations of superconductivity

In this paper, we establish the global fast dynamics for the time‐dependent Ginzburg–Landau equations of superconductivity. We show the squeezing property and the existence of finite‐dimensional exponential attractors for the system. In addition we prove the existence of the global attractor in L² × L² for the Ginzburg–Landau equations in two spatial dimensions. Copyright © 1999 John Wiley & Sons, Ltd.     

Convergence analysis of a coupled method for time‐dependent convection‐diffusion equations

The coupling of two locally mass conservative methods is formulated and analyzed for the time‐dependent convection‐diffusion problem. Finite volume method is used in some subdomains and interior penalty discontinuous Galerkin method is used in other subdomains. Numerical examples show the advantages of the proposed hybrid method, namely an accurate approximation obtained at a reduced computational cost. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 133–157, 2014     

Finite element study of time‐dependent Maxwell's equations in dispersive media

In this article, we consider the time‐dependent Maxwell's equations in a bounded domain when dispersive media are involved. The Crank‐Nicolson scheme is developed to approximate the electric field equation by Nedelec edge elements and is proved to be optimal convergent in energy norm. The analysis is carried out for Debye medium, but the same results hold true for other dispersive media such as plasma and Lorentz medium. Furthermore, our analysis extends straightforward to cases when a dispersive medium and a simple medium (such as air) are coupled. Mathematics Subject Classification (2000): 65N30, 35L15, 78‐08. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Strang‐type preconditioners applied to ordinary and neutral differential‐algebraic equations

This paper deals with boundary‐value methods (BVMs) for ordinary and neutral differential‐algebraic equations. Different from what has been done in Lei and Jin (Lecture Notes in Computer Science, vol. 1988. Springer: Berlin, 2001; 505–512), here, we directly use BVMs to discretize the equations. The discretization will lead to a nonsymmetric large‐sparse linear system, which can be solved by the GMRES method. In order to accelerate the convergence rate of GMRES method, two Strang‐type block‐circulant preconditioners are suggested: one is for ordinary differential‐algebraic equations (ODAEs), and the other is for neutral differential‐algebraic equations (NDAEs). Under some suitable conditions, it is shown that the preconditioners are invertible, the spectra of the preconditioned systems are clustered, and the solution of iteration converges very rapidly. The numerical experiments further illustrate the effectiveness of the methods. Copyright © 2011 John Wiley & Sons, Ltd.     

Regularization of linear difference equations with analytic coefficients and their applications

We propose a regularization method for four-element linear difference equations with analytic coefficients. We study these equations in the class of functions which are holomorphic in the complex plane with a cruciform cut and vanish at infinity. We give several examples illustrating the dependence of the solvability properties of equations on the choice of periodic coefficients. We describe various applications.     

Semi‐linear wave models with power non‐linearity and scale‐invariant time‐dependent mass and dissipation

In this paper we will consider the semi‐linear Cauchy problem for wave models with scale‐invariant time‐dependent mass and dissipation and power non‐linearity. The goal is to study the interplay between the coefficients of the mass and the dissipation term to prove blow‐up results or global existence (in time) of small data energy solutions.