We describe the rif algorithm which uses a nite number of diierentiations and algebraic operations to simplify analytic nonlinear systems of partial diierential equations to what we call reduced involutive form. This form includes the integrability conditions of the system and satisses a constant rank condition. The algorithm is useful for classifying initial value problems for determined pde s...

We consider a system of semilinear partial differential equations (PDEs) with measurable coefficients and nonlinear Neumann boundary condition. then construct sequence penalized PDEs, which converges to our initial problem. Since the we may be discontinuous, use notion solution in [Formula: see text]-viscosity sense. The method is based on backward stochastic their S-tightness. This work motiva...

Whether integrable, partially integrable or nonintegrable, nonlinear partial differential equations (PDEs) can be handled from scratch with essentially the same toolbox, when one looks for analytic solutions in closed form. The basic tool is the appropriate use of the singularities of the solutions, and this can be done without knowing these solutions in advance. Since the elaboration of the si...

We present a control design method for nonlinear partial differential equations (PDEs) based on a combination of gain scheduling and backstepping theory for linear PDEs. A benchmark first-order hyperbolic system with an in-domain nonlinearity is considered first. For this system a nonlinear feedback law, based on gain scheduling, is derived explicitly, and a proof of local exponential stability...

Output least-squares formulation with Tikhonov regularization is one of the most frequently used and reliable methods for parameter identification in PDEs. As the discretized optimization system of the formulation is nonlinear, illconditioned, and constrained with some PDEs, its numerical solution is very time-consuming and often unstable. In this paper, we present a multilevel model correction...

The paper deals with three different Newton algorithms that have recently been worked out in the general frame of affine invariance. Of particular interest is their performance in the numerical solution of discretized boundary value problems (BVPs) for nonlinear partial differential equations (PDEs). Exact Newton methods, where the arising linear systems are solved by direct elimination, and in...

In our previous paper [Ekren, Touzi and Zhang (2015)], we introduced a notion of viscosity solutions for fully nonlinear path-dependent PDEs, extending the semilinear case of Ekren et al. [Ann. Probab. 42 (2014) 204– 236], which satisfies a partial comparison result under standard Lipshitz-type assumptions. The main result of this paper provides a full, well-posedness result under an additional...

The Schwarz Alternating Method can be used to solve elliptic boundary value problems on domains which consist of two or more overlapping subdomains. The solution is approximated by an innnite sequence of functions which results from solving a sequence of elliptic boundary value problems in each of the subdomains. In this paper, proofs of convergence of some Schwarz Alternating Methods for nonli...

In this paper, we introduce and study one-dimensional models for the behavior of pedestrians in a narrow street or corridor. We begin at the microscopic level by formulating a stochastic cellular automata model with explicit rules for pedestrians moving in two opposite directions. Coarse-grained mesoscopic and macroscopic analogs are derived leading to the coupled system of PDEs for the density...

Last week we proved existence of weak solutions of the Dirichlet problem for a class of elliptic operators (in particular, for the Laplace operator), by using the Riesz representation theorem. This method, however, is limited to linear PDEs. It is therefore of interest to study more robust methods, which can be applied also to nonlinear PDEs. One such method is to obtain the solution to the PDE...