We introduce a simple, rigorous, and unified framework for solving nonlinear partial differential equations (PDEs), inverse problems (IPs) involving the identification of parameters in PDEs, using Gaussian processes. The proposed approach: (1) provides natural generalization collocation kernel methods to PDEs IPs; (2) has guaranteed convergence very general class comes equipped with path comput...

In contrast to the all-discrete approaches that dominated image processing in the recent past, in computer vision since the mid 1980’s there have been proposed continuous models for several vision tasks based on partial differential equations (PDEs). The discrete part of such approaches comes only in the corresponding difference equations (numerical algorithms) that approximate the solution of ...

Keywords: Frobenius integrable decompositions Partial differential equations Differential polynomials Bäcklund transformations Ordinary differential equations Nonlinearity a b s t r a c t Frobenius integrable decompositions are presented for a kind of ninth-order partial differential equations of specific polynomial type. Two classes of such partial differential equations possessing Frobenius i...

Significant advances have taken place in the last few years in the development of control designs for nonlinear infinite-dimensional systems. Such systems typically take the form of nonlinear ODEs (ordinary differential equations) with delays and nonlinear PDEs (partial differential equations). In this article we review several representative but general results on nonlinear control in the infi...

This paper is concerned with the Sobolev type weak solutions of one class second order quasilinear parabolic partial differential equations (PDEs, for short). First all, similar to Feng, Wang and Zhao [9] Wu Yu [29], we use a family coupled forward-backward stochastic (FBSDEs, short) which satisfy monotonous assumption represent classical PDEs. Then, based on PDEs approximating PDEs, prove exis...

We show that any classical matrix Q × Q S-integrable Partial Differential Equation (PDE) obtainable as commutativity condition of a certain type of vector fields possesses a family of lower-dimensional reductions represented by the matrix Q × n 0 Q quasilinear first order PDEs solved in [29] by the method of characteristics. In turn, these PDEs admit two types of available particular solutions:...

Significant advances have taken place in the last few years in the development of control designs for nonlinear infinite-dimensional systems. Such systems typically take the form of nonlinear ODEs (ordinary differential equations) with delays and nonlinear PDEs (partial differential equations). In this article we review several representative but general results on nonlinear control in the infi...

Integration factor methods are a class of ‘‘exactly linear part’’ time discretization methods. In [Q. Nie, Y.-T. Zhang, R. Zhao, Efficient semi-implicit schemes for stiff systems, Journal of Computational Physics, 214 (2006) 521–537], a class of efficient implicit integration factor (IIF) methods were developed for solving systems with both stiff linear and nonlinear terms, arising from spatial...

In this paper we develop partial differential equations (PDEs) that model the generation of a large class of morphological filters, the levelings and the openings/closings by reconstruction. These types of filters are very useful in numerous image analysis and vision tasks ranging from enhancement, to geometric feature detection, to segmentation. The developed PDEs are nonlinear functions of th...

By using the geometric concept of PDEs with prescribed curvature representation, we prove that the 1+2 dimensional isotropic Landau-Lifshitz equation is gauge equivalent to a 1+2 dimensional nonlinear Schrödinger-type system. From the nonlinear Schrödinger-type system, we construct blowing up H3(R2)-solutions to the Landau-Lifshitz equation, which reveals the blow up phenomenon of the equation.