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We present a combinatorial method of constructing solutions to the normal ordering of boson operators. Generalizations of standard combinatorial notions - the Stirling and Bell numbers, Bell polynomials and Dobinski relations - lead to…
A unified general approach is presented for construction of solutions of the characteristic initial value problems for various integrable hyperbolic reductions of Einstein's equations for space-times with two commuting isometries in General…
In this short paper, we elucidate how the solution of certain illustrative boundary value problems for the Airy equation $u_t+u_{xxx}=0$ on $[0,1]$ can be expressed as a perturbation of the solution of the purely periodic problem. The…
We study the initial-boundary value problem for the Fokker-Planck equation in an interval with absorbing boundary conditions. We develop a theory of well-posedness of classical solutions for the problem. We also prove that the resulting…
We provide infinitely many solutions of a Dirichlet problem on balls.
In this paper we use some results related to regularity, Betti numbers and reduction of generic initial ideals, showing their stability in passing from an ideal to its initial ideal if the last has some simple properties.
Like many numerical methods, solvers for initial value problems (IVPs) on ordinary differential equations estimate an analytically intractable quantity, using the results of tractable computations as inputs. This structure is closely…
We consider a boundary value problem in unbounded 2D doubly periodic composite with circular inclusions having arbitrary constant conductivities. By introducing complex potentials, the boundary value problem for the Laplace equation is…
In this work, initial-boundary value problems for the time-fractional Airy equation are considered on different intervals. We study the properties of potentials for this equation and, using these properties, construct solutions to the…
We consider the so-called Gross-Pitaevskii equations supplemented with non-standard boundary conditions. We prove two mathematical results concerned with the initial value problem for these equations in Zhidkov spaces.
In this paper a technique is suggested to integrate linear initial boundary value problems with exponential quadrature rules in such a way that the order in time is as high as possible. A thorough error analysis is given for both the…
We prove the existence and multiplicity of periodic solutions of bouncing type for a second-order differential equation with a weak repulsive singularity. Such solutions can be catalogued according to the minimal period and the number of…
We prove some partial results on the periodicity of billiard systems on graphs. The results specialize to the case of $n$ billiards with equal mass on the unit interval or circle traveling at the same speed.
We consider initial/boundary value problems for time-fractional parabolic PDE of order $0<\alpha<1$ with Caputo fractional derivative (also called fractional diffusion equations in the literature). We prove well-posedness of corresponding…
In this paper, we present an averaging method for obtaining quasi-periodic response solutions in perturbed, real analytic, quasi-periodic systems with Diophantine frequency vectors. Under the assumptions that the averaged system possesses a…
We identify many new solvable subcases of the general dynamical system characterized by two autonomous first-order ordinary differential equations with purely quadratic right-hand sides; the solvable character of these dynamical systems…
The relativistic Vlasov-Maxwell system of plasma physics is considered with initial data on a past light cone. This characteristic initial value problem arises in a natural way as a mathematical framework to study the existence of solutions…
In this note, we show a classical result on the local existence and uniqueness of a solution to an initial value problem subject to a Lipschitz condition. We use only elementary tools from mathematical analysis, without involving any…
This paper presents an iterative scheme that converges to the solution of a pseudo-monotone variational inequality problem in the setting of $\mathbb{R}^{n}$. Traditional methods often require projections onto the feasible set…
Nonlinear control-affine systems described by ordinary differential equations with bounded measurable input functions are considered. The solvability of general boundary value problems for these systems is formulated in the sense of…