Abstract:
We investigate the existence of weak solutions to a class of quasilinear elliptic equations with nonlinear Neumann boundary conditions in exterior domains. Problems of this kind arise in various areas of science and technology. An important model case related to the initial data problem in general relativity is presented. As an application of our main result, we deduce the existence of the conformal factor for the Hamiltonian constraint in general relativity in the presence of multiple black holes. We also give a proof for uniqueness in this case.

Abstract:
This article concerns the nonlinear evolution equation $$displaylines{ frac{du(t)}{dt} in A(t)u(t), quad 0 leq s < t < T, cr u(s) = u_0 }$$ in a real Banach space X, where the nonlinear, time-dependent, and multi-valued operator $ A(t) : D(A(t)) subset X o X$ has a time-dependent domain D(A(t)). It will be shown that, under certain assumptions on A(t), the equation has a strong solution. Illustrations are given of solving quasi-linear partial differential equations of parabolic type with time-dependent boundary conditions. Those partial differential equations are studied to a large extent.

Abstract:
We study the asymptotic behavior of positive solutions $u$ of $$ -Delta_p u(x) = V(x) u(x)^{p-1}, quad p>1; x in Omega,$$ and related partial differential inequalities, as well as conditions for existence of such solutions. Here, $Omega$ contains the exterior of a ball in $mathbb{R}^N$ $1 Keywords p-Laplacian --- Riccati --- uncertainty principle.

Abstract:
A system of coupled nonlinear partial differential equations with convective and dispersive terms was modified from Boussinesq-type equations. Through a special formulation, a system of nonlinear partial differential equations was solved alternately and explicitly in time without linearizing the nonlinearity. Coupled compact schemes of sixth order accuracy in space were developed to obtain numerical solutions. Within couple compact schemes, variables and their first and second derivatives were solved altogether. The sixth order accuracy in space is achieved with a memory-saving arrangement of state variables so that the linear system is banded instead of blocked. This facilitates solving very large systems. The efficiency, simplicity, and accuracy make this coupled compact method viable as variational and weighted residual methods. Results were compared with exact solutions which were obtained via devised forcing terms. Error analyses were carried out, and the sixth order convergence in space and second order convergence in time were demonstrated. Long time integration was also studied to show stability and error convergence rates. 1. Introduction Compact difference methods, essentially the implicit versions of finite different methods, are superior to the explicit versions in achieving high order accuracy. High order compact methods directly approximate all derivatives with high accuracy without any variational operation or projection. They are very effective for complex and strongly nonlinear partial differential equations, especially when variational methods are inconvenient to implement. They have smaller stencils than explicit finite difference methods while maintaining high order accuracy. Much development has been achieved in finite difference methods. A large number of papers related to various finite difference algorithms have been published over the past decades. Kreiss [1] was among the first who pioneered the research on compact implicit methods. Hirsh [2] developed and applied a 4th order compact scheme to Burgers’ equation in fluid mechanics. Adam [3] developed 4th order compact schemes for parabolic equations on uniform grids. Hoffman [4] studied the truncation errors of the centered finite difference method on both uniform and nonuniform grids and discussed the accuracy of these schemes after applying grid transformations. Dennis and Hudson [5] studied a 4th order compact scheme to approximate Navier-Stokes type operators. Rai and Moin [6] presented several finite difference solutions for an incompressible turbulence channel flow. They

Abstract:
The compact explicit expressions for formal exact operator solutions to Cauchy problem for sufficiently general systems of nonlinear differential equations (ODEs and PDEs) in the form of chronological operator exponents are given. The variant of exact solutions in the form of ordinary (without chronologization) operator exponents are proposed.

Abstract:
Second initial boundary problem in narrow domains of width $\epsilon\ll 1$ for linear second order differential equations with nonlinear boundary conditions is considered in this paper. Using probabilistic methods we show that the solution of such a problem converges as $\epsilon \downarrow 0$ to the solution of a standard reaction-diffusion equation in a domain of reduced dimension. This reduction allows to obtain some results concerning wave front propagation in narrow domains. In particular, we describe conditions leading to jumps of the wave front.

Abstract:
We use the scale of Besov spaces B^\alpha_{\tau,\tau}(O), \alpha>0, 1/\tau=\alpha/d+1/p, p fixed, to study the spatial regularity of the solutions of linear parabolic stochastic partial differential equations on bounded Lipschitz domains O\subset R^d. The Besov smoothness determines the order of convergence that can be achieved by nonlinear approximation schemes. The proofs are based on a combination of weighted Sobolev estimates and characterizations of Besov spaces by wavelet expansions.

Abstract:
New problem is considered that is to find nonlinear differential equations with special solutions. Method is presented to construct nonlinear ordinary differential equations with exact solution. Crucial step to the method is the assumption that nonlinear differential equations have exact solution which is general solution of the simplest integrable equation. The Riccati equation is shown to be a building block to find a lot of nonlinear differential equations with exact solutions. Nonlinear differential equations of the second, third and fourth order with special solutions are given. Most of these equations are used at the description of processes in physics and in theory of nonlinear waves.

Abstract:
In this article, we consider the setting of single-valued, smoothly varying directions of reflection and non-smooth time-dependent domains whose boundary is H\"{o}lder continuous in time. In this setting, we prove existence and uniqueness of strong solutions to stochastic differential equations with oblique reflection. In the same setting, we also prove, using the theory of viscosity solutions, a comparison principle for viscosity solutions to fully nonlinear second-order parabolic partial differential equations and, as a consequence, we obtain existence and uniqueness for this class of equations as well. Our results are generalizations of two articles by Dupuis and Ishii to the setting of time-dependent domains.