相关论文: Implicit Solutions of PDE's
We show that for any uniformly elliptic fully nonlinear second-order equation with bounded measurable "coefficients" and bounded "free" term one can find an approximating equation which has a unique continuous and having the second…
We consider singular perturbations of eigenvalue problems. We prove that to these problems correspond simple eigenvalues and we study their asymptotic behavior. As a result, we prove global bifurcation results for non uniformly and fully…
Conventional finite-difference schemes for solving partial differential equations are based on approximating derivatives by finite-differences. In this work, an alternative theory is proposed which view finite-difference schemes as…
In this short paper we identify special systems of (an arbitrary number) N of first-order Difference Equations with nonlinear homogeneous polynomials of arbitrary degree M in their right-hand sides, which feature very simple explicit…
Nonlinear second-order ordinary differential equations are common in various fields of science, such as physics, mechanics and biology. Here we provide a new family of integrable second-order ordinary differential equations by considering…
Uniformly regular equilibrium problems are natural generalizations of abstract equilibrium prob lems and they are defined over the uniformly prox-regular nonconvex sets. Some new efficient implicit methods for solving uniformly regular…
A broad set of sufficient conditions consisting of systems of linear partial differential equations is presented which guarantees that the Wronskian determinant solves the Korteweg-de Vries equation in the bilinear form. A systematical…
In multi-phase fluid flow, fluid-structure interaction, and other applications, partial differential equations (PDEs) often arise with discontinuous coefficients and singular sources (e.g., Dirac delta functions). These complexities arise…
Nonlinear systems of partial differential equations (PDEs) may permit several distinct solutions. The typical current approach to finding distinct solutions is to start Newton's method with many different initial guesses, hoping to find…
Motivated by the study of systems of higher order boundary value problems with functional boundary conditions, we discuss, by topological methods, the solvability of a fairly general class of systems of perturbed Hammerstein integral…
In this article, using DiPerna-Lions theory \cite{Di-Li}, we investigate linear second order stochastic partial differential equations with unbounded and degenerate non-smooth coefficients, and obtain several conditions for existence and…
The existence of sufficiently many finite order meromorphic solutions of a differential equation, or difference equation, or differential-difference equation, appears to be a good indicator of integrability. In this paper, we investigate…
A kind of spatial fractional diffusion equations in this paper are studied. Firstly, an L1 formula is employed for the spatial discretization of the equations. Then, a second order scheme is derived based on the resulting semi-discrete…
A simple theorem on proportionality of indefinite real quadratic forms is proved, and is used to clarify the proof of the invariance of the interval in Special Relativity from Einstein's postulate on the universality of the speed of light;…
We solve the local equivalence problem for second order (smooth or analytic) ordinary differential equations. We do so by presenting a {\em complete convergent normal form} for this class of ODEs. The normal form is optimal in the sense…
In this article, we study the set of all solutions of linear differential equations using Hurwitz series. We first obtain explicit recursive expressions for solutions of such equations and study the group of differential automorphisms of…
We obtain exact solutions to the class of parabolic partial differential equations of arbitrary dimensionality and with arbitrary potentials. The solutions are presented in a compact-form: as explicit mathematical expressions consisting of…
The Cauchy problem for fractional derivatives linear systems of ordinary differential equations with constant coefficients is considered, where at first the analytic expressions are given through the matrix exponent of its corresponding…
In this paper, we study about existence and non-existence of finite order transcendental entire solutions of the certain non-linear differential-difference equations. We also study about conjectures posed by Rong et al. and Chen et al.
For a general differential system $\dot x(t) = \sum_{d=1}^3 u_d(t)X_d$, where $X_d$ generates the simple Lie algebra of type $\mathfrak{a}_1$, we compute the explicit solution in terms of iterated integrals of products of $u_d$'s. As a…