Approximate solutions of vector fields and an application to Denjoy–Carleman regularity of solutions of a nonlinear PDE

2021 ◽  
Author(s):  
Nicholas Braun Rodrigues ◽  
Antonio Victor da Silva
Keyword(s):  
Vector Fields ◽  
Approximate Solutions ◽  
Nonlinear Pde ◽  
Regularity Of Solutions

Nonlinear PDE with vector fields

2000 ◽  
Vol 81 (1) ◽  
pp. 111-137 ◽  
Author(s):  
David Holcman
Keyword(s):  
Vector Fields ◽  
Nonlinear Pde

A New Method for Nonlinear Two-Point Boundary Value Problems in Solid Mechanics

2001 ◽  
Vol 68 (5) ◽  
pp. 776-786 ◽  
Author(s):  
L. S. Ramachandra ◽  
D. Roy
Keyword(s):  
Boundary Value Problems ◽  
Vector Fields ◽  
Solid Mechanics ◽  
Boundary Value ◽  
Approximate Solutions ◽  
Algebraic Equations ◽  
Nonlinear Odes ◽  
Post Buckling ◽  
Independent Variable ◽  
Solution Parameters

A local and conditional linearization of vector fields, referred to as locally transversal linearization (LTL), is developed for accurately solving nonlinear and/or nonintegrable boundary value problems governed by ordinary differential equations. The locally linearized vector field is such that solution manifolds of the linearized equation transversally intersect those of the nonlinear BVP at a set of chosen points along the axis of the only independent variable. Within the framework of the LTL method, a BVP is treated as a constrained dynamical system, which in turn is posed as an initial value problem. (IVP) In the process, the LTL method replaces the discretized solution of a given system of nonlinear ODEs by that of a system of coupled nonlinear algebraic equations in terms of certain unknown solution parameters at these chosen points. A higher order version of the LTL method, with improved path sensitivity, is also considered wherein the dimension of the linearized equation needs to be increased. Finally, the procedure is used to determine post-buckling equilibrium paths of a geometrically nonlinear column with and without imperfections. Moreover, deflections of a tip-loaded nonlinear cantilever beam are also obtained. Comparisons with exact solutions, whenever available, and other approximate solutions demonstrate the remarkable accuracy of the proposed LTL method.

Regularity theory of Kolmogorov operator revisited

2020 ◽  
pp. 1-12
Author(s):  
Damir Kinzebulatov
Keyword(s):  
Elliptic Equation ◽  
Vector Fields ◽  
Regularity Theory ◽  
Regularity Of Solutions ◽  
Hölder Regularity ◽  
Kolmogorov Operator ◽  
Holder Regularity ◽  
Critical Order

Abstract We consider Kolmorogov operator $-\Delta +b \cdot \nabla $ with drift b in the class of form-bounded vector fields (containing vector fields having critical-order singularities). We characterize quantitative dependence of the Sobolev and Hölder regularity of solutions to the corresponding elliptic equation on the value of the form-bound of b.

Ck,α-regularity of solutions to quasilinear equations structured on Hörmander’s vector fields

2013 ◽  
Vol 92 ◽  
pp. 13-23 ◽  
Author(s):  
Marco Bramanti ◽  
Maria Stella Fanciullo
Keyword(s):  
Vector Fields ◽  
Regularity Of Solutions ◽  
Quasilinear Equations

scholarly journals The exponential Lie series for continuous semimartingales

2015 ◽  
Vol 471 (2184) ◽  
pp. 20150429 ◽  
Author(s):  
Kurusch Ebrahimi-Fard ◽  
Simon J. A. Malham ◽  
Frederic Patras ◽  
Anke Wiese
Keyword(s):  
Vector Fields ◽  
Approximate Solutions ◽  
Lie Series ◽  
Integration Schemes ◽  
Group Integration ◽  
Quadratic Covariation ◽  
Natural Change ◽  
Lie Group Integration ◽  
Commuting Vector Fields ◽  
Explicit Proof

We consider stochastic differential systems driven by continuous semimartingales and governed by non-commuting vector fields. We prove that the logarithm of the flowmap is an exponential Lie series. This relies on a natural change of basis to vector fields for the associated quadratic covariation processes, analogous to Stratonovich corrections. The flowmap can then be expanded as a series in compositional powers of vector fields and the logarithm of the flowmap can thus be expanded in the Lie algebra of vector fields. Further, we give a direct explicit proof of the corresponding Chen–Strichartz formula which provides an explicit formula for the Lie series coefficients. Such exponential Lie series are important in the development of strong Lie group integration schemes that ensure approximate solutions themselves lie in any homogeneous manifold on which the solution evolves.

Normal Forms and Bifurcation of Planar Vector Fields

1994 ◽  
Author(s):  
Shui-Nee Chow ◽  
Chengzhi Li ◽  
Duo Wang
Keyword(s):  
Vector Fields ◽  
Normal Forms ◽  
Planar Vector Fields ◽  
Planar Vector

MATHEMATICAL MODELS AND ALGORITHMS FOR RECONSTRUCTION OF SINGULAR SUPPORT OF FUNCTIONS AND VECTOR FIELDS BY TOMOGRAPHIC DATA

2015 ◽  
Vol 3 (1) ◽  
pp. 4-44
Author(s):  
E.Yu. Derevtsov ◽  
◽  
S.V. Maltseva ◽  
I.E. Svetov ◽  
◽  
...  
Keyword(s):  
Mathematical Models ◽  
Vector Fields ◽  
Singular Support ◽  
Tomographic Data
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