Asymptotically linear solutions of differential equations via Lyapunov functions

We discuss the existence of solutions with oblique asymptotes to a class of second order nonlinear ordinary differential equations by means of Lyapunov functions. The approach is new in this field and allows for simpler proofs of general results regarding Emden-Fowler like equations.     

Investigation of a system of linear differential equations with random coefficients

We investigate a system of linear differential equations with random coefficients that depend on a periodic Markov process. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 8, pp. 1137–1143, August, 1998.     

Optimization of a system of linear differential equations with random coefficients

We consider a system of differential equations with controls that are linearly contained in the right-hand sides. We establish a necessary condition for the optimal control that minimizes a quadratic functional. Kiev National Economic University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 4, pp. 556–561, April, 1999.     

Ordinary differential equations with fractal noise

The differential equation

for fractal-type functions is determined via fractional calculus. Under appropriate conditions we prove existence and uniqueness of a local solution by means of its representation for certain -functions and . The method is also applied to Itô stochastic differential equations and leads to a general pathwise representation. Finally we discuss fractal sample path properties of the solutions.

     

Ordinary differential equations and their exponentials

In the context of Synthetic Differential Geometry, we discuss vector fields/ordinary differential equations as actions; in particular, we exploit function space formation (exponential spaces) in the category of actions.     

Formulas of Liapunov functions for systems of linear partial differential equations

Formulas of explicit quadratic Liapunov functions for showing asymptotic stability of the system of linear partial differential equations on (0,∞)×Ω, are constructed, where A is an n×n real matrix, u=T(u₁,u₂,…,un), Ω is a bounded domain in Rk with smooth boundary ∂Ω, and Δ denotes the Laplacian operator on Rk with Δu=T(Δu₁,Δu₂,…,Δun). These formulas are also modified and applied to a number of nonautonomous linear and nonlinear systems and models in structural stability, traveling wave, and Navier-Stokes equations.     

Reversible linear differential equations

Let ? be a meromorphic connection on a vector bundle over a compact Riemann surface Γ. An automorphism $\sigma :\Gamma \to \Gamma$ is called a symmetry of ? if the pullback bundle and the pullback connection can be identified with ?. We study the symmetries by means of what we call the Fano group; and, under the hypothesis that ? has a unimodular reductive Galois group, we relate the differential Galois group, the Fano group and the symmetries by means of an exact sequence.     

Indicial rational functions of linear ordinary differential equations with polynomial coefficients

The notion of indicial rational function is introduced for ordinary differential equations with polynomial coefficients and polynomial right-hand sides, and algorithms for its construction are proposed.     

On random fuzzy differential equations

We consider a Cauchy problem in a random fuzzy setting. Under the condition of Lipschitzean right-hand side the existence and uniqueness of the solution is proven, also the continuous dependence on the right-hand side and initial condition is shown. Some kind of boundedness of the solution is established.     

Strong approximation of time-changed stochastic differential equations involving drifts with random and non-random integrators

BIT Numerical Mathematics - The rates of strong convergence for various approximation schemes are investigated for a class of stochastic differential equations (SDEs) which involve a random time...