scholarly journals Direct construction method for conservation laws of partial differential equations Part II: General treatment

2002 ◽  
Vol 13 (5) ◽  
pp. 567-585 ◽  
Author(s):  
STEPHEN C. ANCO ◽  
GEORGE BLUMAN
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Conservation Laws ◽  
General Treatment ◽  
Local Conservation ◽  
Partial Differential ◽  
Direct Construction ◽  
Computational Implementation ◽  
Direct Construction Method ◽  
General Method

This paper gives a general treatment and proof of the direct conservation law method presented in Part I (see Anco & Bluman [3]). In particular, the treatment here applies to finding the local conservation laws of any system of one or more partial differential equations expressed in a standard Cauchy-Kovalevskaya form. A summary of the general method and its effective computational implementation is also given.

scholarly journals Direct construction method for conservation laws of partial differential equations Part I: Examples of conservation law classifications

2002 ◽  
Vol 13 (5) ◽  
pp. 545-566 ◽  
Author(s):  
STEPHEN C. ANCO ◽  
GEORGE BLUMAN
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Conservation Laws ◽  
Conservation Law ◽  
Wave Equations ◽  
Explicit Construction ◽  
Nonlinear Wave Equations ◽  
Local Conservation ◽  
Partial Differential ◽  
Direct Construction

An effective algorithmic method is presented for finding the local conservation laws for partial differential equations with any number of independent and dependent variables. The method does not require the use or existence of a variational principle and reduces the calculation of conservation laws to solving a system of linear determining equations similar to that for finding symmetries. An explicit construction formula is derived which yields a conservation law for each solution of the determining system. In the first of two papers (Part I), examples of nonlinear wave equations are used to exhibit the method. Classification results for conservation laws of these equations are obtained. In a second paper (Part II), a general treatment of the method is given.

scholarly journals Symmetry properties of conservation laws

2016 ◽  
Vol 30 (28n29) ◽  
pp. 1640003 ◽  
Author(s):  
Stephen C. Anco
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Conservation Laws ◽  
Conservation Law ◽  
Conserved Quantity ◽  
Partial Differential ◽  
Standard Formula ◽  
Infinitesimal Symmetry ◽  
Symmetry Properties ◽  
General Method

Symmetry properties of conservation laws of partial differential equations are developed by using the general method of conservation law multipliers. As main results, simple conditions are given for characterizing when a conservation law and its associated conserved quantity are invariant (and, more generally, homogeneous) under the action of a symmetry. These results are used to show that a recent conservation law formula (due to Ibragimov) is equivalent to a standard formula for the action of an infinitesimal symmetry on a conservation law multiplier.

Symmetry-invariant conservation laws of partial differential equations

2017 ◽  
Vol 29 (1) ◽  
pp. 78-117 ◽  
Author(s):  
STEPHEN C. ANCO ◽  
ABDUL H. KARA
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Conservation Laws ◽  
Stokes Equations ◽  
Two Dimensions ◽  
Navier Stokes ◽  
Partial Differential ◽  
Navier Stokes Equations ◽  
General Method

A simple characterization of the action of symmetries on conservation laws of partial differential equations is studied by using the general method of conservation law multipliers. This action is used to define symmetry-invariant and symmetry-homogeneous conservation laws. The main results are applied to several examples of physically interest, including the generalized Korteveg-de Vries equation, a non-Newtonian generalization of Burger's equation, theb-family of peakon equations, and the Navier–Stokes equations for compressible, viscous fluids in two dimensions.

scholarly journals Conservation laws for partial differential equations based on the polynomial characteristic method

2020 ◽  
Vol 24 (4) ◽  
pp. 2529-2534
Author(s):  
Yi Tian ◽  
Kang-Le Wang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Conservation Laws ◽  
Conservation Law ◽  
Differential Polynomial ◽  
Construction Method ◽  
Characteristic Method ◽  
Characteristic Set ◽  
Direct Construction ◽  
Direct Construction Method

In this paper, the direct construction method combined with the differential polynomial characteristic set algorithm is used to complete conservation laws of PDE. The process of the direct construction method is to solve a system of linear determining equations, which is not easy to be solved. This paper uses the differential polynomial characteristic set algorithm to overcome the shortcoming, and constructs an explicit conservation law.

scholarly journals Recent Advances in Symmetry Groups and Conservation Laws for Partial Differential Equations and Applications

2014 ◽  
Vol 2014 ◽  
pp. 1-2 ◽  
Author(s):  
Maria Gandarias ◽  
Mariano Torrisi ◽  
Maria Bruzón ◽  
Rita Tracinà ◽  
Chaudry Masood Khalique
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Conservation Laws ◽  
Symmetry Groups ◽  
Partial Differential ◽  
Recent Advances
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