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Lie Symmetries and Conservation Laws of the Generalised Foam-Drainage Equation

  • Zhi-Yong Zhang EMAIL logo and Kai-Hua Ma
Published/Copyright: January 11, 2017
Zeitschrift für Naturforschung A
From the journal Zeitschrift für Naturforschung A Volume 72 Issue 3

Abstract

We perform a complete Lie point symmetry classification of the generalised foam-drainage equation and then construct an optimal system of one-dimensional subalgebra of the admitted symmetry operators and use them to reduce the equations under study. A power series solution of the reduced equation is constructed. Moreover, we find all multipliers of the equations and apply them to construct conservation laws.

Keywords: Conservation Law; Generalised Foam-Drainage Equation; Optimal System; Symmetry

Acknowledgements

This paper is supported by the National Natural Science Foundation of China (Nos. 11671014, 11301012), Scientic Research Project of Beijing Educational Committee, the Youth Talent Program (No. XN071009) and College Student’s Sci-tech Activities of North China University of Technology.

Appendix

In this section, we show the convergence of the power solution (22). From (24), we get

(A-1)|cn + 2|≤M[|cn|+∑k = 0n|ck||cn + 1 − k|+∑k = 0n∑j = 0k|cn − k||cj + 1||ck + 1 − j| +∑k = 1n∑j = 0k|cn + 2 − k||cjck − j|],

whereM=max{|1λc02|, 2c02}.

Assuming another power series

(A-2)μ=P(z)=∑n = 0∞Pnzn,

where

Pn + 2=M[Pn+∑k = 0nPkPn + 1 − k+∑k = 0n∑j = 0kPn − kPj + 1Pk + 1 − j +∑k = 1n∑j = 0kPn + 2 − kPjPk − j]

and P0=|c0|, P1=|c1|, P2=|(1−2λc12+2λc1)/(2λc0)|. Obviously, we find

|cn|≤Pn   n=0, 1, 2, …

Therefore, the series (A-2) is the majorant series of (22), so we have

μ=P(z)=P0+P1z+P2z2+∑n = 1∞Pn + 2zn + 2=P0+P1z+P2z2 +M[∑n = 1∞Pnzn + 2+∑n = 1∞∑k = 0nPkPn + 1 − kzn + 2 +∑n = 1∞∑k = 0n∑j = 0kPn − kPj + 1Pk + 1 − jzn + 2+∑n = 1∞∑k = 1n∑j = 0kPn + 2 − kPjPk − jzn + 2] =P0+P1z+P2z2+M[P(z)z2+P(z)z(P(z)−P0−P1z) +(P(z)−P0)3+P(z)(P(z)−P0)(P(z)−P0−P1z)].

Let

Θ(z, μ)=μ−P0−P1z−P2z2−M[μz2+ μz(μ−P0−P1z)+(μ−P0)3+ μ(μ−P0)(μ−P0−P1z)]=0.

It is easy to find that Θ(z, μ) is the analytic implicit in the real plane, Θ(0, P0)=0 and Θ′μ(0, P0)=1. According to the implicit function theorem, we conclude μ=P(z) is analytic in a neighbourhood of the point (0, P0) of the plane with a positive radius. This implies the power series (22) converges in a neighbourhood of the plane.

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Received: 2016-9-3
Accepted: 2016-11-18
Published Online: 2017-1-11
Published in Print: 2017-3-1

©2017 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Hydrogen and Carbon Vapour Pressure Isotope Effects in Liquid Fluoroform Studied by Density Functional Theory
  3. Analytic Approximations to Nonlinear Boundary Value Problems Modeling Beam-Type Nano-Electromechanical Systems
  4. Resistance Distances and Kirchhoff Index in Generalised Join Graphs
  5. Residual Symmetries and Interaction Solutions for the Classical Korteweg-de Vries Equation
  6. Nanofluidic Transport over a Curved Surface with Viscous Dissipation and Convective Mass Flux
  7. Reconstruction of f(T) Gravity with Interacting Variable-Generalised Chaplygin Gas and the Thermodynamics with Corrected Entropies
  8. Peristaltic Flow of Rabinowitsch Fluid in a Curved Channel: Mathematical Analysis Revisited
  9. Analysis of the Laminar Newtonian Fluid Flow Through a Thin Fracture Modelled as a Fluid-Saturated Sparsely Packed Porous Medium
  10. Lie Symmetries and Conservation Laws of the Generalised Foam-Drainage Equation
  11. Lie Symmetry Analysis, Analytical Solutions, and Conservation Laws of the Generalised Whitham–Broer–Kaup–Like Equations
  12. Discrete and Semidiscrete Models for AKNS Equation
https://doi.org/10.1515/zna-2016-0336
Keywords for this article
Conservation Law; Generalised Foam-Drainage Equation; Optimal System; Symmetry

Articles in the same Issue

  1. Frontmatter
  2. Hydrogen and Carbon Vapour Pressure Isotope Effects in Liquid Fluoroform Studied by Density Functional Theory
  3. Analytic Approximations to Nonlinear Boundary Value Problems Modeling Beam-Type Nano-Electromechanical Systems
  4. Resistance Distances and Kirchhoff Index in Generalised Join Graphs
  5. Residual Symmetries and Interaction Solutions for the Classical Korteweg-de Vries Equation
  6. Nanofluidic Transport over a Curved Surface with Viscous Dissipation and Convective Mass Flux
  7. Reconstruction of f(T) Gravity with Interacting Variable-Generalised Chaplygin Gas and the Thermodynamics with Corrected Entropies
  8. Peristaltic Flow of Rabinowitsch Fluid in a Curved Channel: Mathematical Analysis Revisited
  9. Analysis of the Laminar Newtonian Fluid Flow Through a Thin Fracture Modelled as a Fluid-Saturated Sparsely Packed Porous Medium
  10. Lie Symmetries and Conservation Laws of the Generalised Foam-Drainage Equation
  11. Lie Symmetry Analysis, Analytical Solutions, and Conservation Laws of the Generalised Whitham–Broer–Kaup–Like Equations
  12. Discrete and Semidiscrete Models for AKNS Equation
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