Extended crystal PDE’s stability,I: The general theory

This work, divided in two parts, follows some our previous works devoted to the algebraic topological characterization of PDE’s. In this first part, the stability of PDE’s is studied in some details in the framework of the geometric theory of PDE’s, and bordism groups theory of PDE’s. In particular we identify criteria to recognize PDE’s that are stable (in extended Ulam sense) and in their regular smooth solutions do not occur finite time unstabilities, (stable extended crystal PDE’s). Applications to some important PDE’s are carefully considered. (In the second part a stable extended crystal PDE, encoding anisotropic incompressible magnetohydrodynamics is obtained Ref. [A. Prástaro, Extended crystal PDE’s (submitted for publication)].)     

Thermal capacity estimates on the Allen-Cahn equation

We consider the Allen-Cahn equation in a well-known scaling regime which gives motion by mean curvature. A well-known transformation of this PDE, using its standing wave, yields a PDE the solution of which is approximately the distance function to an interface moving by mean curvature. We give bounds on this last fact in terms of thermal capacity. Our techniques hinge upon the analysis of a certain semimartingale associated with a certain PDE (the PDE for the approximate distance function) and an analogue of some results by Bañuelos and Øksendal relating lifetimes of diffusions to exterior capacities.

     

Extended crystal PDE’S stability,II: The extended crystal MHD-PDE’S

This paper is the second part of a work devoted to the algebraic topological characterization of PDE’s stability, and its relationship with an important class of PDE’s called extended crystals PDE’s in the sense introduced in [A. Prástaro, Extended crystal PDE’s (submitted for publication)]. In fact, their integral bordism groups can be considered as extensions of subgroups of crystallographic groups. This allows us to identify a characteristic class that measures the obstruction to the existence of global solutions. In part I [A. Prástaro, Extended crystal PDE’s stability, I: The general theory, Math. Comput. Modelling, 49 (9–10) (2009) 1759–1780] we identified criteria to recognize PDE’s that are stable (in extended Ulam sense) and in their regular smooth solutions, finite time instabilities do not occur (stable extended crystal PDE’s). Here, we study in some detail, a new PDE encoding anisotropic incompressible magnetohydrodynamics. Stable extended crystal MHD-PDE’s are obtained, where in their smooth solutions, instabilities do not occur in finite time. These results are considered first for systems without a body energy source, and later, by also introducing a contribution from an energy source, in order to take into account nuclear energy production. A condition in order that solutions satisfy the second principle of thermodynamics is given.     

A Linear PDE Approach to the Bellman Equation of Ergodic Control with Periodic Structure

   Abstract. In this paper we give a new proof of the existence result of Bensoussan [1, Theorem II-6.1] for the Bellman equation of ergodic control with periodic structure. This Bellman equation is a nonlinear PDE, and he constructed its solution by using the solution of a nonlinear PDE. On the contrary, our key idea is to solve two linear PDEs. Hence, we propose a linear PDE approach to this Bellman equation.     

A Linear PDE Approach to the Bellman Equation of Ergodic Control with Periodic Structure

Abstract. In this paper we give a new proof of the existence result of Bensoussan [1, Theorem II-6.1] for the Bellman equation of ergodic control with periodic structure. This Bellman equation is a nonlinear PDE, and he constructed its solution by using the solution of a nonlinear PDE. On the contrary, our key idea is to solve two linear PDEs. Hence, we propose a linear PDE approach to this Bellman equation.     

A third order partial differential equation for isotropic boundary based triangular Bézier surface generation

We approach surface design by solving a linear third order Partial Differential Equation (PDE). We present an explicit polynomial solution method for triangular Bézier PDE surface generation characterized by a boundary configuration. The third order PDE comes from a symmetric operator defined here to overcome the anisotropy drawback of any operator over triangular Bézier surfaces.     

Quantum exotic PDE’s

Following the previous works on the Prástaro’s formulation of algebraic topology of quantum (super) PDE’s, it is proved that a canonical Heyting algebra (integral Heyting algebra) can be associated to any quantum PDE. This is directly related to the structure of its global solutions. This allows us to recognize a new inside in the concept of quantum logic for microworlds. Furthermore, the Prástaro’s geometric theory of quantum PDE’s is applied to the new category of quantum hypercomplex manifolds, related to the well-known Cayley–Dickson construction for algebras. Theorems of existence for local and global solutions are obtained for (singular) PDE’s in this new category of noncommutative manifolds. Finally, the extension of the concept of exotic PDE’s, recently introduced by Prástaro, has been extended to quantum PDE’s. Then a smooth quantum version of the quantum (generalized) Poincaré conjecture is given too. These results extend ones for quantum (generalized) Poincaré conjecture, previously given by Prástaro.     

A third order partial differential equation for isotropic boundary based triangular Bézier surface generation

We approach surface design by solving a linear third order Partial Differential Equation (PDE). We present an explicit polynomial solution method for triangular Bézier PDE surface generation characterized by a boundary configuration. The third order PDE comes from a symmetric operator defined here to overcome the anisotropy drawback of any operator over triangular Bézier surfaces.     

Symmetry reduced and new exact non-traveling wave solutions of potential Kadomtsev-Petviashvili equation with p-power

With the aid of Maple symbolic computation and Lie group method, PKPp equation is reduced to some (1 + 1)-dimensional partial differential equations, in which there are linear PDE with constant coefficients, nonlinear PDE with constant coefficients, and nonlinear PDE with variable coefficients. Using the separation of variables, homoclinic test technique and auxiliary equation methods, we obtain new abundant exact non-traveling solution with arbitrary functions for the PKPp.     

Exact Solutions of Linear Partial Differential Equations with Variable Coefficients

Bäcklund transformations relating the solutions of linear PDE with variable coefficients to those of PDE with constant coefficients are found, generalizing the study of Varley and Seymour [2]. Auto-Bäcklund transformations are also determined. To facilitate the generation of new solutions via Bäcklund transformation, explicit solutions of both classes of the PDE just mentioned are found using invariance properties of these equations and other methods. Some of these solutions are new.     

PDEs for the Gaussian ensemble with external source and the Pearcey distribution

The present paper studies a Gaussian Hermitian random matrix ensemble with external source, given by a fixed diagonal matrix with two eigenvalues ±a. As a first result, the probability that the eigenvalues of the ensemble belong to an interval E satisfies a fourth‐order PDE with quartic nonlinearity; the variables are the eigenvalue a and the boundary of E. This equation enables one to find a PDE for the Pearcey distribution. The latter describes the statistics of the eigenvalues near the closure of a gap, i.e., when the support of the equilibrium measure for large‐size random matrices has a gap that can be made to close. The Gaussian Hermitian random matrix ensemble with external source, described above, has this feature. The Pearcey distribution is shown to satisfy a fourth‐order PDE with cubic nonlinearity. This also gives the PDE for the transition probability of the Pearcey process, a limiting process associated with nonintersecting Brownian motions on ℝ. © 2006 Wiley Periodicals, Inc.     

Planforms in three dimensions

The spatially periodic, steady-state solutions to systems of partial differential equations (PDE) are calledplanforms. There already exists a partial classification of the planforms for Euclidean equivariant systems of PDE inR ² (see [6, 7]), In this article we attempt to give such a classification for Euclidean equivariant systems of PDE inR ³. Based on the symmetry and spatial periodicity of each planform, 59 different planforms are found.We attempt to find the planforms on all lattices inR ³ that are forced to exist near a steady-state bifurcation from a trivial solution. The proof of our classification uses Liapunov-Schmidt reduction with symmetry (which can be used if we assume spatial periodicity of the solutions) and the Equivariant Branching Lemma. The analytical problem of finding planforms for systems of PDE is reduced to the algebraic problem of computing isotropy subgroups with one dimensional fixed point subspaces.The Navier-Stokes equations and reaction-diffusion equations (with constant diffusion coefficients) are examples of systems of PDE that satisfy the conditions of our classifications. In this article, we show that our classification applies to the Kuramoto-Sivashinsky equation.     

Geometric Completeness of Distribution Spaces

This paper introduces a notion of geometric completeness for spaces of distributions, modelled after the notion of a complete variety in algebraic geometry. It is related to the following elimination problem for systems of PDE: consider the set of homogeneous solutions of a system of PDE in some space of distributions. When is the projection of this set onto some of its coordinates also the set of homogeneous solutions of a system of PDE?     

High‐order partial differential equation de‐noising method for vibration signal

A novel approach for 1D vibration signal de‐noising filter using partial differential equation (PDE) is presented. In particular, the numerical solution of higher‐order PDE is generated, and we show that it enables the amplitude‐frequency characteristic in filter to be estimated more accurately, which results in better de‐noising performance in comparison with the low‐order PDE. The de‐noising tests on different degree of artificial noise are conducted. Experimental tests have been rigorously compared with different de‐noising methods to verify the efficacy of the proposed high‐order PDE filter method. Copyright © 2014 John Wiley & Sons, Ltd.     

Mean-field limit for the stochastic Vicsek model

We consider the continuous version of the Vicsek model with noise, proposed as a model for collective behaviour of individuals with a fixed speed. We rigorously derive the kinetic mean-field partial differential equation satisfied when the number N of particles tends to infinity, quantifying the convergence of the law of one particle to the solution of the PDE. For this we adapt a classical coupling argument to the present case in which both the particle system and the PDE are defined on a surface rather than on the whole space R^d. As part of the study we give existence and uniqueness results for both the particle system and the PDE.     

Existence of local‐in‐time classical solutions of a model of flow in a bounded elastic tube

This paper studies the local‐in‐time existence of classical solutions to a hyperbolic system with differential boundary conditions modelling a flow in an elastic tube. The well‐known Lax transformations used for obtaining a priori estimates for conservation laws are difficult to apply here because of the inhomogeneity of the partial differential equations (PDE). Rather, our method relies on a suitable splitting of the original system into the PDE part and the ODE part, the characteristics for the PDE part, appropriate modulus of continuity estimates and a compactness argument. Copyright © 2016 John Wiley & Sons, Ltd.     

Invertible Mappings of Nonlinear PDEs to Linear PDEs through Admitted Conservation Laws

An algorithmic method using conservation law multipliers is introduced that yields necessary and sufficient conditions to find invertible mappings of a given nonlinear PDE to some linear PDE and to construct such a mapping when it exists. Previous methods yielded such conditions from admitted point or contact symmetries of the nonlinear PDE. Through examples, these two linearization approaches are contrasted.      

A PDE representation of the density of the minimal entropy martingale measure in stochastic volatility markets

Under general conditions stated in Rheinländer [An entropy approach to the stein/stein model with correlation. Preprint, 2003, ETH Zürich.], we prove that in a stochastic volatility market the Radon–Nikodym density of the minimal entropy martingale measure (MEMM) can be expressed in terms of the solution of a semilinear PDE. The semilinear PDE is suggested by the dynamic programming approach to the utility indifference pricing problem of contingent claims. One of our main results is the existence and uniqueness of a classical solution of the semilinear PDE in the case of a general stochastic volatility model with additive noise correlated with the asset price. Our results are applied to the Stein–Stein and Heston stochastic volatility models.     

Risk premium and fair option prices under stochastic volatility: the HARA solution

We have solved the problem of finding (HARA) fair option price under a general stochastic volatility model. For a given HARA utility, the ‘risk premium’, i.e., the ‘market price of volatility risk’ is determined via a solution of a certain nonlinear PDE. Equivalently, the fair option price is determined as a solution of an uncoupled system of a non-linear PDE and a Black–Scholes type PDE. To cite this article: S.D. Stojanovic, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Multiple shooting methods for parabolic optimal control problems with control constraints

In the context of ordinary differential equations, shooting techniques are a state-of-the-art solver component, whereas their application in the framework of partial differential equations (PDE) is still at an early stage. We present two multiple shooting approaches for optimal control problems (OCP) governed by parabolic PDE. Direct and indirect shooting for PDE optimal control stem from the same extended problem formulation. Our approach reveals that they are structurally similar but show major differences in their algorithmic realizations. In the presented numerical examples we cover a nonlinear parabolic optimal control problem with additional control constraints. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)