A note on the Balanced method

Recently the Balanced method was introduced as a class of quasi-implicit methods for solving stiff stochastic differential equations. We examine asymptotic and mean-square stability for several implementations of the Balanced method and give a generalized result for the mean-square stability region of any Balanced method. We also investigate the optimal implementation of the Balanced method with respect to strong convergence.

Three-body recombination of ultracold Bose gases using the truncated Wigner method

We apply the truncated Wigner method to the process of three-body recombination in ultracold Bose gases. We find that within the validity regime of the Wigner truncation for two-body scattering, three-body recombination can be treated using a set of coupled stochastic differential equations that include diffusion terms, and can be simulated using known numerical methods. As an example we investigate the behavior of a simple homogeneous Bose gas, finding a very slight increase of the loss rate compared to that obtained by using the standard method.

A variational method and approximations of a Cauchy problem for elliptic equations

A Cauchy problem for general elliptic second-order linear partial differential equations in which the Dirichlet data in H½(?1 ? ?3) is assumed available on a larger part of the boundary ? of the bounded domain O than the boundary portion ?1 on which the Neumann data is prescribed, is investigated using a conjugate gradient method. We obtain an approximation to the solution of the Cauchy problem by minimizing a certain discrete functional and interpolating using the finite diference or boundary element method. The minimization involves solving equations obtained by discretising mixed boundary value problems for the same operator and its adjoint. It is proved that the solution of the discretised optimization problem converges to the continuous one, as the mesh size tends to zero. Numerical results are presented and discussed.

A Cauchy Integral Related to a Robot-safety Device System

We introduce a robot-safety device system attended by two different repairmen. The twin system is characterized by the natural feature of cold standby and by an admissible “risky” state. In order to analyse the random behaviour of the entire system (robot, safety device, repair facility) we employ a stochastic process endowed with probability measures satisfying general Hokstad-type differential equations. The solution procedure is based on the theory of sectionally holomorphic functions, characterized by a Cauchy-type integral defined as a Cauchy principal value in double sense. An application of the Sokhotski-Plemelj formulae determines the long-run availability of the robot-safety device. Finally, we consider the particular but important case of deterministic repair.

Continuous Dependence of Solutions of Quasidifferential Equations with Non-Fixed Time of Impulses

In this article on quasidifferential equation with non-fixed time of impulses we consider the continuous dependence of the solutions on the initial conditions as well as the mappings defined by these equations. We prove general theorems for quasidifferential equations from which follows corresponding results for differential equations, differential inclusion and equations with Hukuhara derivative.

Inhomogeneous Fractional Diffusion Equations

2000 Mathematics Subject Classification: Primary 26A33; Secondary 35S10, 86A05

On a Differential Equation with Left and Right Fractional Derivatives

Mathematics Subject Classification: 26A33; 70H03, 70H25, 70S05; 49S05

Generalized Convolution Transforms and Toeplitz Plus Hankel Integral Equations

Mathematics Subject Classification: 44A05, 44A35

Some Properties of Mittag-Leffler Functions and Matrix-Variate Analogues: A Statistical Perspective

Mathematical Subject Classification 2010:26A33, 33E99, 15A52, 62E15.

Necessary and Sufficient Condition for Oscillations of Neutral Differential Equation

2000 Mathematics Subject Classification: 34K15, 34C10.

Riemann-Hilbert Problems with Canonical Normalization and Families of Commuting Operators

2010 Mathematics Subject Classification: 35Q15, 31A25, 37K10, 35Q58.

Symbolic Solving of Partial Differential Equation Systems and Compatibility Conditions

An algorithm is produced for the symbolic solving of systems of partial differential equations by means of multivariate Laplace–Carson transform. A system of K equations with M as the greatest order of partial derivatives and right-hand parts of a special type is considered. Initial conditions are input. As a result of a Laplace–Carson transform of the system according to initial condition we obtain an algebraic system of equations. A method to obtain compatibility conditions is discussed.

Solvability of an Infinite System of Singular Integral Equations

2000 Mathematics Subject Classification: 45G15, 26A33, 32A55, 46E15.

An impulsive differential equation model for Marek's disease

Many dynamical processes are subject to abrupt changes in state. Often these perturbations can be periodic and of short duration relative to the evolving process. These types of phenomena are described well by what are referred to as impulsive differential equations, systems of differential equations coupled with discrete mappings in state space. In this thesis we employ impulsive differential equations to model disease transmission within an industrial livestock barn. In particular we focus on the poultry industry and a viral disease of poultry called Marek's disease. This system lends itself well to impulsive differential equations. Entire cohorts of poultry are introduced and removed from a barn concurrently. Additionally, Marek's disease is transmitted indirectly and the viral particles can survive outside the host for weeks. Therefore, depopulating, cleaning, and restocking of the barn are integral factors in modelling disease transmission and can be completely captured by the impulsive component of the model. Our model allows us to investigate how modern broiler farm practices can make disease elimination difficult or impossible to achieve. It also enables us to investigate factors that may contribute to virulence evolution. Our model suggests that by decrease the cohort duration or by decreasing the flock density, Marek's disease can be eliminated from a barn with no increase in cleaning effort. Unfortunately our model also suggests that these practices will lead to disease evolution towards greater virulence. Additionally, our model suggests that if intensive cleaning between cohorts does not rid the barn of disease, it may drive evolution and cause the disease to become more virulent.