Related papers: Volterra chain and Catalan numbers
We study, in the periodic setting, the well-posedness of the Cauchy problem associated to the operator $P(t, D_{x}, D_{t}) = D_{t} - a_{2}(t) \Delta_{x} + \sum_{j = 1}^{N} a_{1, j}(t) D_{x_{j}} + a_{0}(t)$, with $T> 0$, $t \in [0, T]$ and…
We consider an hierarchy of integrable 1+2-dimensional equations related to Lie algebra of the vector fields on the line. The solutions in quadratures are constructed depending on $n$ arbitrary functions of one argument. The most…
A Catalan word $w$ is said to be flattened if the subsequence of $w$ obtained by taking the first letter of each weakly increasing run is nondecreasing. Let $\mathcal{F}_n$ denote the set of flattened Catalan words of length $n$, which has…
We consider the following Cauchy problem for weakly coupled systems of semi-linear damped elastic waves with a power source non-linearity in three-dimensions: \begin{equation*} U_{tt}-a^2\Delta U-\big(b^2-a^2\big)\nabla\text{div }…
We give a short proof, that can be used in an introductory real analysis course, that if a function that is defined on the set of real numbers is continuous on a countable dense set, then it is continuous on an uncountable set. This is done…
The main solutions in sense of Kantorovich of nonlinear Volterra operator-integral equations are constructed. Convergence of the successive approximations is established through studies of majorant integral and majorant algebraic equations.…
In this work I consider the abstract Cauchy problems with Caputo fractional time derivative of order $\alpha\in(0,1]$, and discuss the continuity of the respective solutions regarding the parameter $\alpha$. I also present a study about the…
The inhomogenous time-fractional telegraph equation with Caputo derevatives with constant coefficients is considered. For considered equation the general representation of regular solution in rectangular domain is obtained, and the…
We prove the existence of weak solutions for distribution-dependent stochastic Volterra equations under linear growth and continuity conditions on the coefficients and mild regularity assumptions on the kernels, including singular kernels.…
We consider the Cauchy problem for the isentropic compressible Euler-Maxwell equations under general pressure laws in a three-dimensional periodic domain. For any smooth initial electron density away from the vacuum and smooth…
This paper is the continuation of the work "On an inverse problem for finite-difference operators of second order". We consider the Cauchy problem for the Toda lattice in the case when the corresponding L-operator is a Jacobi matrix with…
We consider an one-dimensional inhomogeneous harmonic chain consisting of two different semi-infinite chains of harmonic oscillators. We study the Cauchy problem with random initial data. Under some restrictions on the interaction between…
We study the Cauchy problem for a multidimensional scalar conservation law on the Bohr compactification of $\R^n$. The existence and uniqueness of entropy solutions are established in the general case of merely continuous flux vector. We…
We are concerned with the global existence of entropy solutions of the two-dimensional steady Euler equations for an ideal gas, which undergoes a one-step exothermic chemical reaction under the Arrhenius-type kinetics. The reaction rate…
We consider the Cauchy problem for a $n\times n$ strictly hyperbolic system of balance laws $$ \{{array}{c} u_t+f(u)_x=g(x,u), x \in \mathbb{R}, t>0 u(0,.)=u_o \in L^1 \cap BV(\mathbb{R}; \mathbb{R}^n), | \lambda_i(u)| \geq c > 0 {for all}…
The starting point of analysis of state space models is investigating existence, uniqueness and solution properties such as the semigroup property, and various formulas for the solutions. Several concepts such as the state transition…
Motivated by representation theory we exhibit an interior structure to Catalan sequences and many generalisations thereof. Certain of these coincide with well known (but heretofore isolated) structures. The remainder are new.
We investigate nonlinear stochastic Volterra equations in space and time that are driven by L\'evy bases. Under a Lipschitz condition on the nonlinear term, we give existence and uniqueness criteria in weighted function spaces that depend…
A way to obtain the series solutions of the 1 + 2 dimensional continuous Toda chain is presented.
We consider the random Cauchy problem for the fully nonlocal telegraph equation of power type with the general $(\mathcal{PC}^{\ast})$ type kernel $(a,b)$. This equation can effectively characterize high-frequency signal transmission in…