Differential-Difference Equations. By Richard Bellman and Kenneth L. Cooke. Pp. xvi, 465. 1963. (Academic Press)

Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.

Solving Ordinary Differential Equations

We present a minimal spiking network that can polychronize, that is, exhibit reproducible time-locked but not synchronous firing patterns with millisecond precision, as in synfire braids. The network consists of cortical spiking neurons with axonal conduction delays and spike-timing-dependent plasticity (STDP); a ready-to-use MATLAB code is included. It exhibits sleeplike oscillations, gamma (40 Hz) rhythms, conversion of firing rates to spike timings, and other interesting regimes. Due to the interplay between the delays and STDP, the spiking neurons spontaneously self-organize into groups and generate patterns of stereotypical polychronous activity. To our surprise, the number of coexisting polychronous groups far exceeds the number of neurons in the network, resulting in an unprecedented memory capacity of the system. We speculate on the significance of polychrony to the theory of neuronal group selection (TNGS, neural Darwinism), cognitive neural computations, binding and gamma rhythm, mechanisms of attention, and consciousness as "attention to memories."

/pdf/polychronization-computation-with-spikes-5d651j5d00.pdf

Polychronization: Computation with Spikes

Convex Analysis: (pms-28)

https://ntrs.nasa.gov/api/citations/20010075154/downloads/20010075154.pdf

Additive Runge-Kutta Schemes for Convection-Diffusion-Reaction Equations

1. Introduction 2. Existence and regularity of solutions of DDEs 3. A review of DDE methods 4. The standard approach via continuous ODE methods 5. Continuous Runge-Kutta methods for ODEs 6. Runge-Kutta methods for DDEs 7. Local error estimation and variable stepsize 8. Stability analysis of Runge-Kutta methods for ODEs 9. Stability analysis of DDEs 10. Stability analysis of Runge-Kutta methods for DDEs

Numerical Methods for Delay Differential Equations

The present paper develops a theory of Natural Continuous Extensions (NCEs) for the discrete approximate solution of an ODE given by a Runge-Kutta process. These NCEs are defined in such a way that the continuous solutions furnished by the one-step collocation methods are included.

/pdf/natural-continuous-extensions-of-runge-kutta-methods-unyrp226ng.pdf

Natural continuous extensions of Runge-Kutta methods

Recently the author and others have studied a class of Runge-Kutta methods for DDEs. These methods make use of certain constrained meshes, which allow to get optimal order results. In this paper we study their asymptotic stability properties, using the test equation

$$\begin{gathered} y'(t) = ay(t) + by(t - \tau ),t > 0 \hfill \\ y(t) = g(t)for - \tau \leqq t \leqq 0 \hfill \\ \end{gathered} $$

wherea,b∈ℂ, σ>0, andg(t) is continuous and complex valued. It is known thaty(t)→0 ast→+∞ if |b|<-Re(a). In particular, we show that any Astable one-step collocation method for ODEs inherits the same property when it is applied to DDEs with a constrained mesh (i.e. it is P-stable).

P-stability properties of runge-kutta methods for delay differential equations

Continuous, explicit Runge–Kutta methods with the minimal number of stages are considered. These methods are continuously differentiable if and only if one of the stages is the FSAL evaluation. A characterization of a subclass of these methods is developed for orders 3, 4, and 5. It is shown how the free parameters of these methods can be used either to minimize the continuous truncation error coefficients or to maximize the stability region. As a representative for these methods the fifth-order method with minimized error coefficients is chosen, supplied with an error estimation method, and analysed by using the DETEST software. The results are compared with a similar implementation of the Dormand–Prince 5(4) pair with interpolant, showing a significant advantage in the new method for the chosen problems.

Derivation of efficient, continuous, explicit Runge-Kutta methods

In this paper we consider finite families of complex $n\times n$-matrices. In particular, we focus on those families that satisfy the so-called finiteness conjecture, which was recently disproved in its more general formulation. We conjecture that the validity of the finiteness conjecture for a finite family of nondefective type is equivalent to the existence of an extremal norm in the class of complex polytope norms. However, we have not been able to prove this complex polytope extremality conjecture, but we are able to prove the small complex polytope extremality theorem under some more restrictive hypotheses on the underlying family of matrices. In addition, our theorem assures a certain finiteness property on the number of vertices of the unit ball of the extremal complex polytope norm, which could be very useful for the construction of suitable algorithms aimed at the actual computation of the spectral radius of the family.

Marino Zennaro

Papers

Numerical Methods for Delay Differential Equations

Natural continuous extensions of Runge-Kutta methods

P-stability properties of runge-kutta methods for delay differential equations

Derivation of efficient, continuous, explicit Runge-Kutta methods

Complex Polytope Extremality Results for Families of Matrices