Related papers: A Symmetry-Based Method for Constructing Nonlocall…
A general sufficient condition for the convergence of subsequences of solutions of non-autonomous, nonlinear difference equations and systems is obtained. For higher order equations the delay sizes and patterns play essential roles in…
Solvable structures, likewise solvable algebras of local symmetries, can be used to integrate scalar ODEs by quadratures. Solvable structures, however, are particularly suitable for the integration of ODEs with a lack of local symmetries.…
In this article, we consider an interesting class of optical and other systems in which the interaction or coupling makes the systems to be $\cal{PT}$-symmetric. We aim to compare their dynamical behaviors with that of the usual $\cal{PT}$…
Some systems of parabolic equations with nonlocal initial conditions are studied. The systems arise when considering steady-state solutions to diffusive age-structured cooperative or competing species. Local and global bifurcation…
In this paper, we consider the inverse problem of determining some coefficients within a coupled nonlinear parabolic system, through boundary observation of its non-negative solutions. In the physical setup, the non-negative solutions…
The concept of the derivative-dependent functional separable solution, as a generalization to the functional separable solution, is proposed. As an application, it is used to discuss the generalized nonlinear diffusion equations based on…
In this paper, we consider a new nonlocal approximation to the linear Stokes system with periodic boundary conditions in two and three dimensional spaces . A relaxation term is added to the equation of nonlocal divergence free equation,…
We discuss numerical aspects related to a new class of nonlinear Stochastic Differential Equations in the sense of McKean, which are supposed to represent non conservative nonlinear Partial Differential equations (PDEs). We propose an…
There has been an arising trend of adopting deep learning methods to study partial differential equations (PDEs). In this paper, we introduce a deep recurrent framework for solving time-dependent PDEs without generating large scale data…
Stochastic solutions provide new rigorous results for nonlinear PDE's and, through its local non-grid nature, are a natural tool for parallel computation. There are two different approaches for the construction of stochastic solutions:…
We apply the semi-discrete method, c.f. \emph{N. Halidias and I.S. Stamatiou (2016), On the numerical solution of some non-linear stochastic differential equations using the semi-discrete method, Computational Methods in Applied…
Many important applications are available for nonlinear reaction-diffusion equation especially in the area of biology and engineering. Therefore a mathematical model for Lie symmetry reduction of system of nonlinear reaction-diffusion…
A nonlocal perfectly matched layer (PML) is formulated for the nonlocal wave equation in the whole real axis and numerical discretization is designed for solving the reduced PML problem on a bounded domain. The nonlocal PML poses challenges…
In this paper, symmetry analysis is extended to study nonlocal differential equations, in particular two integrable nonlocal equations, the nonlocal nonlinear Schr\"odinger equation and the nonlocal modified Korteweg--de Vries equation. Lie…
We consider a system of semi-linear partial differential equations with measurable coefficients and a nonlinear Neumann boundary condition. We then construct a sequence of penalized partial differential equations which converges to a…
We derive novel algorithms for optimization problems constrained by partial differential equations describing multiscale particle dynamics, including non-local integral terms representing interactions between particles. In particular, we…
Every linear system of partial differential equations (PDEs) admits a scaling symmetry in its dependent variables. In conjunction with other admitted symmetries of linear type, the associated invariant solution condition poses a linear…
Nonlinear differential equations (DEs) are used in a wide range of scientific problems to model complex dynamic systems. The differential equations often contain unknown parameters that are of scientific interest, which have to be estimated…
Based on nonlocal symmetry method, localized excitations and interactional solutions are investigated for the reduced Maxwell-Bloch equations. The nonlocal symmetries of the reduced Maxwell-Bloch equations are obtained by the truncated…
A direct method for the computation of polynomial conservation laws of polynomial systems of nonlinear partial differential equations (PDEs) in multi-dimensions is presented. The method avoids advanced differential-geometric tools. Instead,…