scholarly journals On an integrable multi-dimensionally consistent 2 n  + 2 n -dimensional heavenly-type equation

2019 ◽  
Vol 475 (2230) ◽  
pp. 20190091 ◽  
Author(s):  
B. G. Konopelchenko ◽  
W. K. Schief
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Vector Fields ◽  
Type Equation ◽  
Higher Dimensions ◽  
Partial Differential ◽  
Integrable Equations ◽  
Infinitesimal Symmetry ◽  
Einstein Spaces ◽  
Algebraic Scheme

Based on the commutativity of scalar vector fields, an algebraic scheme is developed which leads to a privileged multi-dimensionally consistent 2 n  + 2 n -dimensional integrable partial differential equation with the associated eigenfunction constituting an infinitesimal symmetry. The ‘universal’ character of this novel equation of vanishing Pfaffian type is demonstrated by retrieving and generalizing to higher dimensions a great variety of well-known integrable equations such as the dispersionless Kadomtsev–Petviashvili and Hirota equations and various avatars of the heavenly equation governing self-dual Einstein spaces.

GLOBAL C1 MAPS ON GENERAL DOMAINS

2009 ◽  
Vol 19 (05) ◽  
pp. 803-832 ◽  
Author(s):  
ALF EMIL LØVGREN ◽  
YVON MADAY ◽  
EINAR M. RØNQUIST
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Vector Fields ◽  
Two Dimensions ◽  
Arbitrary Lagrangian Eulerian ◽  
Reduced Basis ◽  
Partial Differential ◽  
Harmonic Extension ◽  
Transfinite Interpolation ◽  
Grid Points

In many contexts, there is a need to construct C1 maps from a given reference domain to a family of deformed domains. In our case, the motivation comes from the application of the Arbitrary Lagrangian Eulerian (ALE) method and also the reduced basis element method. In these methods, the maps are used to construct the grid-points needed on the deformed domains, and the corresponding Jacobian of the map is used to map vector fields from one domain to another. In order to keep the continuity of the mapped vector fields, the Jacobian must be continuous, and thus the maps need to be C1. In addition, the constructed grids on the deformed domains should be quality grids in the sense that, for a given partial differential equation defined on any of the deformed domains, the solution should be accurate. Since we are interested in a family of deformed domains, we consider the solutions of the partial differential equation to be a family of solutions governed by the geometry of the domains. Different mapping strategies are discussed and compared: the transfinite interpolation proposed by Gordon and Hall,12 the pseudo-harmonic extension proposed by Gordon and Wixom,13 a new generalization of the Gordon–Hall method (e.g., to general polygons in two dimensions), the harmonic extension, and the mean-valued extension proposed by Floater.8

scholarly journals On resistive magnetohydrodynamic equilibria of an axisymmetric toroidal plasma with flow

2000 ◽  
Vol 64 (5) ◽  
pp. 601-612 ◽  
Author(s):  
G. N. THROUMOULOPOULOS ◽  
H. TASSO
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Incompressible Flows ◽  
Type Equation ◽  
Magnetic Axis ◽  
Order Partial Differential Equation ◽  
Equilibrium Equations ◽  
Partial Differential ◽  
Toroidal Plasma ◽  
Mhd Equilibrium

It is shown that the magnetohydrodynamic (MHD) equilibrium states of an axisymmetric toroidal plasma with finite resistivity and flows parallel to the magnetic field are governed by a second-order partial differential equation for the poloidal magnetic flux function ψ coupled with a Bernoulli-type equation for the plasma density (which are identical in form to the corresponding ideal MHD equilibrium equations) along with the relation Δ*ψ = Vcσ (here Δ* is the Grad–Schlüter–Shafranov operator, σ is the conductivity and Vc is the constant toroidal-loop voltage divided by 2π). In particular, for incompressible flows, the above-mentioned partial differential equation becomes elliptic and decouples from the Bernoulli equation [H. Tasso and G. N. Throumoulopoulos, Phys. Plasma5, 2378 (1998)]. For a conductivity of the form σ = σ(R, ψ) (where R is the distance from the axis of symmetry), several classes of analytic equilibria with incompressible flows can be constructed having qualitatively plausible σ profiles, i.e. profiles with σ taking a maximum value close to the magnetic axis and a minimum value on the plasma surface. For σ = σ(ψ), consideration of the relation Δ*ψ = Vc σ(ψ) in the vicinity of the magnetic axis leads then to a proof of the non-existence of either compressible or incompressible equilibria. This result can be extended to the more general case of non-parallel flows lying within the magnetic surfaces.

scholarly journals Monte-Carlo algorithms for a forward Feynman–Kac-type representation for semilinear nonconservative partial differential equations

2018 ◽  
Vol 24 (1) ◽  
pp. 55-70 ◽  
Author(s):  
Anthony Le Cavil ◽  
Nadia Oudjane ◽  
Francesco Russo
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Monte Carlo ◽  
Numerical Experiments ◽  
Type Equation ◽  
Parabolic Partial Differential Equation ◽  
Partial Differential ◽  
Type Representation ◽  
Monte Carlo Algorithms ◽  
Semilinear Parabolic

Abstract The paper is devoted to the construction of a probabilistic particle algorithm. This is related to a nonlinear forward Feynman–Kac-type equation, which represents the solution of a nonconservative semilinear parabolic partial differential equation (PDE). Illustrations of the efficiency of the algorithm are provided by numerical experiments.

COMPUTING THE CONVEX ENVELOPE USING A NONLINEAR PARTIAL DIFFERENTIAL EQUATION

2008 ◽  
Vol 18 (05) ◽  
pp. 759-780 ◽  
Author(s):  
ADAM M. OBERMAN
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Unstructured Grids ◽  
Nonlinear Partial Differential Equation ◽  
Higher Dimensions ◽  
Convex Envelope ◽  
Computational Results ◽  
Partial Differential ◽  
Local Method ◽  
Nonsmooth Data

A fully nonlinear partial differential equation for the convex envelope was recently introduced by the author. In this paper, the equation is discretized using a finite difference method. The resulting scheme yields an explicit local method to compute the convex envelope. The scheme is shown to converge. Computational results are presented for smooth and nonsmooth data. Extensions to higher dimensions and unstructured grids are discussed.

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