Jason A. Roberts

Reliability of difference analogues to preserve stability properties of stochastic Volterra integro-differential equations

We consider the reliability of some numerical methods in preserving the stability properties of the linear stochastic functional differential equation , where α, β, σ, τ ≥ 0 are real constants, and W(t) is a standard Wiener process. The areas of the regions of asymptotic stability for the class of methods considered, indicated by the sufficient conditions for the discrete system, are shown to be equal in size to each other and we show that an upper bound can be put on the time-step parameter for the numerical method for which the system is asymptotically mean-square stable. We illustrate our results by means of numerical experiments and various stability diagrams. We examine the extent to which the continuous system can tolerate stochastic perturbations before losing its stability properties and we illustrate how one may accurately choose a numerical method to preserve the stability properties of the original problem in the numerical solution. Our numerical experiments also indicate that the quality of the sufficient conditions is very high.

Superfast solution of linear convolutional Volterra equations using QTT approximation

We address a linear fractional differential equation and develop effective solution methods using algorithms for the inversion of triangular Toeplitz matrices and the recently proposed QTT format. The inverses of such matrices can be computed by the divide and conquer and modified Binis algorithms, for which we present the versions with the QTT approximation. We also present an efficient formula for the shift of vectors given in QTT format, which is used in the divide and conquer algorithm. As a result, we reduce the complexity of inversion from the fast Fourier level O ( n log n ) to the speed of superfast Fourier transform, i.e., O ( log 2 n ) . The results of the paper are illustrated by numerical examples.

Bifurcations in numerical methods for volterra integro-differential equations

We are interested in finding approximate solutions to parameter-dependent Volterra integro-differential equations over long time intervals using numerical schemes. This paper concentrates on changes in qualitative behavior (bifurcations) in the solutions and extends the work of Brunner and Lambert and Matthys (who considered only changes in stability behavior) to consider other bifurcations. We begin by considering a one-parameter equation with fading memory separable convolution kernel: we give an analytical discussion of bifurcations in this case and provide details of the behavior of numerical schemes. We extend our analysis to consider an equation with two-parameter fading memory convolution kernel and show the relationship to the classical test equation studied by the earlier authors. We draw attention to the fact that known stability results may not provide a reliable framework for choice of numerical scheme when other changes in qualitative behavior are also of interest. We give bifurcation plots for a variety of methods and show how, for known values of the parameters, stepsizes h>0 may be chosen to preserve the correct qualitative behavior in the numerical solution of the Volterra integro-differential equation.

A note on finite difference methods for nonlinear fractional differential equations with non-uniform meshes

ABSTRACT We consider finite difference methods for solving nonlinear fractional differential equations in the Caputo fractional derivative sense with non-uniform meshes. Under the assumption that the Caputo derivative of the solution of the fractional differential equation is suitably smooth [C. Li, Q. Yi, and A. Chen, Finite difference methods with non-uniform meshes for nonlinear fractional differential equations, J. Comput. Phys. 316 (2016), pp. 614–631] obtained the error estimates of finite difference methods with non-uniform meshes. However, the Caputo derivative of the solution of the fractional differential equation in general has a weak singularity near the initial time. In this paper, we obtain the error estimates of finite difference methods with non-uniform meshes when the Caputo fractional derivative of the solution of the fractional differential equation has lower smoothness. The convergence result shows clearly how the regularity of the Caputo fractional derivative of the solution affect the order of convergence of the finite difference methods. Numerical results are presented that confirm the sharpness of the error analysis.

Detailed error analysis for a fractional adams method with graded meshes

We consider a fractional Adams method for solving the nonlinear fractional differential equation 0CDtαy(t)=f(t,y(t)),α>0

Nonlinear Volterra Integro-Differential Equations Stability and Numerical Stability of θ -Methods

\,^{C}_{0}D^{\alpha }_{t} y(t) = f(t, y(t)), \, \alpha >0

Data aggregation in wireless sensor networks with minimum delay and minimum use of energy: A comparative study

, equipped with the initial conditions y(k)(0)=y0(k),k=0,1,…,⌈α⌉−1

Numerical approaches to bifurcations in solutions to integro-differential equations

y^{(k)} (0) = y_{0}^{(k)}, k=0, 1, \dots , \lceil \alpha \rceil -1

The Numerical Simulation of the Qualitative Behaviour of Volterra Integro-Differential Equations

. Here, α may be an arbitrary positive number and ⌈α⌉ denotes the smallest integer no less than α and the differential operator is the Caputo derivative. Under the assumption 0CDtαy∈C2[0,T]

ASYMPTOTIC STABILITY ANALYSIS OF A STOCHASTIC VOLTERRA INTEGRO-DIFFERENTIAL EQUATION WITH FADING MEMORY

\,^{C}_{0}D^{\alpha }_{t} y \in C^{2}[0, T]