Delay Partial Differential Equations Research Articles

Optimal control problems for stochastic delay evolution equations in Banach spaces

We consider the optimal control for a Banach space valued stochastic delay evolution equation. The existence and uniqueness of the mild solution for the associated Hamilton–Jacobi–Bellman equations are obtained by means of backward stochastic differential equations. An application to optimal control of stochastic delay partial differential equations is also given.

A hybrid approximation method for solving Hutchinson’s equation

The hybrid function approximation method for solving Hutchinson’s equation which is a nonlinear delay partial differential equation, is investigated. The properties of hybrid of block-pulse functions and Lagrange interpolating polynomials based on Legendre–Gauss-type points are presented and are utilized to replace the system of nonlinear delay differential equations resulting from the application of Legendre pseudospectral method, by a system of nonlinear algebraic equations. The validity and applicability of the proposed method are demonstrated through two illustrative examples on Hutchinson’s equation.

A fitted numerical method for a system of partial delay differential equations

We consider a system of two coupled partial delay differential equations (PDDEs) describing the dynamics of two cooperative species. The original system is reduced to a system of ordinary delay differential equations (DDEs) obtained by applying the method of lines. Then we construct a fitted operator finite difference method (FOFDM) to solve this resulting system. The model considered in this paper is very sensitive to small changes in the parameters associated in with the model. Depending on the values of these parameters, the solution can be stable, periodic and/or aperiodic. Such behavior of the solution is exploited via the proposed FOFDM. This FOFDM is analyzed for convergence and it is seen that this method is unconditionally stable and has the accuracy of O(k+h2), where k and h denote time and space step-sizes, respectively. Some numerical results confirming theoretical observations are also presented. These results are comparable with those obtained in the literature.

On the Use of Delay Equations in Engineering Applications

This paper is a review of applications of delay differential equations to different areas of engineering science. Starting with a general overview of delay models, we present some recent results on the use of retarded, advanced and neutral delay differential equations. An emerging area for modeling with the help of delay equations is real-time dynamic substructuring, or hybrid testing. We introduce the main idea of this technique together with the latest advances. Special emphasis is given to the development of the theory and applications of partial delay differential equations. The review concludes with a summary of some open problems and questions concerning the analysis of spatially extended delayed systems.

Exponential stability of energy solutions to stochastic partial differential equations with variable delays and jumps

In this work, we investigate stochastic partial differential equations with variable delays and jumps. We derive by estimating the coefficients functions in the stochastic energy equality some sufficient conditions for exponential stability and almost sure exponential stability of energy solutions, and generalize the results obtained by Taniguchi [T. Taniguchi, The exponential stability for stochastic delay partial differential equations, J. Math. Anal. Appl. 331 (2007) 191–205] and Wan and Duan [L. Wan, J. Duan, Exponential stability of non-autonomous stochastic partial differential equations with finite memory, Statist. Probab. Lett. 78 (5) (2008) 490–498] to cover a class of more general stochastic partial differential equations with jumps. Finally, an illustrative example is established to demonstrate our established theory.

Frequent Oscillatory Behavior of Delay Partial Difference Equations with Positive and Negative Coefficients

This paper is concerned with a class of nonlinear delay partial difference equations with positive and negative coefficients, which also contains forcing terms. By making use of frequency measures, some new oscillatory criteria are established.

Frequent Oscillatory Behavior of Delay Partial Difference Equations with Positive and Negative Coefficients

Frequent Oscillatory Behavior of Delay Partial Difference Equations with Positive and Negative Coefficients

Operator splittings and spatial approximations for evolution equations

The convergence of various operator splitting procedures, such as the sequential, the Strang and the weighted splitting, is investigated in the presence of a spatial approximation. To this end the relevant notions and results of numerical analysis are presented, a variant of Chernoff’s product formula is proved and the general Trotter–Kato approximation theorem is used. The methods are applied to an abstract partial delay differential equation.

Stability in distribution of mild solutions to stochastic partial differential delay equations with jumps

The existence, uniqueness and some sufficient conditions for stability in distribution of mild solutions to stochastic partial differential delay equations with jumps are presented. The principle technique of our investigation is to construct a proper approximating strong solution system and carry out a limiting type of argument to pass on stability of strong solutions to mild ones. As a consequence, stability results of Basak et al . ( Basak et al . 1999 J. Math. Anal. Appl. 202 , 604–622) and Yuan et al . ( Yuan et al . 2003 Syst. Control Lett. 50 , 195–207) are generalized to cover a class of much more general stochastic partial differential delay equations with jumps in infinite dimensions. In contrast to the almost sure exponential stability in Ichikawa ( Ichikawa 1982 J. Math. Anal. Appl. 90 , 12–44) and Luo & Liu ( Luo & Liu 2008 Stoch. Proc. Appl. 118 , 864–895) and the moment exponential stability in Luo & Liu, we present a new result on the stability in distribution of mild solutions. Finally, an example is given to demonstrate the applicability of our work.

Delays and Oscillations in Networks of Spiking Neurons: A Two-Timescale Analysis

Oscillations are a ubiquitous feature of many neural systems, spanning many orders of magnitude in frequency. One of the most prominent oscillatory patterns, with possible functional implications, is that occurring in the mammalian thalamocortical system during sleep. This system is characterized by relatively long delays (reaching up to 40 msec) and gives rise to low-frequency oscillatory waves. Motivated by these phenomena, we study networks of excitatory and inhibitory integrate-and-fire neurons within a Fokker-Planck delay partial differential equation formalism and establish explicit conditions for the emergence of oscillatory solutions, and for the amplitude and period of the ensuing oscillations, for relatively large values of the delays. When a two-timescale analysis is employed, the full partial differential equation is replaced in this limit by a discrete time iterative map, leading to a relatively simple dynamic interpretation. This asymptotic result is shown numerically to hold, to a good approximation, over a wide range of parameter values, leading to an accurate characterization of the behavior in terms of the underlying physical parameters. Our results provide a simple mechanistic explanation for one type of slow oscillation based on delayed inhibition, which may play an important role in the slow spindle oscillations occurring during sleep. Moreover, they are consistent with experimental findings related to human motor behavior with visual feedback.

Flow invariance for nonlinear partial differential delay equations

Several fundamental results on existence, flow-invariance, regularity, and linearized stability of solutions to the nonlinear partial differential delay equationu˙(t)+Bu(t)∋F(ut),t≥0,u0=φ,\dot {u}(t) + Bu(t) \ni F(u_t), t\geq 0, u_0 = \varphi ,withB⊂X×XB\subset X\times Xω−\omega -accretive, are developed for a general Banach spaceX.X.In contrast to existing results, with the history-responseFFglobally defined and, at least, Lipschitz on bounded sets, the results are tailored for situations withFFdefined on (possibly) thin subsets of the initial-history spaceEEonly, and are applied to place several classes of population models in their naturalL1−L^1-setting. The main result solves the open problem of a subtangential condition for flow-invariance of solutions in the fully nonlinear case, paralleling those known for the cases of (a) no delay, (b) ordinary delay equations withB≡0,B\equiv 0,and (c) the semilinear case.

ON EXPONENTIAL STABILITY FOR STOCHASTIC DELAY PARTIAL DIFFERENTIAL EQUATIONS

This paper is concerned with the exponential stability of energy solutions to a nonlinear stochastic delay partial differential equations with finite delay in separable Hilbert spaces. Some exponential stability criteria are obtained by constructing the Lyapunov function. As an application, one example is also given to illustrate our results.

Interval oscillation theorems for second order nonlinear partial delay differential equations

Using the integral averaging method and the generalized Riccati technique, we derive new interval oscillation criteria for second order nonlinear partial delay differential equations. These results are different from most known ones in the sense that they are based on information only on a sequence of subintervals of (0,∞), rather than on the whole (0,∞). Our results are of a high degree of generality and sharper than the existing results in literature.

Frequent oscillatory solutions of a nonlinear partial difference equation

This paper is concerned with a class of nonlinear delay partial difference equations with variable coefficients, which may change sign. By making use of frequency measures, some new oscillatory criteria are established.

From Boltzmann transport equation to single-phase-lagging heat conduction

In the present work the single-phase-lagging heat conduction model is re-derived analytically from the Boltzmann transport equation. In contrast to the Maxwell–Cattaneo law (CV model), it is Galilean invariant in the moving media. Based on this model, the governing equation of the microscale heat conduction is established, which is formulated into a delay partial differential equation. The corresponding initial and boundary conditions are prescribed. The thermal oscillation of the single-phase-lagging heat conduction is investigated.

Flow invariance for solutions to nonlinear nonautonomous partial differential delay equations

We investigate the problem of existence and flow invariance of mild solutions to nonautonomous partial differential delay equations u ˙ ( t ) + B ( t ) u ( t ) ∋ F ( t , u t ) , t ⩾ s , u s = φ , where B ( t ) is a family of nonlinear multivalued, α-accretive operators with D ( B ( t ) ) possibly depending on t, and the operators F ( t , . ) being defined—and Lipschitz continuous—possibly only on “thin” subsets of the initial history space E. The results are applied to population dynamics models. We also study the asymptotic behavior of solutions to this equation. Our analysis will be based on the evolution operator associated to the equation in the initial history space E.

Delay partial differential equations

Delay partial differential equations

Analytical and numerical stability of neutral delay integro-differential equations and neutral delay partial differential equations

This paper is concerned with the analytical and numerical stability of neutral delay integro-differential equations (NDIDEs) and neutral delay partial differential equations (NDPDEs). We study the delay-dependent stability of the real coefficient linear test equations for NDIDEs. Furthermore, we prove that the trapezium rule can preserve the delay-dependent stability of the test equations considered. We also discuss the delay-dependent stability of the continuous problems, the semi-discrete problems and the fully discrete problems of linear NDPDEs. Some numerical experiments are given to confirm the theoretical results.

Frequent oscillation in a nonlinear partial difference equation

This paper is concerned with a class of nonlinear delay partial difference equations with variable coefficients, which may change sign. By making use of frequency measures, some new oscillatory criteria are established. This is the first time oscillation of these partial difference equations is discussed by employing frequency measures.

Modelling real-time dynamic substructuring using partial delay differential equations

Real-time dynamic substructuring is a new component testing method for simulating the dynamics of complex engineering systems. The physical component is tested within a computer-generated ‘virtual’ environment using real-time control techniques. Delays in communication which occur between the component and the virtual environment can potentially destabilize the simulation. In this paper, the mechanism for this instability is examined using a beam-oscillator system as a case study. We will show how the stability and the amplitude response of the system change with the time delay. Numerical simulations of the reduced system as well as a full-delayed beam equation are performed. A series of experimental tests is carried out on a beam-oscillator system. Comparison of the theoretical, numerical and experimental results is presented and these agree remarkably well.