Approximation properties for solutions to non‐Lipschitz stochastic differential equations with Lévy noise

In this paper, we consider the non‐Lipschitz stochastic differential equations and stochastic functional differential equations with delays driven by Lévy noise, and the approximation theorems for the solutions to these two kinds of equations will be proposed respectively. Non‐Lipschitz condition is much weaker condition than the Lipschitz one. The simplified equations will be defined to make its solutions converge to that of the corresponding original equations both in the sense of mean square and probability, which constitute the approximation theorems. Copyright © 2014 John Wiley & Sons, Ltd.     

The improved split‐step θ methods for stochastic differential equation

Two improved split‐step θ methods, which, respectively, named split‐step composite θ method and modified split‐step θ‐Milstein method, are proposed for numerically solving stochastic differential equation of Itô type. The stability and convergence of these methods are investigated in the mean‐square sense. Moreover, an approach to improve the numerical stability is illustrated by choices of parameters of these two methods. Some numerical examples show the accordance between the theoretical and numerical results. Further numerical tests exhibit not only the Hamiltonian‐preserving property of the improved split‐step θ methods for a stochastic differential system but also the positivity‐preserving property of the modified split‐step θ‐Milstein method for the Cox–Ingersoll–Ross model. Copyright © 2013 John Wiley & Sons, Ltd.     

Split least‐squares finite element methods for non‐Fickian flow in porous media

In this article, we introduce two least‐squares finite element procedures for parabolic integro‐differential equations arising in the modeling of non‐Fickian flow in porous media. By selecting the least‐squares functional properly the presented procedure can be split into two independent subprocedures, one subprocedure is for the primitive unknown and the other is for the flux. The optimal order convergence analysis is established. Numerical examples are given to show the efficiency of the introduced schemes. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

A non‐linear and non‐local boundary condition for a diffusion equation in petroleum engineering

This article deals with a boundary value problem for Laplace equation with a non‐linear and non‐local boundary condition. This problem comes from petroleum engineering and is used to obtain an estimation of well productivity. The non‐linear and non‐local boundary condition is written on the well boundary. On the outer reservoir boundaries, we have both Dirichlet and Neumann conditions. In this paper, we prove the existence and uniqueness of a solution to this problem. The existence is proved by Schauder theorem and the uniqueness is obtained under more restricted conditions, when the involved operator is a contraction. Copyright © 2005 John Wiley & Sons, Ltd.     

On non‐autonomous integro‐differential‐algebraic evolutionary problems

In this article, we show that a technique for showing well‐posedness results for evolutionary equations in the sense of Picard and McGhee [Picard, McGhee, Partial Differential Equations: A unified Hilbert Space Approach, DeGruyter, Berlin, 2011] established in [Picard, Trostorff, Wehowski, Waurick, On non‐autonomous evolutionary problems. J. Evol. Equ. 13:751‐776, 2013] applies to a broader class of non‐autonomous integro‐differential‐algebraic equations. Using the concept of evolutionary mappings, we prove that the respective solution operators do not depend on certain parameters describing the underlying spaces in which the well‐posedness results are established. Copyright © 2014 John Wiley & Sons, Ltd.     

Exponential mean-square stability of the improved split-step theta methods for non-autonomous stochastic differential equations

We consider the mean-square stability of the so-called improved split-step theta method for stochastic differential equations. First, we study the mean-square stability of the method for linear test equations with real parameters. When θ 3/2, the improved split-step theta methods can reproduce the mean-square stability of the linear test equations for any step sizes h 0. Then, under a coupled condition on the drift and diffusion coefficients, we consider exponential mean-square stability of the method for nonlinear non-autonomous stochastic differential equations. Finally, the obtained results are supported by numerical experiments.     

Mild solutions of non‐Lipschitz stochastic integrodifferential evolution equations

In this work, we study the existence and uniqueness of mild solutions for stochastic partial integrodifferential equations under local non‐Lipschitz conditions on the coefficients. Our analysis makes use of the theory of resolvent operators as developed by R. Grimmer as well as a stopping time technique. Our results complement and improve several earlier related works. An example is provided to illustrate the theoretical results. Copyright © 2016 John Wiley & Sons, Ltd.     

Strong convergence of the split‐step θ‐method for stochastic age‐dependent population equations

In this paper, we constructed the split‐step θ (SSθ)‐method for stochastic age‐dependent population equations. The main aim of this paper is to investigate the convergence of the SS θ‐method for stochastic age‐dependent population equations. It is proved that the proposed method is convergent with strong order 1/2 under given conditions. Finally, an example is simulated to verify the results obtained from the theory, and comparative analysis with Euler method is given, the results show the higher accuracy of the SS θ‐method. Copyright © 2014 John Wiley & Sons, Ltd.     

Ritz–Galerkin method for solving a parabolic equation with non‐local and time‐dependent boundary conditions

The paper is devoted to the investigation of a parabolic partial differential equation with non‐local and time‐dependent boundary conditions arising from ductal carcinoma in situ model. Approximation solution of the present problem is implemented by the Ritz–Galerkin method, which is a first attempt at tackling parabolic equation with such non‐classical boundary conditions. In the process of dealing with the difficulty caused by integral term in non‐local boundary condition, we use a trick of introducing the transition function G(x,t) to convert non‐local boundary to another non‐classical boundary, which can be handled with the Ritz–Galerkin method. Illustrative examples are included to demonstrate the validity and applicability of the technique in this paper. Copyright © 2015 John Wiley & Sons, Ltd.     

Some non‐standard problems in viscous flow

Energy bounds are derived for Dirichlet type boundary value problems for the Navier–Stokes and Stokes equations when a combination of the solution values initially and at a later time is prescribed. The bounds are obtained by means of a differential inequality and imply uniqueness and continuous data dependence of the solutions for a range of values of the parameter in the non‐standard auxiliary condition. Copyright © 2004 John Wiley & Sons, Ltd.     

Exact boundary controllability for non‐autonomous quasilinear wave equations

In this paper, we first show that quite different from the autonomous case, the exact boundary controllability for non‐autonomous wave equations possesses various possibilities. Then we adopt a constructive method to establish the exact boundary controllability for one‐dimensional non‐autonomous quasilinear wave equations with various types of boundary conditions. Finally, we apply the results to multi‐dimensional quasilinear wave equation with rotation invariance. Copyright © 2007 John Wiley & Sons, Ltd.     

Modified Hermitian and skew‐Hermitian splitting methods for non‐Hermitian positive‐definite linear systems

To further study the Hermitian and non‐Hermitian splitting methods for a non‐Hermitian and positive‐definite matrix, we introduce a so‐called lopsided Hermitian and skew‐Hermitian splitting and then establish a class of lopsided Hermitian/skew‐Hermitian (LHSS) methods to solve the non‐Hermitian and positive‐definite systems of linear equations. These methods include a two‐step LHSS iteration and its inexact version, the inexact Hermitian/skew‐Hermitian (ILHSS) iteration, which employs some Krylov subspace methods as its inner process. We theoretically prove that the LHSS method converges to the unique solution of the linear system for a loose restriction on the parameter α. Moreover, the contraction factor of the LHSS iteration is derived. The presented numerical examples illustrate the effectiveness of both LHSS and ILHSS iterations. Copyright © 2007 John Wiley & Sons, Ltd.     

Oscillation and non‐oscillation results for solutions of perturbed half‐linear equations

The purpose of this paper is to describe the oscillatory properties of second‐order Euler‐type half‐linear differential equations with perturbations in both terms. All but one perturbations in each term are considered to be given by finite sums of periodic continuous functions, while coefficients in the last perturbations are considered to be general continuous functions. Since the periodic behavior of the coefficients enables us to solve the oscillation and non‐oscillation of the considered equations, including the so‐called critical case, we determine the oscillatory properties of the equations with the last general perturbations. As the main result, we prove that the studied equations are conditionally oscillatory in the considered very general setting. The novelty of our results is illustrated by many examples, and we give concrete new corollaries as well. Note that the obtained results are new even in the case of linear equations.     

Iterative operator‐splitting methods for nonlinear differential equations and applications

In this article, we consider iterative operator‐splitting methods for nonlinear differential equations with bounded and unbounded operators. The main feature of the proposed idea is the embedding of Newton's method for solving the split parts of the nonlinear equation at each step. The convergence properties of such a mixed method are studied and demonstrated. We confirm with numerical applications the effectiveness of the proposed scheme in comparison with the standard operator‐splitting methods by providing improved results and convergence rates. We apply our results to deposition processes. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1026–1054, 2011     

General non‐isospectral integrable coupling of AKNS equations associated with

By means of the real Lie algebra and compatible condition of spectral problems, a general non‐isospectral AKNS equations is constructed. Furthermore, by enlarging spectral problems, the corresponding non‐isospectral integrable coupling of AKNS equations associated with is also obtained in this paper.     

A three‐dimensional finite element model for the control of certain non‐linear bioreactors

This paper deals with the dynamics of non‐linear distributed parameter fixed‐bed bioreactors. The model consists of a pair of non‐linear partial differential (evolution) equations. The true spatially three‐dimensional situation is considered instead of the usual one‐dimensional approximation. This enables one to take into account the effects of flow profiles and the true location of the measurement transducer. The (output) evolution of the corresponding open‐loop control system is simulated. Furthermore, the associated closed‐loop system with respect to the relevant output function is considered. Especially, the asymptotic output tracking is found to be successful by applying the usual process based on the state feedback linearization. Copyright © 2000 John Wiley & Sons, Ltd.     

Riesz basis property of root vectors of non‐self‐adjoint operators generated by aircraft wing model in subsonic airflow

This paper is the third in a series of several works devoted to the asymptotic and spectral analysis of a model of an aircraft wing in a subsonic air flow. This model has been developed in the Flight Systems Research Center of UCLA and is presented in the works by Balakrishnan. The model is governed by a system of two coupled integro‐differential equations and a two‐parameter family of boundary conditions modeling the action of the self‐straining actuators. The differential parts of the above equations form a coupled linear hyperbolic system; the integral parts are of the convolution type. The system of equations of motion is equivalent to a single operator evolution–convolution equation in the energy space. The Laplace transform of the solution of this equation can be represented in terms of the so‐called generalized resolvent operator, which is an operator‐valued function of the spectral parameter. This generalized resolvent operator is a finite‐meromorphic function on the complex plane having the branch cut along the negative real semi‐axis. Its poles are precisely the aeroelastic modes and the residues at these poles are the projectors on the generalized eigenspaces. In the first two papers (see [33, 34]) and in the present one, our main object of interest is the dynamics generator of the differential parts of the system. This generator is a non‐self‐adjoint operator in the energy space with a purely discrete spectrum. In the first paper, we have shown that the spectrum consists of two branches, and have derived their precise spectral asymptotics with respect to the eigenvalue number. In the second paper, we have derived the asymptotical approximations for the mode shapes. Based on the asymptotical results of the first two papers, in the present paper, we (a) prove that the set of the generalized eigenvectors of the aforementioned differential operator is complete in the energy space; (b) construct the set of vectors which is biorthogonal to the set of the generalized eigenvectors in the case when there might be not only eigenvectors but associate vectors as well; and (c) prove that the set of the generalized eigenvectors forms a Riesz basis in the energy space. To prove the main result of the paper, we made use of the Nagy–Foias functional model for non‐self‐adjoint operators. The results of all three papers will be important for the reconstruction of the solution of the original initial‐boundary‐value problem from its Laplace transform in the forthcoming papers. Copyright © 2000 John Wiley & Sons, Ltd.     

Flow of a non‐linear (density‐gradient‐dependent) viscous fluid with heat generation,viscous dissipation and radiation

In this paper, we study the flow of a compressible (density‐gradient‐dependent) non‐linear fluid down an inclined plane, subject to radiation boundary condition. The convective heat transfer is also considered where a source term, similar to the Arrhenius type reaction, is included. The non‐dimensional forms of the equations are solved numerically and the competing effects of conduction, dissipation, heat generation and radiation are discussed. Copyright © 2008 John Wiley & Sons, Ltd.     

Existence of one non‐trivial anti‐periodic solution for second‐order impulsive differential inclusions

The existence of one non‐trivial solution for a second‐order impulsive differential inclusion is established. More precisely, a recent critical point result is exploited, in order to prove the existence of a determined open interval of positive eigenvalues for which the considered problem admits at least one non‐trivial anti‐periodic solution. Copyright © 2017 John Wiley & Sons, Ltd.     

Local and non‐local boundary quenching

The quenching problem is examined for a one‐dimensional heat equation with a non‐linear boundary condition that is of either local or non‐local type. Sufficient conditions are derived that establish both quenching and non‐quenching behaviour. The growth rate of the solution near quenching is also given for a power‐law non‐linearity. The analysis is conducted in the context of a nonlinear Volterra integral equation that is equivalent to the initial–boundary value problem. Copyright © 1999 John Wiley & Sons, Ltd.