Explicit pseudo two-step exponential Runge–Kutta methods for the numerical integration of first-order differential equations

Numerical Algorithms - This paper is devoted to the explicit pseudo two-step exponential Runge–Kutta (EPTSERK) methods for the numerical integration of first-order ordinary differential...     

On the differential properties of the support function of the∈-subdifferential of a convex function

We study differential properties of the support function of the-subdifferential of a convex function; applications in algorithmics are also given.     

Solvability of a periodic boundary-value problem for a system of first-order ordinary differential equations with (β,γ,δ)-comparison pairs

Existence theorems are proved for an incomplete set of upper and lower functions. Bibliography: 1 title. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 248, 1998, pp. 231–241. Translated by L. Yu. Kolotilina.     

Numerical simulation of the solution of a stochastic differential equation driven by a Lévy process

The Euler scheme is a well-known method of approximation of solutions of stochastic differential equations (SDEs). A lot of results are now available concerning the precision of this approximation in case of equations driven by a drift and a Brownian motion. More recently, people got interested in the approximation of solutions of SDEs driven by a general Lévy process. One of the problem when we use Lévy processes is that we cannot simulate them in general and so we cannot apply the Euler scheme. We propose here a new method of approximation based on the cutoff of the small jumps of the Lévy process involved. In order to find the speed of convergence of our approximation, we will use results about stability of the solutions of SDEs.     

Finite-difference modeling à la Mickens of the distribution of the stopping time in a stochastic differential equation

Departing from a general stochastic differential equation with Brownian diffusion, we establish that the distribution of probability of the stopping time is governed by a parabolic partial differential equation. A particular form of the problem under investigation may be associated to a stochastic generalization of the well-known Paris’ law from structural mechanics, in which case, the solution of the boundary-value problem represents the probability distribution of the hitting time. An implicit, convergent and probability-based discretization to approximate the solution of the boundary-value problem is proposed in this work. Using a convenient vector representation of our scheme, we prove that the method preserves the most relevant properties of a probability distribution function, namely, the non-negativity, the boundedness from above by 1, and the monotonicity. In addition, we establish that our method is a convergent technique, and provide some illustrative comparisons against known exact solutions.     

Some properties of the Itô–Wiener expansion of the solution of a stochastic differential equation and local times

In this paper, we use the formula for the Itô–Wiener expansion of the solution of the stochastic differential equation proven by Krylov and Veretennikov to obtain several results concerning some properties of this expansion. Our main goal is to study the Itô–Wiener expansion of the local time at the fixed point for the solution of the stochastic differential equation in the multidimensional case (when standard local time does not exist even for Brownian motion). We show that under some conditions the renormalized local time exists in the functional space defined by the _L2$L_2$-norm of the action of some smoothing operator.     

A canonical system of two differential equations with periodic coefficients and the Poincaré-Denjoy theory of differential equations on a torus

The passage from Cartesian to polar coordinates in a canonical system with periodic coefficients gives rise to a nonlinear differential equation whose right-hand side is periodic in time and the polar angle and thus this equation can be regarded as a differential equation on a torus. In accord with Poincaré-Denjoy theory, the behavior of a solution to a differential equation on a torus is characterized by the rotation number and some homeomorphic mapping of a circle onto itself. We study connections of strong stability (instability) of a canonical system, including the membership in the nth stability (instability) domain, with the rotation number and fixed points of this mapping.     

On general solution of a first-order non-linear differential equation of the formx(dy/dx)=y (λ+f (x,y)) with negative rational λ

Summary This paper studies an equation of the form x dy/dx=y(+f(x, y)),f(0,0)=0, where is a negative rational number andf(x, y) is a holomorphic function of (x, y) near (0, 0). It is known that by a formal transformation this equation is formally reduced to an equation of the form (R). The simplest case such that ==0 was studied by H. Dulac. The general case was studied by M. Hukuhara and he obtained an analytic expression (but not convergent) for a general solution. We will discuss how to construct a convergent one.Dedicated to Professor Taro Yoshizawa on his sixtieth birthday     

Conditions of smoothness for the distribution density of a solution of a multidimensional linear stochastic differential equation with lévy noise

We obtain a sufficient condition of smoothness for the distribution density of a multidimensional Ornstein–Uhlenbeck process with Lévy noise, i.e., for the solution of a linear stochastic differential equation with Lévy noise.     

Cauchy–Gelfand problem and the inverse problem for a first-order quasilinear equation

Gelfand’s problem on the large time asymptotics of the solution of the Cauchy problem for a first-order quasilinear equation with initial conditions of the Riemann type is considered. Exact asymptotics in the Cauchy–Gelfand problem are obtained and the initial data parameters responsible for the localization of shock waves are described on the basis of the vanishing viscosity method with uniform estimates without the a priori monotonicity assumption for the initial data.     

Application of the differential transformation method for the solution of the hyperchaotic Rössler system

The aim of this paper is to investigate the accuracy of the differential transformation method (DTM) for solving the hyperchaotic Rössler system, which is a four-dimensional system of ODEs with quadratic nonlinearities. Comparisons between the DTM solutions and the fourth-order Runge–Kutta (RK4) solutions are made. The DTM scheme obtained from the DTM yields an analytical solution in the form of a rapidly convergent series. The direct symbolic-numeric scheme is shown to be efficient and accurate.     

A sufficient condition for the existence of a positive solution for a nonlinear fractional differential equation with the Riemann–Liouville derivative

In this work, by means of the fixed point theorem in a cone, we establish the existence result for a positive solution to a kind of boundary value problem for a nonlinear differential equation with a Riemann–Liouville fractional order derivative. An example illustrating our main result is given. Our results extend previous work in the area of boundary value problems of nonlinear fractional differential equations [C. Goodrich, Existence of a positive solution to a class of fractional differential equations, Appl. Math. Lett. 23 (2010) 1050–1055].     

Generalization of the lions theory for first-order evolution differential equations with discontinuous operators and with α ∈ [1/2, 1]

We prove existence, uniqueness, and smoothness theorems for weak solutions of the problem $$ du(t)/dt + A(t)u(t) = f(t), t \in ]0,T[; u(0) = u_0 \in H, $$ where, for almost all t, the linear unbounded operators A(t) with domains D(A(t)) depending on t are closed and maximal accretive and have bounded inverses A ^?1(t) discontinuous with respect to t in the Hilbert space H. There exists an α ∈ [1/2, 1] such that the following is true in H for almost all t: the power A ^α (t) is subordinate to the power A* ^α (t) of the adjoint operators A*(t), the operators A ^α (t) and A* ^α (t) do not form an obtuse angle, and the domains D(A* ^α (t)) of the operators A* ^α (t) are not increasing with respect to t. This paper is the first to prove the well-posedness of the mixed problem for the multidimensional linearized Korteweg-de Vries equation smooth in time with boundary conditions piecewise constant in time.     

Mixing “in the sense of ibragimov.” Estimate for the rate of approach of a family of integral functionals of a solution of a differential equation with periodic coefficients to a family of wiener processes. Some applications. II

In the first part of the present paper, we established estimates for the rate of approach of the integrals of a family of “physical” white noises to a family of Wiener processes. We use this result to establish the estimate for the rate of approach of a family of solutions of ordinary differential equations perturbed by some “physical” white noises to a family of solutions of the corresponding It? equations. We consider both the case where the coefficient of random perturbation is separated from zero and the case where it is not separated from zero.     

Church–Rosser property of a simple reduction for full first-order classical natural deduction

A system of typed terms which corresponds with the classical natural deduction with one conclusion and full logical symbols is defined. Church–Rosser property of the system is proved using an extended method of parallel reduction.     

Mixing “In the sense of Ibragimov.” Estimate for the rate of approach of a family of integral functionals of a solution of a differential equation with periodic coefficients to a family of wiener processes. Some applications. I

We prove that a bounded 1-periodic function of a solution of a time-homogeneous diffusion equation with 1-periodic coefficients forms a process that satisfies the condition of uniform strong mixing. We obtain an estimate for the rate of approach of a certain normalized integral functional of a solution of an ordinary time-homogeneous stochastic differential equation with 1-periodic coefficients to a family of Wiener processes in probability in the metric of space C [0, T]. As an example, we consider an ordinary differential equation perturbed by a rapidly oscillating centered process that is a 1-periodic function of a solution of a time-homogeneous stochastic differential equation with 1-periodic coefficients. We obtain an estimate for the rate of approach of a solution of this equation to a solution of the corresponding It? stochastic equation.     

Solvability of a system of equations for the Fourier–Chebyshev coefficients when solving ordinary differential equations by the Chebyshev series method

A solvability theorem for a system of equations with respect to approximate values of Fourier–Chebyshev coefficients is formulated. This theorem is a theoretical justification for numerical solution of ordinary differential equations using Chebyshev series.