I. Naeem

Comparison of Different Approaches to Construct First Integrals for Ordinary Differential Equations

Different approaches to construct first integrals for ordinary differential equations and systems of ordinary differential equations are studied here. These approaches can be grouped into three categories: direct methods, Lagrangian or partial Lagrangian formulations, and characteristic (multipliers) approaches. The direct method and symmetry conditions on the first integrals correspond to first category. The Lagrangian and partial Lagrangian include three approaches: Noether’s theorem, the partial Noether approach, and the Noether approach for the equation and its adjoint as a system. The characteristic method, the multiplier approaches, and the direct construction formula approach require the integrating factors or characteristics or multipliers. The Hamiltonian version of Noether’s theorem is presented to derive first integrals. We apply these different approaches to derive the first integrals of the harmonic oscillator equation. We also study first integrals for some physical models. The first integrals for nonlinear jerk equation and the free oscillations of a two-degree-of-freedom gyroscopic system with quadratic nonlinearities are derived. Moreover, solutions via first integrals are also constructed.

First integrals for a general linear system of two second-order ODEs via a partial Lagrangian

The partial Noether operators and first integrals of a general system of two linear second-order ordinary differential equations (ODEs) with variable coefficients are studied by means of a partial Lagrangian. The canonical form for the general system of two second-order ordinary differential equations is invoked and all cases of this system are discussed with respect to partial Noether operators. We also tabulate the results for the special case b(x) = c(x) of the system which was considered elsewhere using a Lagrangian and a partial Lagrangian. The first integrals are obtained explicitly by exploiting a Noether-like theorem with the help of partial Noether operators. This study gives a new way to construct first integrals for systems without a variational principle as not all linear equations have a Lagrangian. Physical applications to conservative and oscillator mechanical systems are given.

Reductions and New Exact Solutions of ZK, Gardner KP, and Modified KP Equations via Generalized Double Reduction Theorem

We study here the Lie symmetries, conservation laws, reductions, and new exact solutions of () dimensional Zakharov-Kuznetsov (ZK), Gardner Kadomtsev-Petviashvili (GKP), and Modified Kadomtsev-Petviashvili (MKP) equations. The multiplier approach yields three conservation laws for ZK equation. We find the Lie symmetries associated with the conserved vectors, and three different cases arise. The generalized double reduction theorem is then applied to reduce the third-order ZK equation to a second-order ordinary differential equation (ODE) and implicit solutions are established. We use the Sine-Cosine method for the reduced second-order ODE to obtain new explicit solutions of ZK equation. The Lie symmetries, conservation laws, reductions, and exact solutions via generalized double reduction theorem are computed for the GKP and MKP equations. Moreover, for the GKP equation, some new explicit solutions are constructed by applying the first integral method to the reduced equations.

Nonclassical Symmetry Analysis of Boundary Layer Equations

The nonclassical symmetries of boundary layer equations for two-dimensional and radial flows are considered. A number of exact solutions for problems under consideration were found in the literature, and here we find new similarity solution by implementing the SADE package for finding nonclassical symmetries.

Partial Noether Operators and First Integrals for a System with two Degrees of Freedom

Abstract We construct all partial Noether operators corresponding to a partial Lagrangian for a system with two degrees of freedom. Then all the first integrals are obtained explicitly by utilizing a Noether-like theorem with the help of the partial Noether operators. We show how the first integrals can be constructed for the system without the need of a variational principle although the Lagrangian L = y′2/2 + z′2/2—v(y, z) does exist for the system. Our objective is twofold: one is to see the effectiveness of the partial Noether approach and the other to determine all the first integrals of the system under study which have not been reported before. Thus, we deduce a complete classification of the potentials v(y, z) for which first integrals exist. This can give rise to further studies on systems which are not Hamiltonian via partial Noether operators and the construction of first integrals from a partial Lagrangian viewpoint.

Exact Solutions of Generalized Modified Boussinesq, Kuramoto-Sivashinsky, and Camassa-Holm Equations via Double Reduction Theory

We find exact solutions of the Generalized Modified Boussinesq (GMB) equation, the Kuromoto-Sivashinsky (KS) equation the and, Camassa-Holm (CH) equation by utilizing the double reduction theory related to conserved vectors. The fourth order GMB equation involves the arbitrary function and mixed derivative terms in highest derivative. The partial Noether’s approach yields seven conserved vectors for GMB equation and one conserved for vector KS equation. Due to presence of mixed derivative term the conserved vectors for GMB equation derived by the Noether like theorem do not satisfy the divergence relationship. The extra terms that constitute the trivial part of conserved vectors are adjusted and the resulting conserved vectors satisfy the divergence property. The double reduction theory yields two independent solutions and one reduction for GMB equation and one solution for KS equation. For CH equation two independent solutions are obtained elsewhere by double reduction theory with the help of conserved Vectors.

Conservation Laws of Some Physical Models via Symbolic Package GeM

We study the conservation laws of evolution equation, lubrication models, sinh-Poisson equation, Kaup-Kupershmidt equation, and modified Sawada-Kotera equation. The symbolic software GeM (Cheviakov (2007) and (2010)) is used to derive the multipliers and conservation law fluxes. Software GeM is Maple-based package, and it computes conservation laws by direct method and first homotopy and second homotopy formulas.

First Integrals for Two Linearly Coupled Nonlinear Duffing Oscillators

We investigate Noether and partial Noether operators of point type corresponding to a Lagrangian and a partial Lagrangian for a system of two linearly coupled nonlinear Duffing oscillators. Then, the first integrals with respect to Noether and partial Noether operators of point type are obtained explicitly by utilizing Noether and partial Noether theorems for the system under consideration. Moreover, if the partial Euler-Lagrange equations are independent of derivatives, then the partial Noether operators become Noether point symmetry generators for such equations. The difference arises in the gauge terms due to Lagrangians being different for respective approaches. This study points to new ways of constructing first integrals for nonlinear equations without regard to a Lagrangian. We have illustrated it here for nonlinear Duffing oscillators.

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.

Group classification and exact solutions of generalized modified Boussinesq equation

We present symmetry classification and exact solutions of generalized modified Boussinesq (GMB) equation. The direct method of group classification is utilized to determine four different functional forms of . The GMB equation admits two-dimensional principle algebra for arbitrary and the algebra extends to three-dimensional for other forms of . Similarity reductions are made in each case and exact solutions are derived.