398 research outputs found
Seven common errors in finding exact solutions of nonlinear differential equations
We analyze the common errors of the recent papers in which the solitary wave
solutions of nonlinear differential equations are presented. Seven common
errors are formulated and classified. These errors are illustrated by using
multiple examples of the common errors from the recent publications. We show
that many popular methods in finding of the exact solutions are equivalent each
other. We demonstrate that some authors look for the solitary wave solutions of
nonlinear ordinary differential equations and do not take into account the well
- known general solutions of these equations. We illustrate several cases when
authors present some functions for describing solutions but do not use
arbitrary constants. As this fact takes place the redundant solutions of
differential equations are found. A few examples of incorrect solutions by some
authors are presented. Several other errors in finding the exact solutions of
nonlinear differential equations are also discussed.Comment: 42 page
Nonlinear maximum principles for dissipative linear nonlocal operators and applications
We obtain a family of nonlinear maximum principles for linear dissipative
nonlocal operators, that are general, robust, and versatile. We use these
nonlinear bounds to provide transparent proofs of global regularity for
critical SQG and critical d-dimensional Burgers equations. In addition we give
applications of the nonlinear maximum principle to the global regularity of a
slightly dissipative anti-symmetric perturbation of 2d incompressible Euler
equations and generalized fractional dissipative 2d Boussinesq equations
Classification of integrable super-systems using the SsTools environment
A classification problem is proposed for supersymmetric evolutionary PDE that
satisfy the assumptions of nonlinearity and nondegeneracy. Four classes of
nonlinear coupled boson-fermion systems are discovered under the homogeneity
assumption |f|=|b|=|D_t|=1/2. The syntax of the Reduce package SsTools, which
was used for intermediate computations, and the applicability of its procedures
to the calculus of super-PDE are described.Comment: MSC 35Q53,37K05,37K10,81T40; PACS 02.30.Ik,02.70.Wz,12.60.Jv; Comput.
Phys. Commun. (2007), 26 pages (accepted
On the relation between standard and -symmetries for PDEs
We give a geometrical interpretation of the notion of -prolongations of
vector fields and of the related concept of -symmetry for partial
differential equations (extending to PDEs the notion of -symmetry for
ODEs). We give in particular a result concerning the relationship between
-symmetries and standard exact symmetries. The notion is also extended to
the case of conditional and partial symmetries, and we analyze the relation
between local -symmetries and nonlocal standard symmetries.Comment: 25 pages, no figures, latex. to be published in J. Phys.
Memory embedded non-intrusive reduced order modeling of non-ergodic flows
Generating a digital twin of any complex system requires modeling and
computational approaches that are efficient, accurate, and modular. Traditional
reduced order modeling techniques are targeted at only the first two but the
novel non-intrusive approach presented in this study is an attempt at taking
all three into account effectively compared to their traditional counterparts.
Based on dimensionality reduction using proper orthogonal decomposition (POD),
we introduce a long short-term memory (LSTM) neural network architecture
together with a principal interval decomposition (PID) framework as an enabler
to account for localized modal deformation, which is a key element in accurate
reduced order modeling of convective flows. Our applications for convection
dominated systems governed by Burgers, Navier-Stokes, and Boussinesq equations
demonstrate that the proposed approach yields significantly more accurate
predictions than the POD-Galerkin method, and could be a key enabler towards
near real-time predictions of unsteady flows
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