We analytically and numerically study a fourth order PDE modeling rough crystal surface diffusion on the macroscopic level. discuss existence of solutions globally in time long dynamics for model. The PDE, originally derived by second author, is continuum limit microscopic model dynamics, given Markov jump process with Metropolis type transition rates. outline convergence argument, which depend...

I report on a contribution to the point symmetry classification problem for second-order partial differential equations (PDEs) in z(x, y), i.e. to an overview over all possible symmetry groups admitted by this class of equations. The article also contains a concise introduction into classical symmetry analysis. Sophus Lie (1842–1899) determined all continuous transformation groups of the 2D pla...

Recently, the use of the heatlike equation was extended to the projective case in order to find a projective analysis of curves and images; unfortunately, this formulation leads to a fifth-order partial differential equation (PDE) that is not easy to implement. Thanks to the use of a three-dimensional (3-D)homogeneous representation of a picture, we present here an alternative. Roughly speaking...

In rod outer segments the light activation of cGMP phosphodiesterase (PDE alpha beta gamma 2) is accomplished by removal of the gamma inhibitory subunit (PDE gamma) from the PDE alpha beta catalytic subunits. A light activation of the inositol signaling pathway also occurs, but there is little information linking these two signal transduction pathways. Here we report that protein kinase C (PKC)...

Finite time singularity formation in a fourth order nonlinear parabolic partial differential equation (PDE) is analyzed. The PDE is a variant of a ubiquitous model found in the field of microelectromechanical systems (MEMS) and is studied on a one-dimensional (1D) strip and the unit disc. The solution itself remains continuous at the point of singularity while its higher derivatives diverge, a ...

We discuss three symbolic approaches for the generation of a finite difference scheme of a general single partial differential equation (PDE). We concentrate on the case of a linear PDE with constant coefficients and prove, that these three approaches are equivalent. We systematically use another symbolic technique, namely the cylindrical algebraic decomposition, in order to derive conditions f...

Geometric modelling using Partial Differential Equations (PDEs) has been gradually recognised due to its smooth instinct, as well as the ability to generate a variety of geometric shapes by intuitively manipulating a relatively small set of PDE boundary curves. In this paper we explore and demonstrate the feasibility of the PDE method in facial geometry parameterisation. The geometry of a gener...

Recently we have created a multitime maximum principle gathering together some concepts in Mechanics, Field Theory, Differential Geometry, and Control Theory. The basic tools of our theory are variational PDE systems, adjoint PDE systems, Hamiltonian PDE systems, duality, multitime maximum principle, incavity on manifolds etc. Now we justify the multitime maximum principle for curvilinear integ...

In this paper we propose a state estimation approach for linear parabolic Partial Differential Equations (PDE) with uncertain parameters. It is based on an extension of the Galerkin projection method. The extended method models projection coefficients, representing the state of the PDE in some basis, by means of a Differential-Algebraic Equation (DAE). The original estimation problem for the PD...

A perception and PDE (partial di erential equation) based nonlinear transformation is proposed to process spoken words in noisy environment. The transformation was designed to reduce noise through time adaptation and spectral enhancement by evolving a focusing quadratic fourth order nonlinear PDE (the Cahn-Hillard equation). Constant, low SNR signals were adaptively decreased to reduce noise. O...