Partial Lagrangian Approach Research Articles

Lie Symmetry Classification, Optimal System, and Conservation Laws of Damped Klein–Gordon Equation with Power Law Non-Linearity

We used the classical Lie symmetry method to study the damped Klein–Gordon equation (Kge) with power law non-linearity utt+α(u)ut=(uβux)x+f(u). We carried out a complete Lie symmetry classification by finding forms for α(u) and f(u). This led to various cases. Corresponding to each case, we obtained one-dimensional optimal systems of subalgebras. Using the subalgebras, we reduced the Kge to ordinary differential equations and determined some invariant solutions. Furthermore, we obtained conservation laws using the partial Lagrangian approach.

Discrete Gradient Methods for Solving SIRI Epidemic Model Numerically While Preserving First Integrals

In this paper, we use discrete gradient methods for numerical solutions of mathematical models of epidemiology: a susceptible infected recovered infected (SIRI) model. The numerical results mimic the true and qualitatively correct behavior of the SIRI model. The first integrals (also known as the constant of motion) of the SIRI model obtained in literature using partial Lagrangian approach are well preserved by the discrete gradient method. The graphical results verify the advantages of the proposed schemes.

On the conservation laws of modified KdV-KP equation

In this paper we study conservation laws of modified Korteweg-de Vries-Kadomtsev Petviashvili equation (mKdv-KP). This equation does not have a classical Lagrangian so one can not apply Noether's celebrated theorem. However, we show that using the partial Lagrangian approach and multiplier method one can obtain a number of local and nonlocal conservation laws for underlying equation.

Conserved Quantities for the General Linear Heat Equation

In this paper, conserved quantities are computed for a class of evolution equation by using the partial Noether approach [2]. The partial Lagrangian approach is applied to the considered equation, infinite many conservation laws are obtained depending on the coefficients of equation for each n. These results give potential systems for the family of considered equation, which are further helpful to compute the exact solutions.

First integrals and exact solutions of the SIRI and tuberculosis models

The first integrals and exact solutions of mathematical models of epidemiology: a susceptible‐infected‐recovered‐infected (SIRI) model and a tuberculosis model with demographic growth are analyzed. These models are represented by systems of first‐order nonlinear ordinary differential equations, and this system is replaced by one which contains a second‐order ordinary differential equation. The partial Lagrangian approach is then utilized to derive the first integrals of these models. Several cases arise. Then, we utilize the derived first integrals to construct exact solutions for the models under investigation and determine new solutions. The dynamic properties of these models are studied too. Copyright © 2016 John Wiley & Sons, Ltd.

A Partial Lagrangian Approach to Mathematical Models of Epidemiology

This paper analyzes the first integrals and exact solutions of mathematical models of epidemiology via the partial Lagrangian approach by replacing the three first-order nonlinear ordinary differential equations by an equivalent system containing one second-order equation and a first-order equation. The partial Lagrangian approach is then utilized for the second-order ODE to construct the first integrals of the underlying system. We investigate the SIR and HIV models. We obtain two first integrals for the SIR model with and without demographic growth. For the HIV model without demography, five first integrals are established and two first integrals are deduced for the HIV model with demography. Then we utilize the derived first integrals to construct exact solutions to the models under investigation. The dynamic properties of these models are studied too. Numerical solutions are derived for SIR models by finite difference method and are compared with exact solutions.

On symmetry group classification of a fin equation

Abstract We deal with the Noether symmetry classification of a nonlinear fin equation, in which thermal conductivity and heat transfer coefficient are assumed to be functions of the temperature. In this study Noether symmetries of the fin equation are investigated using the partial Lagrangian approach. This classification includes Noether symmetries, first integrals and some invariant solutions with respect to different choices of thermal conductivity and heat transfer coefficient functions.

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

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.

Approximate conservation laws of nonlinear perturbed heat and wave equations

We construct approximate conservation laws for non-variational nonlinear perturbed (1+1) heat and wave equations by utilizing the partial Lagrangian approach. These perturbed nonlinear heat and wave equations arise in a number of important applications which are reviewed. Approximate symmetries of these have been obtained in the literature. Approximate partial Noether operators associated with a partial Lagrangian of the underlying perturbed heat and wave equations are derived herein. These approximate partial Noether operators are then used via the approximate version of the partial Noether theorem in the construction of approximate conservation laws of the underlying perturbed heat and wave equations.

Potential symmetries and conservation laws for generalized quasilinear hyperbolic equations

Based on the Lie group method, the potential symmetries and invariant solutions for generalized quasilinear hyperbolic equations are studied. To obtain the invariant solutions in an explicit form, the physically interesting situations with potential symmetries are focused on, and the conservation laws for these equations in three physically interesting cases are found by using the partial Lagrangian approach.

Conserved quantities for a class of (1 + n)-dimensional linear evolution equation

A special type of (1+n)-dimensional linear evolution equation is considered. A class of the equations generated by the Fokker–Planck equation becomes the subcase of the considered equation. Conserved vectors using the partial Lagrangian approach is derived in terms of the coefficients of the discussed equation. Derived results are used for the different models from different sciences. We also discuss the conservation laws of the heat equation on curved manifolds and in different coordinate systems. Potential systems are also obtained for some models. At last conclusion is given.

Lie point symmetries, partial Noether operators and first integrals of the Painlevé–Gambier equations

We utilize the Lie–Tressé linearization method to obtain linearizing point transformations of certain autonomous nonlinear second-order ordinary differential equations contained in the Painlevé–Gambier classification. These point transformations are constructed using the Lie point symmetry generators admitted by the underlying Painlevé–Gambier equations. It is also shown that those Painlevé–Gambier equations which have a few Lie point symmetries and hence are not linearizable by this method can be integrated by a quadrature. Moreover, by making use of the partial Lagrangian approach we obtain time dependent and time independent first integrals for these Painlevé–Gambier equations which have not been reported in the earlier literature. A comparison of the results obtained in this paper is made with the ones obtained using the generalized Sundman linearization method.

Variational approaches to conservation laws for a nonlinear evolution equation with time dependent coefficients

The conservation laws of a nonlinear evolution equation of time dependent variable coefficients of damping and dispersion is studied. The equation under consideration is not derivable from a variational principle which means that one cannot appeal to the Noether theorem to determine the conservation laws. We utilize the new conservation theorem (N.H. Ibragimov, [8]) and the partial Lagrangian approach (A.H. Kara, F.M. Mahomed, [13]) to construct local, and infinite number of nonlocal conservation laws (due to the transformation of the dependent variable) of the underlying equation.

Approximate First Integrals for a System of Two Coupled Van Der Pol Oscillators with Linear Diffusive Coupling

The approximate partial Noether operators for a system of two coupled van der Pol oscillators with linear diffusive coupling are presented via a partial Lagrangian approach. The underlying system of two equations, in general, do not admit a standard Lagrangian. However, the approximate first integrals are constructed by utilization of the partial Noether’s theorem with the help of approximate partial Noether operators associated with a partial Lagrangian. These approximate partial Noether operators are not approximate symmetries of the system under study and they do not form an approximate Lie algebra. Moreover, we show how approximate first integrals can be constructed for perturbed ordinary differential equations (ODEs) without making use of a standard Lagrangian.

Conservation laws of KdV equation with time dependent coefficients

We determine conservation laws of the generalized KdV equation of time dependent variable coefficients of the linear damping and dispersion. The underlying equation is not derivable from a variational principle and hence one cannot use Noether’s theorem here to construct conservation laws as there is no Lagrangian. However, we show that by utilizing the new conservation theorem and the partial Lagrangian approach one can construct a number of local and nonlocal conservation laws for the underlying equation.

On symmetries, conservation laws and invariant solutions of the foam-drainage equation

This study deals with symmetry group properties and conservation laws of the foam-drainage equation. Firstly, we study the classical Lie symmetries, optimal systems, similarity reductions and similarity solutions of the foam-drainage equation which are obtained through the Lie group method of infinitesimal transformations. Secondly, using the new general theorem on non-local conservation laws and partial Lagrangian approach, local and non-local conservation laws are also studied and, finally, non-classical symmetries are derived.

Conservation laws for a class of soil water equations

In this paper, we consider a class of nonlinear partial differential equations which model soil water infiltration, redistribution and extraction in a bedded soil profile irrigated by a line source drip irrigation system. By using the nonlocal conservation theorem method and the partial Lagrangian approach, conservation laws are presented. It is observed that both approaches lead to the nontrivial and infinite conservation laws.

On the conservation laws and invariant solutions of the mKdV equation

In this paper, we consider modified Korteweg–de Vries (mKdV) equation. By using the nonlocal conservation theorem method and the partial Lagrangian approach, conservation laws for the mKdV equation are presented. It is observed that only nonlocal conservation theorem method lead to the nontrivial and infinite conservation laws. In addition, invariant solution is obtained by utilizing the relationship between conservation laws and Lie-point symmetries of the equation.