Observation of vector solitary waves in soft laminates using a finite volume method

**differential**

**equations**whose analytical solution is seldom trackable, hence emerges the need for suitable numerical solvers. Expand abstract.

**differential**

**equations**whose analytical solution is seldom trackable, hence emerges the need for suitable numerical solvers. Based on a finite volume method in one space dimension, we here develop a designated scheme for nonlinear waves with two coupled components that propagate in soft laminates. We apply our scheme to a periodic laminate made of two alternating compressible Gent layers, and consider two cases. In one case, we analyze a motion whose component along the lamination direction is coupled to a component in the layers plane, to discover vector solitary waves in a continuum medium. In the second case, we analyze a motion with two coupled components in the plane of the layers, and observe the propagation of linearly polarized solitary waves, followed by a single circularly polarized wave. The framework we developed offers a platform for further investigation of these waves and their extension to higher dimensional problems.

4/10 relevant

engrXiv

Pursuit Game for an Infinite System of First-Order **Differential**
**Equations** with Negative Coefficients

**differential**game described by an infinite system of first-order differential

**equations**in Hilbert space. Expand abstract.

**differential**game described by an infinite system of first-order

**differential**

**equations**in Hilbert space. The control functions of players are subject to geometric constraints. The pursuer attempts to bring the system from a given initial state to the origin for a finite time and the evader's purpose is opposite. We obtain a guaranteed pursuit time and construct a strategy for pursuer.

10/10 relevant

arXiv

Convergence of delay **equations** driven by a H\"older continuous function
of order $\beta\in(\frac13,\frac12)$

**differential**

**equations**driven by a H\"older continuous function of order $\beta \in (\frac13,\frac12)$ converges with the supremum norm to the solution for the

**equation**without delay. As an application, we discuss the applications to stochastic

**differential**

**equations**.

5/10 relevant

arXiv

Equilibrium and sensitivity analysis of a spatio-temporal host-vector epidemic model

**differential**

**equations**is complementarily characterized via a theoretical study of its equilibrium states and a numerical study of its transitive phase using global sensitivity analysis. Expand abstract.

**differential**

**equations**is complementarily characterized via a theoretical study of its equilibrium states and a numerical study of its transitive phase using global sensitivity analysis. The results are discussed in terms of implications concerning the surveillance and control of the disease over a medium-to-long temporal horizon.

4/10 relevant

arXiv

On the tridiagonalization of systems of coupled linear **differential**
**equations**

**differential**

**equations**with non-constant coefficients by appealing only to `well-behaved' distributions? Expand abstract.

**differential**

**equations**with non-constant coefficients by appealing only to `well-behaved' distributions? In this work we show that if all the non-constant coefficients are smooth functions then tridiagonalisation is always possible using only piecewise smooth functions and isolated Dirac delta distributions. As a corollary, we formally establish the convergence and good behavior of the recently published Lanczos-like algorithm for solving arbitrary linear

**differential**systems with smooth coefficients via tridiagonalization. This is a key piece in evaluating the hitherto elusively difficult ordered exponential function, both formally and numerically.

10/10 relevant

arXiv

Strong solutions of stochastic **differential** **equations** with coefficients
in mixed-norm spaces

**equation**s in mixed-norm spaces, we prove the existence and uniqueness of strong solutions to stochastic

**differential**

**equations**driven by Brownian motion with coefficients in spaces with mixed-norm, which extends Krylov and R\"ockner's result in [11] and Zhang's result in [18]. Expand abstract.

**equations**in mixed-norm spaces, we prove the existence and uniqueness of strong solutions to stochastic

**differential**

**equations**driven by Brownian motion with coefficients in spaces with mixed-norm, which extends Krylov and R\"ockner's result in [11] and Zhang's result in [18].

10/10 relevant

arXiv

Singular Initial Value Problems for Scalar Quasi-Linear Ordinary
**Differential** **Equations**

**differential**

**equations**where the initial condition corresponds to an impasse point of the

**equation**. With a

**differential**geometric approach, we reduce the problem to questions in dynamical systems theory. As an application, we discuss in detail second-order

**equations**of the form $g(x)u''=f(x,u,u')$ with an initial condition imposed at a simple zero of $g$. This generalises results by Liang and also makes them more transparent via our geometric approach.

10/10 relevant

arXiv

Optimally weighted loss functions for solving PDEs with Neural Networks

**differential**

**equations**, with the number of data points being updated adaptively. Expand abstract.

**differential**equations, giving rise to the framework of physics informed neural networks. We introduce a generalization for these methods that manifests as a scaling parameter which balances the relative importance of the different constraints imposed by partial

**differential**

**equations**. A mathematical motivation of these generalized methods is provided, which shows that for linear and well-posed partial

**differential**equations, the functional form is convex. We then derive a choice for the scaling parameter that is optimal with respect to a measure of relative error. Because this optimal choice relies on having full knowledge of analytical solutions, we also propose a heuristic method to approximate this optimal choice. The proposed methods are compared numerically to the original methods on a variety of model partial

**differential**equations, with the number of data points being updated adaptively. For several problems, including high-dimensional PDEs the proposed methods are shown to significantly enhance accuracy.

7/10 relevant

arXiv

On the asymptotic behavior of solutions to time-fractional elliptic
**equations** driven a multiplicative white noise

**differential**

**equations**, Ito's formula and establishing a new weighted norm associated with a Lyapunov-Perron operator defined from this representation... Expand abstract.

**equations**driven a multiplicative noise. By combining the eigenfunction expansion method for symmetry elliptic operators, the variation of constant formula for strong solutions to scalar stochastic fractional

**differential**equations, Ito's formula and establishing a new weighted norm associated with a Lyapunov-Perron operator defined from this representation of solutions, we show the asymptotic behaviour of solutions to these systems in mean square sense. As a consequence, we also prove existence, uniqueness and the convergence rate of their solutions.

4/10 relevant

arXiv

An invariant subbundle of the KZ connection mod $p$ and reducibility of $\hat{sl}_2$ Verma modules mod $p$

**differential**

**equations**over $\mathbb C$ in the case, when its multidimensional hypergeometric solutions are one-dimensional integrals. We also consider the same

**differential**

**equations**over a finite field $\mathbb F_p$. We study the space of polynomial solutions of these

**differential**

**equations**over $\mathbb F_p$, constructed in a previous work by V. Schechtman and the author. The module of these polynomial solutions defines an invariant subbundle of the associated KZ connection modulo $p$. We describe the algebraic

**equations**for that subbundle and argue that the

**equations**correspond to highest weight vectors of the associated $\hat{sl}_2$ Verma modules over the field $\mathbb F_p$.

7/10 relevant

arXiv