Isogroup classification and group-invariant solutions of the nonlinear diffusion-convection equation T = (D1(T)Tx)x − D2′(T)Tx

1997 ◽  
Vol 35 (1) ◽  
pp. 1-14 ◽  
Author(s):  
K. Vijayakumar
Keyword(s):  
Nonlinear Diffusion ◽  
Invariant Solutions ◽  
Group Invariant ◽  
Convection Equation ◽  
Group Invariant Solutions ◽  
Diffusion Convection

Group Invariant Solutions and Conservation Laws of the Fornberg– Whitham Equation

2014 ◽  
Vol 69 (8-9) ◽  
pp. 489-496 ◽  
Author(s):  
Mir Sajjad Hashemi ◽  
Ali Haji-Badali ◽  
Parisa Vafadar
Keyword(s):  
Exact Solutions ◽  
Reduction Method ◽  
Lie Symmetry ◽  
First Integrals ◽  
Invariant Solutions ◽  
Analysis Method ◽  
Lie Symmetry Analysis ◽  
Group Invariant ◽  
Group Invariant Solutions ◽  
Whitham Equation

In this paper, we utilize the Lie symmetry analysis method to calculate new solutions for the Fornberg-Whitham equation (FWE). Applying a reduction method introduced by M. C. Nucci, exact solutions and first integrals of reduced ordinary differential equations (ODEs) are considered. Nonlinear self-adjointness of the FWE is proved and conserved vectors are computed

scholarly journals First Integrals, Integrating Factors, and Invariant Solutions of the Path Equation Based on Noether and -Symmetries

2013 ◽  
Vol 2013 ◽  
pp. 1-15 ◽  
Author(s):  
Gülden Gün ◽  
Teoman Özer
Keyword(s):  
Governing Equation ◽  
First Integrals ◽  
Invariant Solutions ◽  
Group Invariant ◽  
Minimum Drag ◽  
Symmetry Properties ◽  
Group Invariant Solutions ◽  
Work First ◽  
Partial Lagrangian ◽  
Integrating Factors

We analyze Noether and -symmetries of the path equation describing the minimum drag work. First, the partial Lagrangian for the governing equation is constructed, and then the determining equations are obtained based on the partial Lagrangian approach. For specific altitude functions, Noether symmetry classification is carried out and the first integrals, conservation laws and group invariant solutions are obtained and classified. Then, secondly, by using the mathematical relationship with Lie point symmetries we investigate -symmetry properties and the corresponding reduction forms, integrating factors, and first integrals for specific altitude functions of the governing equation. Furthermore, we apply the Jacobi last multiplier method as a different approach to determine the new forms of -symmetries. Finally, we compare the results obtained from different classifications.

Perturbed invariant subspaces and approximate generalized functional variable separation solution for nonlinear diffusion–convection equations with weak source

2018 ◽  
Vol 32 (07) ◽  
pp. 1850093
Author(s):  
Ya-Rong Xia ◽  
Shun-Li Zhang ◽  
Xiang-Peng Xin
Keyword(s):  
Nonlinear Diffusion ◽  
Invariant Subspaces ◽  
Complete Classification ◽  
Variable Separation ◽  
Weak Source ◽  
Convection Equation ◽  
Functional Variable ◽  
Diffusion Convection

In this paper, we propose the concept of the perturbed invariant subspaces (PISs), and study the approximate generalized functional variable separation solution for the nonlinear diffusion–convection equation with weak source by the approximate generalized conditional symmetries (AGCSs) related to the PISs. Complete classification of the perturbed equations which admit the approximate generalized functional separable solutions (AGFSSs) is obtained. As a consequence, some AGFSSs to the resulting equations are explicitly constructed by way of examples.

Optimal systems and their group-invariant solutions to geodesic equations

2019 ◽  
Vol 16 (09) ◽  
pp. 1950135
Author(s):  
Bismah Jamil ◽  
Tooba Feroze ◽  
Muhammad Safdar
Keyword(s):  
Nonlinear Systems ◽  
Separation Of Variables ◽  
Noether Symmetries ◽  
Invariant Solutions ◽  
One Dimensional ◽  
Group Invariant ◽  
First Order ◽  
Geodesic Equations ◽  
Integrating Factor ◽  
Group Invariant Solutions

We find one-dimensional optimal systems of the Lie subalgebras of Noether symmetries associated with systems of geodesic equations. Further, we find invariants corresponding to each element of the derived optimal system. The derived invariants are shown to reduce systems of geodesic equations (nonlinear systems of quadratically semi-linear second-order ordinary differential equations (ODEs)) to nonlinear systems of first-order ODEs. The resulting systems are solved via known methods (e.g. separation of variables, integrating factor, etc.). In some cases, we provide exact solutions of these systems of geodesic equations.

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