Constructing Low-Dimensional Dynamical Systems of Nonlinear Partial Differential Equations Using Optimization

2013 ◽  
Vol 8 (11) ◽  
Author(s):  
Jun Shuai ◽  
Xuli Han
Keyword(s):  
Partial Differential Equations ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Partial Differential ◽  
Low Dimensional

Dynamical Systems and Nonlinear Partial Differential Equations

1989 ◽  
Author(s):  
Jack Hale ◽  
Constantine M. Dafermos ◽  
John Mallet-Paret ◽  
Panagiotis E. Souganidis ◽  
Walter Strauss
Keyword(s):  
Partial Differential Equations ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Partial Differential

Dynamical Systems and Nonlinear Partial Differential Equations

1993 ◽  
Author(s):  
Constantine M. Dafermos ◽  
John Mallet-Paret ◽  
Panagiotis E. Souganidis ◽  
Walter Strauss
Keyword(s):  
Partial Differential Equations ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Partial Differential

scholarly journals Optimal Combination of EEFs for the Model Reduction of Nonlinear Partial Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Jun Shuai ◽  
Xuli Han
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Model Reduction ◽  
Nonlinear Partial Differential Equations ◽  
Optimization Techniques ◽  
Optimal Combination ◽  
Nonlinear Pdes ◽  
Partial Differential ◽  
Tiny Amount ◽  
Low Dimensional

Proper orthogonal decomposition is a popular approach for determining the principal spatial structures from the measured data. Generally, model reduction using empirical eigenfunctions (EEFs) can generate a relatively low-dimensional model among all linear expansions. However, the neglectful modes representing only a tiny amount of energy will be crucial in the modeling for certain type of nonlinear partial differential equations (PDEs). In this paper, an optimal combination of EEFs is proposed for model reduction of nonlinear partial differential equations (PDEs), obtained by the basis function transformation from the initial EEFs. The transformation matrix is derived from straightforward optimization techniques. The present new EEFs can keep the dynamical information of neglectful modes and generate a lower-dimensional and more precise dynamical system for the PDEs. The numerical example shows its effectiveness and feasibility for model reduction of the nonlinear PDEs.

Lyapunov-type inequalities for partial differential equations with 𝑝-Laplacian

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Robert Stegliński
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Nonlinear Partial Differential Equations ◽  
Dirichlet Boundary Conditions ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Lyapunov Inequalities ◽  
Funct Anal

Abstract The aim of this paper is to extend results from [A. Cañada, J. A. Montero and S. Villegas, Lyapunov inequalities for partial differential equations, J. Funct. Anal. 237 (2006), 1, 176–193] about Lyapunov-type inequalities for linear partial differential equations to nonlinear partial differential equations with 𝑝-Laplacian with zero Neumann or Dirichlet boundary conditions.

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