The purpose of this note is to give a necessary and sufficient condition for the coexistence of positive solutions to a rather general type of elliptic predator-prey system of the Dirichlet problem on the bounded domain omega when omega is a subset of Rn is large. The result is that the partial differential equation system possesses positive coexistence if and only if the corresponding ordinary...

We put a direct new method to construct the rational Jacobi elliptic solutions for nonlinear differential difference equations which may be called the rational Jacobi elliptic functions method. We use the rational Jacobi elliptic function method to construct many new exact solutions for some nonlinear differential difference equations in mathematical physics via the lattice equation and the dis...

Reaction–diffusion systems are used to model many chemical and biological phenomena in the natural world [25, 26], and systems of coupled partial differential equations are also used in other physical models such as nonlinear Schrödinger systems in multi-component Bose–Einstein condensates and nonlinear optics [19, 23]. The steady-state solutions or standing-wave solutions of such systems of no...

For a second-order particle system in Rd subject to locally-in-space pairwise annihilation, we prove scaling limit for its empirical measure on position and velocity towards degenerate elliptic partial differential equation. Crucial ingredients are Green’s function estimates the associated hypoelliptic operator an Itô-Tanaka trick.

the numerical methods are of great importance for approximating the solutions of nonlinear ordinary or partial differential equations, especially when the nonlinear differential equation under consideration faces difficulties in obtaining its exact solution. in this latter case, we usually resort to one of the efficient numerical methods. in this paper, the chebyshev collocation method is sugge...

An abstract robust multilevel method for solving symmetric positive definite systems resulting from discretizing elliptic partial differential equations is developed. The term “robust” refers to the convergence rate of the method being independent of discretization parameters, i.e., the problem size, and problem parameters. Important instances of such problem parameters are in particular (highl...

Elliptic partial differential equations are important both from application and analysis points of views. In this paper we apply the Closest Point Method to solving elliptic equations on general curved surfaces. Based on the closest point representation of the underlying surface, we formulate an embedding equation for the surface elliptic problem, then discretize it using standard finite differ...

Abstract The current work focuses on the solutions of Kadomtsev and Petviashvili (KP) equation, which models nonlinear waves in a dispersive medium. modified auxiliary equation approach is utilized to find analytical KP equation. Consequently, set including Jacobi elliptic solitary periodic obtained. geometry derived plotted with an appropriate choice parameters. It can be seen that proposed me...

In this article we define an Eulerian level set method for a partial differential equation on an evolving hypersurface Γ (t) contained in a domain Ω ⊂ R. The key idea is based on formulating the partial differential equations on all level set surfaces of a prescribed time dependent function Φ whose zero level set is Γ (t). The resulting equation is then solved in one dimension higher but can be...

An abstract robust multilevel method for solving symmetric positive definite systems resulting from discretizing elliptic partial differential equations is developed. The term “robust” refers to the convergence rate of the method being independent of discretization parameters, i.e., the problem size, and problem parameters. Important instances of such problem parameters are in particular (highl...