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We reconsider the theory of scattering for some long range Hartree equations with potential |x|^-gamma with 1/2 < gamma < 1. More precisely we study the local Cauchy problem with infinite initial time, which is the main step in the…
In this paper, we study a fast and linearized finite difference method to solve the nonlinear time-fractional wave equation with multi fractional orders. We first propose a discretization to the multi-term Caputo derivative based on the…
We present several second-order linear differential equations that are associated to a particular Riccati equation with only one constant parameter in its coefficients through the technique of supersymmetric factorizations and through a…
Stable computational algorithms for the approximate solution of the Cauchy problem for nonstationary problems are based on implicit time approximations. Computational costs for boundary value problems for systems of coupled multidimensional…
We employ weak hypocoercivity methods to study the long-term behavior of operator semigroups generated by degenerate Kolmogorov operators with variable second-order coefficients, which solve the associated abstract Cauchy problem. We prove…
Given a Hilbert space, we investigate the well-posedness of the Cauchy problem for the wave equation for operators with discrete non-negative spectrum acting on it. We consider the cases when the time-dependent propagation speed is regular,…
We study first-order symmetrizable hyperbolic $N\times N$ systems in a spacetime cylinder whose lateral boundary is totally characteristic. In local coordinates near the boundary at $x=0$, these systems take the form \[ \partial_t u +…
This paper extends the model reduction method by the operator projection to the one-dimensional special relativistic Boltzmann equation. The derivation of arbitrary order globally hyperbolic moment system is built on our careful study of…
We define a class of pseudo-differential operators in a completely new way, which is called the abstract operators and expounded systematically the theory of abstract operators. By combining abstract operators with the Laplace transform, we…
This paper is devoted to the homogenization of weakly coupled cooperative parabolic systems in strong convection regime with purely periodic coefficients. Our approach is to factor out oscillations from the solution via principal…
A problem of a wave identification is formulated. An example is considered in conditions of one-dimensional Cauchy problem for conventional string equation in matrix form and its inhomogeneous two-component version. The acoustic and…
We study a class of Tricomi-type partial differential equations previously investigated in [28]. Firstly, we generalize the representation formula for the solution obtained there by allowing the coefficient in front of the second-order…
In this article, we prove the convergence of a semi-discrete numerical method applied to a general class of nonlocal nonlinear wave equations where the nonlocality is introduced through the convolution operator in space. The most important…
We study a class of hyperbolic Cauchy problems, associated with linear operators and systems with polynomially bounded coefficients, variable multiplicities and involutive characteristics, globally defined on R^n. We prove well-posedness in…
We consider a wave equation with a nonlocal logarithmic damping depending on a small parameter $\theta \in (0,1/2)$. This research is a counter part of that was initiated by Charao-D'Abbicco-Ikehata considered in [5] for the large parameter…
In this paper, we study higher order hyperbolic pseudo-differential equations with variable multiplicities. We work in arbitrary space dimension and we assume that the principal part is time-dependent only. We identify sufficient conditions…
In this article we describe the novel method to construct fundamental solutions for operators with variable coefficients. That method was introduced in "A note on the fundamental solution for the Tricomi-type equation in the hyperbolic…
We consider an abstract second order linear equation with a strong dissipation, namely a friction term which depends on a power of the "elastic" operator. In the homogeneous case, we investigate the phase spaces in which the initial value…
Schemes with the second-order approximation in time are considered for numerical solving the Cauchy problem for an evolutionary equation of first order with a self-adjoint operator. The implicit two-level scheme based on the Pad\'{e}…
We propose a generalized finite element method for the strongly damped wave equation with highly varying coefficients. The proposed method is based on the localized orthogonal decomposition introduced and is designed to handle independent…