Dissipativity of θ-methods for nonlinear Volterra delay-integro-differential equations
Journal of Computational and Applied Mathematics
Computers & Mathematics with Applications
Stability of linear multistep methods for delay integro-differential equations
Computers & Mathematics with Applications
The extended one-leg methods for nonlinear neutral delay-integro-differential equations
Applied Numerical Mathematics
Applied Numerical Mathematics
Stabilization of Second Order Evolution Equations with Unbounded Feedback with Time-Dependent Delay
SIAM Journal on Control and Optimization
Delay-dependent stability analysis of numerical methods for stochastic delay differential equations
Journal of Computational and Applied Mathematics
Linearized oscillation theory for a nonlinear equation with a distributed delay
Mathematical and Computer Modelling: An International Journal
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This paper is concerned with the study of the stability of ordinary and partial differential equations with both fixed and distributed delays, and with the study of the stability of discretizations of such differential equations. We start with a delay-dependent asymptotic stability analysis of scalar ordinary differential equations with real coefficients. We study the exact stability region of the continuous problem as a function of the parameters of the model. Next, it is proved that a time discretization based on the trapezium rule can preserve the asymptotic stability for the considered set of test problems. In the second part of the paper, we study delay partial differential equations. The stability region of the fully continuous problem is analyzed first. Then a semidiscretization in space is applied. It is shown that the spatial discretization leads to a reduction of the stability region when the standard second-order central difference operator is employed to approximate the diffusion operator. Finally we consider the delay-dependent stability of the fully discrete problem, where the partial differential equation is discretized both in space and in time. Some numerical examples and further discussions are given.