scholarly journals Classification of the Group Invariant Solutions for Contaminant Transport in Saturated Soils under Radial Uniform Water Flows

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
M. M. Potsane ◽  
R. J. Moitsheki
Keyword(s):  
Contaminant Transport ◽  
Saturated Soils ◽  
Lie Point Symmetries ◽  
Invariant Solutions ◽  
Deterministic Models ◽  
Group Invariant ◽  
Long Run ◽  
Stream Functions ◽  
Group Invariant Solutions ◽  
The One

The transport of chemicals through soils to the groundwater or precipitation at the soils surfaces leads to degradation of these resources. Serious consequences may be suffered in the long run. In this paper, we consider macroscopic deterministic models describing contaminant transport in saturated soils under uniform radial water flow backgrounds. The arising convection-dispersion equation given in terms of the stream functions is analyzed using classical Lie point symmetries. A number of exotic Lie point symmetries are admitted. Group invariant solutions are classified according to the elements of the one-dimensional optimal systems. We analyzed the group invariant solutions which satisfy the physical boundary conditions.

scholarly journals On optimal system, exact solutions and conservation laws of the modified equal-width equation

2018 ◽  
Vol 3 (2) ◽  
pp. 409-418 ◽  
Author(s):  
Chaudry Masood Khalique ◽  
Oke Davies Adeyemo ◽  
Innocent Simbanefayi
Keyword(s):  
Wave Propagation ◽  
Conservation Laws ◽  
Optimal System ◽  
Lie Point Symmetries ◽  
Invariant Solutions ◽  
Nonlinear Media ◽  
One Dimensional ◽  
Group Invariant ◽  
Group Invariant Solutions ◽  
Dimensional Wave

AbstractIn this paper we study the modified equal-width equation, which is used in handling simulation of a single dimensional wave propagation in nonlinear media with dispersion processes. Lie point symmetries of this equation are computed and used to construct an optimal system of one-dimensional subalgebras. Thereafter using an optimal system of one-dimensional subalgebras, symmetry reductions and new group-invariant solutions are presented. The solutions obtained are cnoidal and snoidal waves. Furthermore, conservation laws for the modified equal-width equation are derived by employing two different methods, the multiplier method and Noether approach.

scholarly journals Symmetry Solutions and Conservation Laws for the 3D Generalized Potential Yu-Toda-Sasa-Fukuyama Equation of Mathematical Physics

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2058
Author(s):  
Chaudry Masood Khalique ◽  
Karabo Plaatjie ◽  
Oageng Lawrence Diteho
Keyword(s):  
Conservation Laws ◽  
Elliptic Integral ◽  
Three Dimensional ◽  
Mathematical Physics ◽  
Lie Point Symmetries ◽  
Invariant Solutions ◽  
Conservation Of Energy ◽  
Group Invariant ◽  
Generalized Potential ◽  
Group Invariant Solutions

In this paper we study the fourth-order three-dimensional generalized potential Yu-Toda-Sasa-Fukuyama (gpYTSF) equation by first computing its Lie point symmetries and then performing symmetry reductions. The resulting ordinary differential equations are then solved using direct integration, and exact solutions of gpYTSF equation are obtained. The obtained group invariant solutions include the solution in terms of incomplete elliptic integral. Furthermore, conservation laws for the gpYTSF equation are derived using both the multiplier and Noether’s methods. The multiplier method provides eight conservation laws, while the Noether’s theorem supplies seven conservation laws. These conservation laws include the conservation of energy and mass.

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.

scholarly journals Two Isospectral-Nonisospectral Super-Integrable Hierarchies and Related Invariant Solutions

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1797
Author(s):  
Huanhuan Lu ◽  
Yufeng Zhang
Keyword(s):  
Integrable Hierarchies ◽  
Integrable Hierarchy ◽  
Lie Point Symmetries ◽  
Invariant Solutions ◽  
Group Invariant ◽  
Loop Algebras ◽  
Zero Curvature ◽  
Group Invariant Solutions ◽  
Fokker Plank Equation ◽  
Coupled Equation

In this article, we adopt two kinds of loop algebras corresponding to the Lie algebra B(0,1) to introduce two line spectral problems with different numbers of even and odd superfunctions. Through generalizing the time evolution λt to a polynomial of λ, two isospectral-nonisospectral super integrable hierarchies are derived in terms of Tu scheme and zero-curvature equation. Among them, the first super integrable hierarchy is further reduced to generalized Fokker–Plank equation and special bond pricing equation, as well as an explicit super integrable system under the choice of specific parameters. More specifically, a super integrable coupled equation is derived and the corresponding integrable properties are discussed, including the Lie point symmetries and one-parameter Lie symmetry groups as well as group-invariant solutions associated with characteristic equation.

Lie Symmetries, One-Dimensional Optimal System and Group Invariant Solutions for the Ripa System

2019 ◽  
Vol 20 (6) ◽  
pp. 713-723 ◽  
Author(s):  
Pabitra Kumar Pradhan ◽  
Manoj Pandey
Keyword(s):  
Adjoint Representation ◽  
Group Classification ◽  
Optimal System ◽  
Invariant Solutions ◽  
Horizontal Temperature Gradient ◽  
One Dimensional ◽  
Group Invariant ◽  
Group Invariant Solutions ◽  
The One ◽  
Complete Symmetry Group

AbstractA complete symmetry group classification for the system of shallow water equations with the horizontal temperature gradient, also known as Ripa system, is presented. A rigorous and systematic procedure based on the general invariants of the adjoint representation is used to construct the one-dimensional optimal system of the Lie algebra. The complete inequivalence class of the group invariant solutions are obtained by using the one-dimensional optimal system. One such solution of the Ripa system is used to study the evolutionary behaviour of the discontinuity wave.

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.

scholarly journals Group Classification of a General Bond-Option Pricing Equation of Mathematical Finance

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Tanki Motsepa ◽  
Chaudry Masood Khalique ◽  
Motlatsi Molati
Keyword(s):  
Option Pricing ◽  
Three Dimensional ◽  
Mathematical Finance ◽  
Group Classification ◽  
Lie Point Symmetries ◽  
Group Invariant ◽  
Black Scholes ◽  
Group Invariant Solutions ◽  
Bond Option

We carry out group classification of a general bond-option pricing equation. We show that the equation admits a three-dimensional equivalence Lie algebra. We also show that some of the values of the constants which result from group classification give us well-known models in mathematics of finance such as Black-Scholes, Vasicek, and Cox-Ingersoll-Ross. For all such values of these arbitrary constants we obtain Lie point symmetries. Symmetry reductions are then obtained and group invariant solutions are constructed for some cases.

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