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We revisit the theory of first-order quasilinear systems with diagonalizable principal part and only real eigenvalues, what is commonly referred to as strongly hyperbolic systems. We provide a self-contained and simple proof of local…
We prove some sharp regularity results for solutions of classical first order hyperbolic initial boundary value problems. Our two main improvements on the existing litterature are weaker regularity assumptions for the boundary data and…
We find a one-parameter family of variables which recast the 3+1 Einstein equations into first-order symmetric-hyperbolic form for any fixed choice of gauge. Hyperbolicity considerations lead us to a redefinition of the lapse in terms of an…
Nonlinear second-order ordinary differential equations are common in various fields of science, such as physics, mechanics and biology. Here we provide a new family of integrable second-order ordinary differential equations by considering…
We study a parabolic initial-boundary-value problem for a system of two differential equations with two boundary conditions of different orders, the Dirichlet and Neumann ones. It occurs specifically in the heat-mass transfer theory. We…
A class of second-order differential equations commonly arising in physics applications are considered, and their explicit hypergeometric solutions are provided. Further, the relationship with the Generalized and Universal Associated…
We consider hyperbolic equations with time-dependent coefficients and develop an abstract framework to derive the asymptotic behaviour of the representation of solutions for large times. We are dealing with generic situations where the…
An extension of the finite and infinite Lie groups properties of complex numbers and functions of complex variable is proposed. This extension is performed exploiting hypercomplex number systems that follow the elementary algebra rules. In…
A new formulation of boundary value problems in gradient elasticity is presented in this work. The main outcome is the construction of partial differential systems of second order, which are typically equivalent with the well known fourth…
In recent work, we used pseudo-differential theory to establish conditions that the initial-boundary value problem for second order systems of wave equations be strongly well-posed in a generalized sense. The applications included the…
In this paper, we establish the second order estimates of solutions to the first initial-boundary value problem for general Hessian type fully nonlinear parabolic equations on Riemannian manifolds. The techniques used in this article can…
In this paper we study the finite element approximation of systems of second-order nonlinear hyperbolic equations. The proposed numerical method combines a $hp$-version discontinuous Galerkin finite element approximation in the time…
We study a scalar hyperbolic partial differential equation with non-linear terms similar to those of the equations of general relativity. The equation has a number of non-trivial analytical solutions whose existence rely on a delicate…
We consider a family of linear singularly perturbed PDE relying on a complex perturbation parameter $\epsilon$. As in a former study of the authors (A. Lastra, S. Malek, Parametric Gevrey asymptotics for some nonlinear initial value Cauchy…
We prove global second-order regularity for a class of quasilinear elliptic equations, both with homogeneous Dirichlet and Neumann boundary conditions. A condition on the integrability of the second fundamental form on the boundary of the…
This is the author Master's Thesis and its main purpose is to demonstrate that it is possible to formulate Einstein's field equations as an initial value problem. The first chapter concerns the hyperbolic equations theory. The definition of…
We present the growing mode solutions of cosmological perturbations to the second order in the matter dominated era. We also present several gauge-invariant combinations of perturbation variables to the second order in most general fluid…
We establish the second-order moment asymptotics for a parabolic Anderson model $\partial_{t}u=(\Delta+\xi)u$ in the hyperbolic space with a regular, stationary Gaussian potential $\xi$. It turns out that the growth and fluctuation…
This is the first in a series of articles on the numerical solution of Friedrich's conformal field equations for Einstein's theory of gravity. We will discuss in this paper why one should be interested in applying the conformal method to…
We propose an elegant formulation of parafermionic algebra and parasupersymmetry of arbitrary order in quantum many-body systems without recourse to any specific matrix representation of parafermionic operators and any kind of deformed…