Coupled nonlinear stochastic differential equations

Systems of n coupled linear or nonlinear differential equations which may be deterministic or stochastic are solved by methods of the first author and his co-workers. Examples include two coupled Riccati equations, coupled linear equations, stochastic coupled equations with product terms, and n coupled stochastic differential equations.     

Approximation for non-smooth functionals of stochastic differential equations with irregular drift

This paper aims at developing a systematic study for the weak rate of convergence of the Euler–Maruyama scheme for stochastic differential equations with very irregular drift and constant diffusion coefficients. We apply our method to obtain the rates of approximation for the expectation of various non-smooth functionals of both stochastic differential equations and killed diffusion. We also apply our method to the study of the weak approximation of reflected stochastic differential equations whose drift is Hölder continuous.     

Coupled differential equations and coupled boundary conditions

It is demonstrated that the high accuracy for approximations requiring only a few terms which is typical of the decomposition method for nonlinear stochastic operator equations, or special cases (linear or deterministic), holds for coupled equations and coupled boundary conditions as well.     

Lower norm error estimates for approximate solutions of differential equations with non-smooth coefficients

Summary In this paper, we derive error estimates inL _p-norm, 1p, for the ₂-Finite Element approximation to solutions of boundary value problems, where the coefficients are functions of bounded variation. The ₂-Finite Element Method was introduced in [3] and was shown to be effective for problems with non-smooth coefficient.The results of this paper form a part of a Ph.D. thesis written at the University of Maryland under the direction of Professor J.E. Osborn     

Coupled differential algebraic equations in the simulation of flexible multibody systems with hydrodynamic force elements

The mathematical modelling of elastohydrodynamic bearings in combustion engines leads to a coupled system of (partial) differential algebraic equations, which is represented by a flexible multibody system model of the engine including crankshaft and bearing and by the Reynolds equation that describes the non-linear effects in the fluid film. The hydrodynamic forces depend strongly on the position and the elastic deformation of crankshaft and bearing shell. We discuss the influence of the spatial discretization on accuracy and numerical effort. Since a fine spatial discretization substantially slows down the numerical solution, we propose a semi-analytical method based on singular perturbation theory to speed-up time integration. Numerical tests for a simplified benchmark problem illustrate the advantages of this approach. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On implicit systems of differential equations

This article considers implicit systems of differential equations. The implicit systems that are considered are given by polynomial relations on the coordinates of the indeterminate function and the coordinates of the time derivative of the indeterminate function. For such implicit systems of differential equations, we are concerned with computing algebraic constraints such that on the algebraic variety determined by the constraint equations the original implicit system of differential equations has an explicit representation. Our approach is algebraic. Although there have been a number of articles that approach implicit differential equations algebraically, all such approaches have relied heavily on linear algebra. The approach of this article is different, we have no linearity requirements at all, instead we rely on algebraic geometry. In particular, we use birational mappings to produce an explicit system. The methods developed in this article are easily implemented using various computer algebra systems.     

Solvable systems of linear differential equations

The asymptotic iteration method (AIM) is an iterative technique used to find exact and approximate solutions to second-order linear differential equations. In this work, we employed AIM to solve systems of two first-order linear differential equations. The termination criteria of AIM will be re-examined and the whole theory is re-worked in order to fit this new application. As a result of our investigation, an interesting connection between the solution of linear systems and the solution of Riccati equations is established. Further, new classes of exactly solvable systems of linear differential equations with variable coefficients are obtained. The method discussed allow to construct many solvable classes through a simple procedure.     

Strong convergence of split-step backward Euler method for stochastic differential equations with non-smooth drift

In this paper, we are concerned with the numerical approximation of stochastic differential equations with discontinuous/nondifferentiable drifts. We show that under one-sided Lipschitz and general growth conditions on the drift and global Lipschitz condition on the diffusion, a variant of the implicit Euler method known as the split-step backward Euler (SSBE) method converges with strong order of one half to the true solution. Our analysis relies on the framework developed in [D. J. Higham, X. Mao and A. M. Stuart, Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM Journal on Numerical Analysis, 40 (2002) 1041-1063] and exploits the relationship which exists between explicit and implicit Euler methods to establish the convergence rate results.     

Bifurcations of non-smooth systems

Non-smooth systems (namely piecewise-smooth systems) have received much attention in the last decade. Many contributions in this area show that theory and applications (to electronic circuits, mechanical systems, …) are relevant to problems in science and engineering. Specially, new bifurcations have been reported in the literature, and this was the topic of this minisymposium. Thus both bifurcation theory and its applications were included. Several contributions from different fields show that non-smooth bifurcations are a hot topic in research. Thus in this paper the reader can find contributions from electronics, energy markets and population dynamics. Also, a carefully-written specific algebraic software tool is presented.     

Flag systems and ordinary differential equations

We discuss the Monge problem in the theory of ordinary differential equations and prove the Cartan criterion for first order Monge systems with the added hypothesis of homogeneity. Next, we examine Hilbert's counter-example and finally give a brief account on the two special cases involving flag systems of length two and three. Entrata in Redazione il 1 agosto 1998.     

Evolution systems for paraxial wave equations of Schrödinger-type with non-smooth coefficients

We prove existence of strongly continuous evolution systems in L² for Schrödinger-type equations with non-Lipschitz coefficients in the principal part. The underlying operator structure is motivated from models of paraxial approximations of wave propagation in geophysics. Thus, the evolution direction is a spatial coordinate (depth) with additional pseudodifferential terms in time and low regularity in the lateral space variables. We formulate and analyze the Cauchy problem in distribution spaces with mixed regularity. The key point in the evolution system construction is an elliptic regularity result, which enables us to precisely determine the common domain of the generators. The construction of a solution with low regularity in the coefficients is the basis for an inverse analysis which allows to infer the lack of lateral regularity in the medium from measured data.