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We are concerned with fully nonlinear possibly degenerate elliptic partial differential equations (PDEs) with superlinear terms with respect to $Du$. We prove several comparison principles among viscosity solutions which may be unbounded…
In this paper, we construct and analyse the symmetries and conservation laws (conserved densities) of a model of a nonlinear Scrodinger equation with PT-symmetric potentials and inhomogeneity.
We expand a partial difference equation (P$\Delta$E) on multiple lattices and obtain the P$\Delta$E which governs its far field behaviour. The perturbative--reductive approach is here performed on well known nonlinear P$\Delta$Es, both…
We present a short review of the evolution of the methodology of the Method of simplest equation for obtaining exact particular solutions of nonlinear partial differential equations (NPDEs) and the recent extension of a version of this…
Large deviations of conservative interacting particle systems, such as the zero range process, about their hydrodynamic limit and their respective rate functions lead to the analysis of the skeleton equation; a degenerate…
In this work, we have presented a simple analytical approximation scheme for generic non-linear FBSDEs. By treating the interested system as the linear decoupled FBSDE perturbed with non-linear generator and feedback terms, we have shown…
In this paper, we mainly study the integrability of 1+1 dimensional dispersive long-wave equation. Firstly, the Lie symmetry analysis of the equation is carried out in the first part. And the optimal system of the equation is obtained…
This paper develops methods for simplifying systems of partial differential equations that have families of conservation laws which depend on functions of the independent or dependent variables. In some cases, such methods can be combined…
We present a contribution to the field of system identification of partial differential equations (PDEs), with emphasis on discerning between competing mathematical models of pattern-forming physics. The motivation comes from developmental…
An explicit formula to find symmetry recursion operators for partial differential equations (PDEs) is obtained from new results connecting variational integrating factors and non-variational symmetries. The formula is special case of a…
We show that the so-called hidden potential symmetries considered in a recent paper [Gandarias M., Physica A, 2008, V.387, 2234-2242] are ordinary potential symmetries that can be obtained using the method introduced by Bluman and…
We study nonlocal conservation laws with a discontinuous flux function of regularity $\mathsf{L}^{\infty}(\mathbb{R})$ in the spatial variable and show existence and uniqueness of weak solutions in…
Physics Informed Neural Networks is a numerical method which uses neural networks to approximate solutions of partial differential equations. It has received a lot of attention and is currently used in numerous physical and engineering…
In recent work, we have proposed a theory for the derivation of an exact nonlinear dispersion relation for elastic wave propagation which here we consider for a thin rod (linearly nondispersive) and a thick rod (linearly dispersive). The…
In this paper we prove that the unique entropy solution to a scalar nonlinear conservation law with strictly monotone velocity and nonnegative initial condition can be rigorously obtained as the large particle limit of a microscopic…
We show for a variety of classes of conservative PDEs that discrete gradient methods designed to have a conserved quantity (here called energy) also have a time-discrete conservation law. The discrete conservation law has the same conserved…
This work deals with the exponential stabilization of a system of three semilinear parabolic partial differential equations (PDEs), written in a strict feedforward form. The diffusion coefficients are considered distinct and the PDEs are…
This article presents an identification methodology to capture general relationships, with application to piecewise nonlinear approximations of model predictive control for constrained (non)linear systems. The mathematical formulation…
Motivated by the mathematics literature on the algebraic properties of so-called polynomial vector flows, we propose a technique for approximating nonlinear differential equations by linear differential equations. Although the idea of…
Noether's theorem connects symmetries to invariants in continuous systems, however its extension to discrete systems has remained elusive. Recognizing the lowest-order finite difference as the foundation of local continuity, a viable method…