Related papers: Unveiling the physics of partial differential equa…
A system of first-order differential equations for a particle with nonzero mass and spin $S = 1$ is constructed. As distinct from the Proca-Duffin-Kemmer (PDK) equations, the system has the form of the dynamical equation…
We give simple proofs of hypoelliptic estimates for some models of kinetic equations with a fractional order diffusion part. The proofs are based on energy estimates together with F. Bouchut and B. Perthame previous ideas.
There is a tendency to write the equations of general relativity as a first order symmetric system of time dependent partial differential equations. However, for numerical reasons, it might be advantageous to use a second order formulation…
This paper presents an observation that under reasonable conditions, many partial differential equations from mathematical physics possess three structural properties. One of them can be understand as a variant of the celebrated Onsager…
We propose a natural gradient term for a class of second-order partial differential equations of the form \begin{equation}\nonumber M(x,Du,D^2u)+g(u)N(x,Du, D^2u)+f(x,u)=0 \;\;\mbox{in}\;\; \Omega, \end{equation} where…
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…
The solution in hyperspherical coordinates for $N$ dimensions is given for a general class of partial differential equations of mathematical physics including the Laplace, wave, heat and Helmholtz, Schr\"{o}dinger, Klein-Gordon and…
The main subject of this paper is the study of analytic second order linear partial differential equations. We aim to solve the classical equations and some more, in the real or complex analytical case. This is done by introducing methods…
We overview a series of recent works addressing numerical simulations of partial differential equations in the presence of some elements of randomness. The specific equations manipulated are linear elliptic, and arise in the context of…
The discovery of partial differential equations (PDEs) is a challenging task that involves both theoretical and empirical methods. Machine learning approaches have been developed and used to solve this problem; however, it is important to…
Additive noise in Partial Differential equations, in particular those of fluid mechanics, has relatively natural motivations. The aim of this work is showing that suitable multiscale arguments lead rigorously, from a model of fluid with…
Many physical systems are described by partial differential equations (PDEs). Determinism then requires the Cauchy problem to be well-posed. Even when the Cauchy problem is well-posed for generic Cauchy data, there may exist characteristic…
A long and intense debate in philosophy is concerned with the question whether there can be haecceistic differences between possible worlds, that is, nonqualitative differences that only arise from different de re representations. According…
We describe a way of solving a partial differential equation using the differential invariants of its point symmetries. By first solving its quotient PDE, which is given by the differential syzygies in the algebra of differential…
Almost all theories of physics have expressed physical laws by means of differential equations. One can ask: why differential equations? What is special about them? This article addresses these questions and is presented as an inquiry-based…
The numerical solution of differential equations can be formulated as an inference problem to which formal statistical approaches can be applied. However, nonlinear partial differential equations (PDEs) pose substantial challenges from an…
Constructing the Theory of Everything (TOE) is an elusive goal of today's physics. Goedel's incompleteness theorem seems to forbid physics axiomatization, a necessary part of the TOE. The purpose of this contribution is to show how physics…
The indeterministic character of physical laws is generally considered to be the most important consequence of quantum physics. A deterministic point of view, however, together with the possibility of well defined Hamiltonian trajectories,…
In this paper we create a model of particle motion on a three-dimensional lattice using discrete random walk with small steps. We rigorously construct a probability space of the particle trajectories. Unlike deterministic approach in…
In this article, the axioms presented in the first one are reformulated according to the special theory of relativity. Using these axioms, quantum mechanic's relativistic equations are obtained in the presence of electromagnetic fields for…