A Posterior Model Reduction Method Based on Weighted Residue for Linear Partial Differential Equations

2014 ◽  
Vol 684 ◽  
pp. 34-40
Author(s):  
Jie Sha ◽  
Li Xiang Zhang ◽  
Chui Jie Wu
Keyword(s):  
Dynamical System ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Linear Partial Differential Equation ◽  
Galerkin Projection ◽  
Lagrangian Multiplier ◽  
Partial Differential ◽  
Projection Equation ◽  
Dimensional Dynamical System ◽  
Low Dimensional

This paper is concerned with a new model reduced method based on optimal large truncated low-dimensional dynamical system, by which the solution of linear partial differential equation (PDE) is able to be approximate with highly accuracy. The method proposed is based on the weighted residue of PDE under consideration, and the weighted residue is used as an alternative optimal control condition (POT-WR) while solving the PDE. A set of bases is constructed to describe a dynamical system required in case. The Lagrangian multiplier is introduced to eliminate the constraints of the Galerkin projection equation, and the penalty function is used to remove the orthogonal constraint. According to the extreme principle, a set of the ordinary differential equations is obtained by taking the variational operation on generalized optimal function. A conjugate gradients algorithm on FORTRAN code is developed to solve these ordinary differential equations with Fourier polynomials as the initial bases for iterations. The heat transfer equation under a potential initial condition is used to verify the method proposed. Good agreement between the simulations and the analytical solutions of example was obtained, indicating that the POT-WR method presented in this paper provides the most effective posterior way of capturing the dominant characteristics of an infinite-dimensional dynamical system with only finitely few bases.

A Novel Low-Dimensional Method for Analytically Solving Partial Differential Equations

2015 ◽  
Vol 7 (6) ◽  
pp. 754-779 ◽  
Author(s):  
Jie Sha ◽  
Lixiang Zhang ◽  
Chuijie Wu
Keyword(s):  
Dynamical System ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Galerkin Projection ◽  
Reference Database ◽  
Lagrangian Multiplier ◽  
Partial Differential ◽  
Dimensional Dynamical System ◽  
Low Dimensional

AbstractThis paper is concerned with a low-dimensional dynamical system model for analytically solving partial differential equations (PDEs). The model proposed is based on a posterior optimal truncated weighted residue (POT-WR) method, by which an infinite dimensional PDE is optimally truncated and analytically solved in required condition of accuracy. To end that, a POT-WR condition for PDE under consideration is used as a dynamically optimal control criterion with the solving process. A set of bases needs to be constructed without any reference database in order to establish a space to describe low-dimensional dynamical system that is required. The Lagrangian multiplier is introduced to release the constraints due to the Galerkin projection, and a penalty function is also employed to remove the orthogonal constraints. According to the extreme principle, a set of ordinary differential equations is thus obtained by taking the variational operation of the generalized optimal function. A conjugate gradient algorithm by FORTRAN code is developed to solve the ordinary differential equations. The two examples of one-dimensional heat transfer equation and nonlinear Burgers’ equation show that the analytical results on the method proposed are good agreement with the numerical simulations and analytical solutions in references, and the dominant characteristics of the dynamics are well captured in case of few bases used only.

scholarly journals Galois theory of aigebraic and differential equations

1996 ◽  
Vol 144 ◽  
pp. 1-58 ◽  
Author(s):  
Hiroshi Umemura
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Galois Theory ◽  
Linear Partial Differential Equation ◽  
Linear Algebraic Group ◽  
Partial Differential ◽  
Linear Ordinary Differential Equations ◽  
Infinite Dimensional ◽  
Linear Partial Differential Equations

This paper will be the first part of our works on differential Galois theory which we plan to write. Our goal is to establish a Galois Theory of ordinary differential equations. The theory is infinite dimensional by nature and has a long history. The pioneer of this field is S. Lie who tried to apply the idea of Abel and Galois to differential equations. Picard [P] realized Galois Theory of linear ordinary differential equations, which is called nowadays Picard-Vessiot Theory. Picard-Vessiot Theory is finite dimensional and the Galois group is a linear algebraic group. The first attempt of Galois theory of a general ordinary differential equations which is infinite dimensional, is done by the thesis of Drach [D]. He replaced an ordinary differential equation by a linear partial differential equation satisfied by the first integrals and looked for a Galois Theory of linear partial differential equations. It is widely admitted that the work of Drach is full of imcomplete definitions and gaps in proofs. In fact in a few months after Drach had got his degree, Vessiot was aware of the defects of Drach’s thesis. Vessiot took the matter serious and devoted all his life to make the Drach theory complete. Vessiot got the grand prix of the academy of Paris in Mathematics in 1903 by a series of articles.

CHAOS AND HYPERCHAOS IN A CLASS OF SIMPLE CELLULAR NEURAL NETWORKS MODELED BY O.D.E.

2006 ◽  
Vol 16 (09) ◽  
pp. 2729-2736 ◽  
Author(s):  
XIAO-SONG YANG ◽  
YAN HUANG
Keyword(s):  
Neural Networks ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Lyapunov Exponents ◽  
Cellular Neural Networks ◽  
New Class ◽  
Connection Matrices ◽  
Low Dimensional

This paper presents a new class of chaotic and hyperchaotic low dimensional cellular neural networks modeled by ordinary differential equations with some simple connection matrices. The chaoticity of these neural networks is indicated by positive Lyapunov exponents calculated by a computer.

A Novel General and Robust Method Based on NAOP for Solving Nonlinear Ordinary Differential Equations and Partial Differential Equations by Cellular Neural Networks

2013 ◽  
Vol 135 (3) ◽  
Author(s):  
Jean Chamberlain Chedjou ◽  
Kyandoghere Kyamakya
Keyword(s):  
Neural Networks ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Cellular Neural Networks ◽  
Nonlinear Ordinary Differential Equations ◽  
Van Der Pol ◽  
Partial Differential ◽  
Nonlinear Odes ◽  
First Order

This paper develops and validates through a series of presentable examples, a comprehensive high-precision, and ultrafast computing concept for solving nonlinear ordinary differential equations (ODEs) and partial differential equations (PDEs) with cellular neural networks (CNN). The core of this concept is a straightforward scheme that we call "nonlinear adaptive optimization (NAOP),” which is used for a precise template calculation for solving nonlinear ODEs and PDEs through CNN processors. One of the key contributions of this work is to demonstrate the possibility of transforming different types of nonlinearities displayed by various classical and well-known nonlinear equations (e.g., van der Pol-, Rayleigh-, Duffing-, Rössler-, Lorenz-, and Jerk-equations, just to name a few) unto first-order CNN elementary cells, and thereby enabling the easy derivation of corresponding CNN templates. Furthermore, in the case of PDE solving, the same concept also allows a mapping unto first-order CNN cells while considering one or even more nonlinear terms of the Taylor's series expansion generally used in the transformation of a PDE in a set of coupled nonlinear ODEs. Therefore, the concept of this paper does significantly contribute to the consolidation of CNN as a universal and ultrafast solver of nonlinear ODEs and/or PDEs. This clearly enables a CNN-based, real-time, ultraprecise, and low-cost computational engineering. As proof of concept, two examples of well-known ODEs are considered namely a second-order linear ODE and a second order nonlinear ODE of the van der Pol type. For each of these ODEs, the corresponding precise CNN templates are derived and are used to deduce the expected solutions. An implementation of the concept developed is possible even on embedded digital platforms (e.g., field programmable gate array (FPGA), digital signal processor (DSP), graphics processing unit (GPU), etc.). This opens a broad range of applications. Ongoing works (as outlook) are using NAOP for deriving precise templates for a selected set of practically interesting ODEs and PDEs equation models such as Lorenz-, Rössler-, Navier Stokes-, Schrödinger-, Maxwell-, etc.

scholarly journals Hidden and Not So Hidden Symmetries

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
P. G. L. Leach ◽  
K. S. Govinder ◽  
K. Andriopoulos
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Recent Work ◽  
Nonlocal Symmetry ◽  
Partial Differential ◽  
Hidden Symmetries ◽  
Contact Symmetry

Hidden symmetries entered the literature in the late Eighties when it was observed that there could be gain of Lie point symmetry in the reduction of order of an ordinary differential equation. Subsequently the reverse process was also observed. Such symmetries were termed “hidden”. In each case the source of the “new” symmetry was a contact symmetry or a nonlocal symmetry, that is, a symmetry with one or more of the coefficient functions containing an integral. Recent work by Abraham-Shrauner and Govinder (2006) on the reduction of partial differential equations demonstrates that it is possible for these “hidden” symmetries to have a point origin. In this paper we show that the same phenomenon can be observed in the reduction of ordinary differential equations and in a sense loosen the interpretation of hidden symmetries.

scholarly journals On classical symmetries of ordinary differential equations related to stationary integrable partial differential equations

2021 ◽  
Vol 41 (5) ◽  
pp. 685-699
Author(s):  
Ivan Tsyfra
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Theoretical Method ◽  
Partial Differential ◽  
Classical Symmetry ◽  
Kdv Equations ◽  
Classical Symmetries

We study the relationship between the solutions of stationary integrable partial and ordinary differential equations and coefficients of the second-order ordinary differential equations invariant with respect to one-parameter Lie group. The classical symmetry method is applied. We prove that if the coefficients of ordinary differential equation satisfy the stationary integrable partial differential equation with two independent variables then the ordinary differential equation is integrable by quadratures. If special solutions of integrable partial differential equations are chosen then the coefficients satisfy the stationary KdV equations. It was shown that the Ermakov equation belong to a class of these equations. In the framework of the approach we obtained the similar results for generalized Riccati equations. By using operator of invariant differentiation we describe a class of higher order ordinary differential equations for which the group-theoretical method enables us to reduce the order of ordinary differential equation.

Some Invariant Solutions of Two-Dimensional Elastodynamics in Linear Homogeneous Isotropic Materials

2013 ◽  
Vol 5 (2) ◽  
pp. 212-221
Author(s):  
Houguo Li ◽  
Kefu Huang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Theoretical Method ◽  
Two Dimensional ◽  
Invariant Solutions ◽  
Partial Differential ◽  
Isotropic Materials ◽  
Group Theoretical Method ◽  
Infinitesimal Operators

AbstractInvariant solutions of two-dimensional elastodynamics in linear homogeneous isotropic materials are considered via the group theoretical method. The second order partial differential equations of elastodynamics are reduced to ordinary differential equations under the infinitesimal operators. Three invariant solutions are constructed. Their graphical figures are presented and physical meanings are elucidated in some cases.

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