Related papers: Representation of continuum equations in physical …
In continuum mechanics, the equations of motion for mixtures are derived through the use of Hamilton's extended principle which regards the mixture as a collection of distinct continua. The internal energy is assumed to be a function of…
Random invariant manifolds are geometric objects useful for understanding complex dynamics under stochastic influences. Under a nonuniform hyperbolicity or a nonuniform exponential dichotomy condition, the existence of random pseudo-stable…
We formulate stochastic partial differential equations on Riemannian manifolds, moving surfaces, general evolving Riemannian manifolds (with appropriate assumptions) and Riemannian manifolds with random metrics, in the variational setting…
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…
Three types of equations of mathematical physics, namely, the equations, which describe any physical processes, the equations of mechanics and physics of continuous media, and field-theory equations are studied in this paper. In the first…
In recent literature several derivations of incompressible Navier-Stokes type equations that model the dynamics of an evolving fluidic surface have been presented. These derivations differ in the physical principles used in the modeling…
Complexes and cohomology, traditionally central to topology, have emerged as fundamental tools across applied mathematics and the sciences. This survey explores their roles in diverse areas, from partial differential equations and continuum…
Using classical differential geometry, the problem of elastic curves and surfaces in the presence of long-range interactions $\Phi$, is posed. Starting from a variational principle, the balance of elastic forces and the corresponding…
Exterior calculus and moving frames are used to describe curved elastic shells. The kinematics follow from the Lie-derivative on forms whereas the dynamics via stress-forms.
Consider briefly the equations of fluid dynamics-they describe the enormous wealth of detail in all the interacting physical elements of a fluid flow-whereas in applications we want to deal with a description of just that which is…
It is pointed out that the current form of extrinsic equation of motion for a particle constrained to remain on a hypersurface is in fact a half-finished version for it is established without regard to the fact that the particle can never…
Continuum solvation methods can provide an accurate and inexpensive embedding of quantum simulations in liquid or complex dielectric environments. Notwithstanding a long history and manifold applications to isolated systems in open boundary…
In this paper, we discuss a general framework for multicontinuum homogenization. Multicontinuum models are widely used in many applications and some derivations for these models are established. In these models, several macroscopic…
A large number of powerful, high-quality, and open-source simulation packages exist to efficiently perform molecular dynamics simulations, and their prevalence has greatly accelerated discoveries across a wide range of scientific domains.…
We assign some kind of invariant manifolds to a given integrable PDE (its discrete or semi-discrete variant). First, we linearize the equation around its arbitrary solution $u$. Then we construct a differential (respectively, difference)…
We obtain an exact strain consistency equation for full, elastic, and plastic strain characteristics that have a clear physical meaning and are naturally related to stresses. The dynamic equations are represented in a form that does not use…
The role of mathematical models in physics has for longer been well established. The issue of their proper building and use appears to be less clear. Examples in this regard from relativity and quantum mechanics are mentioned. Comments…
The paper gives an introduction to rate equations in nonlinear continuum mechanics which should obey specific transformation rules. Emphasis is placed on the geometrical nature of the operations involved in order to clarify the different…
Some of recent developments, including recent results, ideas, techniques, and approaches, in the study of degenerate partial differential equations are surveyed and analyzed. Several examples of nonlinear degenerate, even mixed, partial…
Exact equations describing flexoelectric deformation in solids, derived previously within the framework of a continuum media theory, are partial differential equations of the fourth order. They are too complex to be used in the cases…