相关论文: Path Integral Solution of Linear Second Order Part…

In this paper we are concerned with the initial boundary value problems of linear and semi-linear parabolic equations with mixed boundary conditions on non-cylindrical domains in spatial-temporal space. We obtain the existence of a weak…

An approach to infinite dimensional integration which unifies the case of oscillatory integrals and the case of probabilistic type integrals is presented. It provides a truly infinite dimensional construction of integrals as linear…

By means of topological methods, we provide new results on the existence, non-existence, localization and multiplicity of nontrivial solutions for systems of perturbed Hammerstein integral equations. In order to illustrate our theoretical…

We derive level set version of partial uniform ellipticity for symmetric concave functions. This suggests an effective approach to investigate second order fully nonlinear equations of elliptic and parabolic type.

Time-periodic solutions to partial differential equations of parabolic type corresponding to an operator that is elliptic in the sense of Agmon-Douglis-Nirenberg are investigated. In the whole- and half-space case we construct an explicit…

We establish the solvability of second order divergence type parabolic systems in Sobolev spaces. The leading coefficients are assumed to be only measurable in one spatial direction on each small parabolic cylinder with the spatial…

Path integral solutions with kinetic coupling potentials $\propto p_1p_2$ are evaluated. As examples I give a Morse oscillator, i.e., a model in molecular physics, and the double pendulum in the harmonic approximation. The former is solved…

It is shown that the non-homogeneous Dirichlet and Neuman problems for the $2^{nd}$-order Seiberg-Witten equation admit a regular solution once the $\mathcal{H}$-condition (described in the article) is satisfied. The approach consist in…

The Dirichlet-Neumann method is a common domain decomposition method for nonoverlapping domain decomposition and the method has been studied extensively for linear elliptic equations. However, for nonlinear elliptic equations, there are…

This paper is devoted to prove the existence of positive solutions of a second order differential equation with a nonhomogeneous Dirichlet conditions given by a parameter dependence integral. The studied problem is a nonlocal perturbation…

We define here the category of partial differential equations. Special cases of morphisms from an object (equation) are symmetries of the equation and reductions of the equation by a symmetry groups, but there are many other morphisms. We…

We introduce a framework for solving a class of parabolic partial differential equations on triangle mesh surfaces, including the Hamilton-Jacobi equation and the Fokker-Planck equation. PDE in this class often have nonlinear or stiff terms…

In this work, we develop an efficient solver based on neural networks for second-order elliptic equations with variable coefficients and singular sources. This class of problems covers general point sources, line sources and the combination…

We investigate existence and uniqueness of bounded solutions of parabolic equations with unbounded coefficients in $M\times \mathbb R_+$, where $M$ is a complete noncompact Riemannian manifold. Under specific assumptions, we establish…

We introduce a hybrid Cole-Hopf-Darboux transformation to relate solutions of nonlinear and linear second order differential equations and derive a sufficient condition for this correspondence. In particular we show that solutions of some…

We present two algorithms for computing hypergeometric solutions of second order linear differential operators with rational function coefficients. Our first algorithm searches for solutions of the form \[ \exp(\int r \,…

Using exhaustion method and finite differences a new method to solve system of partial differential equations and is presented. This method allows design algorithm to solve linear and nonlinear systems in irregular domains. Applying this…

A multi-cube method is developed for solving systems of elliptic and hyperbolic partial differential equations numerically on manifolds with arbitrary spatial topologies. It is shown that any three-dimensional manifold can be represented as…

This paper introduces a convenient solution space for the uniformly elliptic fully nonlinear path dependent PDEs. It provides a wellposedness result under standard Lipschitz-type assumptions on the nonlinearity and an additional assumption…

For a linear, strictly elliptic second order differential operator in divergence form with bounded, measurable coefficients on a Lipschitz domain $\Omega$ we show that solutions of the corresponding elliptic problem with Robin and thus in…