scholarly journals Group invariant solutions of certain partial differential equations

2021 ◽  
Vol 315 (1) ◽  
pp. 235-254
Author(s):  
Jaime Ripoll ◽  
Friedrich Tomi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Invariant Solutions ◽  
Partial Differential ◽  
Group Invariant ◽  
Group Invariant Solutions

scholarly journals λ-Symmetry and μ-Symmetry Reductions and Invariant Solutions of Four Nonlinear Differential Equations

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1138
Author(s):  
Yu-Shan Bai ◽  
Jian-Ting Pei ◽  
Wen-Xiu Ma
Keyword(s):  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Order Partial Differential Equation ◽  
Invariant Solutions ◽  
Specific Group ◽  
Partial Differential ◽  
Group Invariant ◽  
Dimensional Diffusion ◽  
Group Invariant Solutions ◽  
Integrating Factors

On one hand, we construct λ-symmetries and their corresponding integrating factors and invariant solutions for two kinds of ordinary differential equations. On the other hand, we present μ-symmetries for a (2+1)-dimensional diffusion equation and derive group-reductions of a first-order partial differential equation. A few specific group invariant solutions of those two partial differential equations are constructed.

OPTIMAL SYSTEMS AND GROUP INVARIANT SOLUTIONS FOR A MODEL ARISING IN FINANCIAL MATHEMATICS

2009 ◽  
Vol 14 (4) ◽  
pp. 495-502 ◽  
Author(s):  
Bienvenue Feugang Nteumagne ◽  
Raseelo J. Moitsheki
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Financial Mathematics ◽  
Invariant Solutions ◽  
Pricing Model ◽  
One Dimensional ◽  
Group Invariant ◽  
Optimal System Of Subalgebras ◽  
Group Invariant Solutions ◽  
The One

We consider a bond‐pricing model described in terms of partial differential equations (PDEs). Classical Lie point symmetry analysis of the considered PDEs resulted in a number of point symmetries being admitted. The one‐dimensional optimal system of subalgebras is constructed. Following the symmetry reductions, we determine the group‐invariant solutions.

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.

scholarly journals A Technique to Classify the Similarity Solutions of Nonlinear Partial (Integro-)Differential Equations. I. Optimal Systems of Solvable Lie Subalgebras

1992 ◽  
Vol 47 (11) ◽  
pp. 1161-1174 ◽  
Author(s):  
H. Kötz
Keyword(s):  
Differential Equations ◽  
Symmetry Group ◽  
Lie Group ◽  
Similarity Solutions ◽  
Point Symmetry ◽  
Invariant Solutions ◽  
Point Symmetry Group ◽  
Group Invariant ◽  
Group Invariant Solutions ◽  
Lie Point Symmetry

Abstract Lie group analysis is a powerful tool for obtaining exact similarity solutions of nonlinear (integro-) differential equations. In order to calculate the group-invariant solutions one first has to find the full Lie point symmetry group admitted by the given (integro-)differential equations and to determine all the subgroups of this Lie group. An effective, systematic means to classify the similarity solutions afterwards is an "optimal system", i.e. a list of group-invariant solutions from which every other such solution can be derived. The problem to find optimal systems of similarity solutions leads to that to "construct" the optimal systems of subalgebras for the Lie algebra of the known Lie point symmetry group. Our aim is to demonstrate a practicable technique for determining these optimal subalgebraic systems using the invariants relative to the group of the inner automorphisms of the Lie algebra in case of a finite-dimensional Lie point symmetry group. Here, we restrict our attention to optimal subsystems of solvable Lie subalgebras. This technique is applied to the nine-dimensional real Lie point symmetry group admitted by the two-dimensional non-stationary ideal magnetohydrodynamic equations

The reducibility of partially invariant solutions of systems of partial differential equations

1995 ◽  
Vol 6 (4) ◽  
pp. 329-354 ◽  
Author(s):  
Jeffrey Ondich
Keyword(s):  
Differential Equations ◽  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
Invariant Solution ◽  
Elliptic Systems ◽  
Invariant Solutions ◽  
Hyperbolic Conservation ◽  
Group Invariant ◽  
Partially Invariant Solution ◽  
Group Invariant Solutions

Ovsiannikov's partially invariant solutions of differential equations generalize Lie's group invariant solutions. A partially invariant solution is only interesting if it cannot be discovered more readily as an invariant solution. Roughly, a partially invariant solution that can be discovered more directly by Lie's method is said to be reducible. In this paper, I develop conditions under which a partially invariant solution or a class of such solutions must be reducible, and use these conditions both to obtain non-reducible solutions to a system of hyperbolic conservation laws, and to demonstrate that some systems have no non-reducible solutions. I also demonstrate that certain elliptic systems have no non-reducible solutions.

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