Related papers: Approximation of arbitrarily high-order PDEs by fi…
In this paper we focus on the Cahn-Hilliard equation with dynamic boundary conditions, by adding two hyperbolic relaxation terms to the system. We verify that the energy of the total system is decreasing with time. By adding two…
We show that many important natural science models in their mathematical formulation can be reduced to non-strictly hyperbolic systems of the same kind. This allows the same methods to be applied to them so that some essential results…
We consider the parabolic Lam\'{e} system on a bounded domain. We focus on two types of inequalities for higher-order derivatives of solutions. The first is related to an $L^p$-$L^p$ estimate locally in time in the Lebesgue space setting,…
In the first part of planned series of papers the formal general solutions to selection of 80 examples of different types of second order nonlinear PDEs in two independent variables with constant parameters are given. The main goal here is…
We introduce novel high order well-balanced finite volume methods for the full compressible Euler system with gravity source term. They require no a priori knowledge of the hydrostatic solution which is to be well-balanced and are not…
This paper develops validated computational methods for studying infinite dimensional stable manifolds at equilibrium solutions of parabolic PDEs, synthesizing disparate errors resulting from numerical approximation. To construct our…
The scaling of the exact solution of a hyperbolic balance law generates a family of scaled problems in which the source term does not depend on the current solution. These problems are used to construct a sequence of solutions whose…
We give tight upper and lower bounds of the cardinality of the index sets of certain hyperbolic crosses which reflect mixed Sobolev-Korobov-type smoothness and mixed Sobolev-analytic-type smoothness in the infinite-dimensional case where…
We examine robustness of exponential dichotomies of boundary value problems for general linear first-order one-dimensional hyperbolic systems. The boundary conditions are supposed to be of types ensuring smoothing solutions in finite time,…
This paper investigates the quantitative homogenization of first-order ODEs. For single-scale scalar ODEs, we obtain a sharp $O(\varepsilon)$ convergence rate and characterize the effective constant. In the multi-scale setting, our results…
We develop symbolic methods of asymptotic approximations for solutions of linear ordinary differential equations and use to them stabilize numerical calculations. Our method follows classical analysis for first-order systems and…
Motivated by important applications in image processing, we study a class of second-order geometric quasilinear hyperbolic partial differential equations (PDEs). This is inspired by the recent development of second-order damping systems…
High-dimensional partial differential equations (PDEs) are ubiquitous in economics, science and engineering. However, their numerical treatment poses formidable challenges since traditional grid-based methods tend to be frustrated by the…
Geophysical flow simulations using hyperbolic shallow water moment equations require an efficient discretization of a potentially large system of PDEs, the so-called moment system. This calls for tailored model order reduction techniques…
In this paper, it is shown that any well-posed 2nd order PDE can be reformulated as a well-posed first order least squares system. This system will be solved by an adaptive wavelet solver in optimal computational complexity. The…
A backstepping-based compensator design is developed for a system of $2\times2$ first-order linear hyperbolic partial differential equations (PDE) in the presence of an uncertain long input delay at boundary. We introduce a transport PDE to…
A new method for the Lie group classification of differential equations is proposed. It is based of the determination of all possible cases of linear dependence of certain indeterminate appearing in the determining equations of symmetries…
The quasi-periodic homogenization for some classes of first-order Hamilton-Jaconi-Bellman equation is studied in this paper. The cell problem of the quasi-periodic homogenization satisfies the non-resonance condition, under which the…
It was shown recently that the constraints on the initial data for Einstein's equations may be posed as an evolutionary problem [9]. In one of the proposed two methods the constraints can be replaced by a first order symmetrizable…
We present a Bethe approximation to study lattice models of linear polymers. The approach is variational in nature and based on the cluster variation method (CVM). We focus on a model with $(i)$ a nearest neighbor attractive energy…