Limiting Bourgain–Brezis estimates for systems of linear differential equations: Theme and variations

J. Bourgain and H. Brezis have obtained in 2002 some new and surprising estimates for systems of linear differential equations, dealing with the endpoint case L ¹ of singular integral estimates and the critical Sobolev space \({W^{1,n}(\mathbb{R}^n)}\) . This paper presents an overview of the results, further developments over the last ten years and challenging open problems.     

Runge–Kutta methods for jump–diffusion differential equations

In this paper we consider Runge–Kutta methods for jump–diffusion differential equations. We present a study of their mean-square convergence properties for problems with multiplicative noise. We are concerned with two classes of Runge–Kutta methods. First, we analyse schemes where the drift is approximated by a Runge–Kutta ansatz and the diffusion and jump part by a Maruyama term and second we discuss improved methods where mixed stochastic integrals are incorporated in the approximation of the next time step as well as the stage values of the Runge–Kutta ansatz for the drift. The second class of methods are specifically developed to improve the accuracy behaviour of problems with small noise. We present results showing when the implicit stochastic equations defining the stage values of the Runge–Kutta methods are uniquely solvable. Finally, simulation results illustrate the theoretical findings.     

Hyers–Ulam stability of linear partial differential equations of first order

In this work, we will prove the Hyers–Ulam stability of linear partial differential equations of first order.     

Legendre–Gauss collocation methods for ordinary differential equations

In this paper, we propose two efficient numerical integration processes for initial value problems of ordinary differential equations. The first algorithm is the Legendre–Gauss collocation method, which is easy to be implemented and possesses the spectral accuracy. The second algorithm is a mixture of the collocation method coupled with domain decomposition, which can be regarded as a specific implicit Legendre–Gauss Runge–Kutta method, with the global convergence and the spectral accuracy. Numerical results demonstrate the spectral accuracy of these approaches and coincide well with theoretical analysis.      

Runge–Kutta Lawson schemes for stochastic differential equations

In this paper, we present a framework to construct general stochastic Runge–Kutta Lawson schemes. We prove that the schemes inherit the consistency and convergence properties of the underlying Runge–Kutta scheme, and confirm this in some numerical experiments. We also investigate the stability properties of the methods and show for some examples, that the new schemes have improved stability properties compared to the underlying schemes.

     

Dichotomies for linear differential equations with delays: the Carathéodory case

Summary This work is concerned with differential equations with delays, on [0, ∞[, of the form +Mu=r, and the corresponding homogeneous equation +Mu=0, where u and r take values in a Banach space E, and M is a ? memory ?. The main results relate the existence of dichotomies or exponential dichotomies of the solutions of the homogeneous equation (a kind of conditional stability) to the admissibility of certain pairs of function spaces for the inhomogeneous equation. The ? memory ? M is quite general: it is a linear mapping that takes continuous E-valued functions on [−1, ∞[ into locally integrable functions on [0, ∞[ in such a way that Mu on [a, b] ⊂ [0, ∞[ depends only on u on [a −1, b], and satisfies a reasonable boundedness condition. Entrata in Redazione il 10 settembre 1973. This work was partially supported by NSF Grant GP-33364X.     

On the pathwise exponential stability of non–linear stochastic partial differential equations

Sufficient conditions to get exponential stability for the sample paths (with probability one) of a non–linear monotone stochastic Partial Differential Equation are proved. In fact, we improve a stability criterion established in Chow [3] since, under the same hypotheses, we get pathwise exponential stability instead of stability of sample paths     

Continuous two-step Runge–Kutta methods for ordinary differential equations

New classes of continuous two-step Runge-Kutta methods for the numerical solution of ordinary differential equations are derived. These methods are developed imposing some interpolation and collocation conditions, in order to obtain desirable stability properties such as A-stability and L-stability. Particular structures of the stability polynomial are also investigated.     

Implicit–explicit predictor–corrector schemes for nonlinear parabolic differential equations

In the present paper, a family of predictor–corrector (PC) schemes are developed for the numerical solution of nonlinear parabolic differential equations. Iterative processes are avoided by use of the implicit–explicit (IMEX) methods. Moreover, compared to the predictor schemes, the proposed methods usually have superior accuracy and stability properties. Some confirmation of these are illustrated by using the schemes on the well-known Fisher’s equation.     

On subnormal solutions of second order linear periodic differential equations

In this paper, we investigate the existence and the form of subnormal solution for a class of second order periodic linear differential equations, estimate the growth properties of all solutions, and answer the question raised by Gundersen and Steinbart.     

The truncated Euler–Maruyama method for stochastic differential delay equations

The numerical solutions of stochastic differential delay equations (SDDEs) under the generalized Khasminskii-type condition were discussed by Mao (Appl. Math. Comput. 217, 5512–5524 2011), and the theory there showed that the Euler–Maruyama (EM) numerical solutions converge to the true solutions in probability. However, there is so far no result on the strong convergence (namely in L ^p ) of the numerical solutions for the SDDEs under this generalized condition. In this paper, we will use the truncated EM method developed by Mao (J. Comput. Appl. Math. 290, 370–384 2015) to study the strong convergence of the numerical solutions for the SDDEs under the generalized Khasminskii-type condition.     

Rational Jacobi elliptic solutions for nonlinear differential–difference lattice equations

In this work, we present a direct new method for constructing the rational Jacobi elliptic solutions for nonlinear differential–difference equations, which may be called the rational Jacobi elliptic function method. We use the rational Jacobi elliptic function method to construct many new exact solutions for some nonlinear differential–difference equations in mathematical physics via the lattice equation. The proposed method is more effective and powerful for obtaining the exact solutions for nonlinear differential–difference equations.     

Oscillation criteria for second-order superlinear Emden–Fowler neutral differential equations

We study oscillatory behavior of solutions to a class of second-order superlinear Emden–Fowler neutral differential equations. New oscillation theorems are presented and their efficiency is illustrated.     

Stability of IMEX Runge–Kutta methods for delay differential equations

Stability of IMEX (implicit–explicit) Runge–Kutta methods applied to delay differential equations (DDEs) is studied on the basis of the scalar test equation du/dt=λu(t)+μu(t-τ)$\mathrm{d}u/\mathrm{d}t=\lambda u(t)+\mu u(t-\tau )$, where τ$\tau$ is a constant delay and λ,μ$\lambda ,\mu$ are complex parameters. More specifically, P-stability regions of the methods are defined and analyzed in the same way as in the case of the standard Runge–Kutta methods. A new IMEX method which possesses a superior stability property for DDEs is proposed. Some numerical examples which confirm the results of our analysis are presented.