Partial Noether Research Articles

Numerical investigation on the thermal response of an unsteady magnetohydrodynamics straight porous fin

Fins are frequently used in many heat transfer applications and also for the control and prevention of thermal damage in electronic and mechanical equipment. Thus, the current investigation focuses on estimating the thermal performance, efficiency, and effectiveness of MHD straight porous fin with temperature-dependent heat transfer coefficient, thermal conductivity, and emissivity. The finite-difference approximation is utilized to solve the governing nonlinear differential equation and numerical solutions are endorsed by analytical solutions of the steady-state model using the partial Noether method, Adomian decomposition method, and homotopy perturbation method. From the model predictions, it is noticed that the rise in geometry parameter, magnetic parameter, Rayleigh number, time, and Biot number rises the heat transfer rate of the system. In addition, fin efficiency and effectiveness are directly correlated with boiling parameter, thermal conductivity parameter, and porosity.

Lie symmetry analysis and conservation laws of non-linear (2+1) elastic wave equation

The elastic wave propagation in inelastic media gives rise to non-linear wave equation. We study such a nonlinear wave in two dimensions using Murnaghan model. Lie symmetries, invariant exact solutions and conservation laws using the Noether theorem have been found. The nonlinear elastic wave equation with a damping term has been dealt with using the partial Noether approach.

The analysis of conservation laws, symmetries and solitary wave solutions of Burgers–Fisher equation

In this paper, the conservation laws, significant symmetries’ application, and traveling wave solutions are obtained for Burger–Fisher equation (BFE). Conservation laws have a great importance for partial and fractional differential equations and their solutions, especially in physics implementations. The conservation theorem and partial Noether approach are implemented for conservation laws for this equation, and the extended sinh-Gordon expansion method (esGEM) is presented for new solitary wave solutions. All obtained conservation laws are trivial conservation laws. The new and comprehensive solitary wave solutions of the equation by the esGEM are also obtained.

THERMAL PERFORMANCE ANALYSIS OF CONVECTIVE-RADIATIVE FIN WITH TEMPERATURE-DEPENDENT THERMAL CONDUCTIVITY IN THE PRESENCE OF UNIFORM MAGNETIC FIELD USING PARTIAL NOETHER METHOD

In this paper, thermal performance of convective-radiative straight fin with temperature-dependent thermal conductivity in the presence of uniform magnetic field is analyzed using partial Noether method. The exact analytical solution is used to investigate the effects of magnetic field, convective, radiative, thermo-geometric and thermal conductivity (non-linear) parameters on the thermal performance of the fin. The results reveal that as the magnetic, convective and radiative parameters increase, the temperature of the fin decreases rapidly and by implication, the rate of heat transfer through the fin increases. The study provides a platform for comparison of results of any other method of analysis of the problem with the results of the exact analytical solutions in this paper. Also, such an analytical tool is valuable as a design and optimization approach for large scale (not necessarily in size) finned heat exchangers where each fin/row are analytically analyzed and where the surrounding fluid is influenced by a magnetic field.

NEW EXACT SOLUTIONS OF A GENERALISED BOUSSINESQ EQUATION WITH DAMPING TERM AND A SYSTEM OF VARIANT BOUSSINESQ EQUATIONS VIA DOUBLE REDUCTION THEORY

The conservation laws of a generalised Boussinesq (GB) equation with damping term are derived via the partial Noether approach. The derived conserved vectors are adjusted to satisfy the divergence condition. We use the definition of the association of symmetries of partial differential equations with conservation laws and the relationship between symmetries and conservation laws to find a double reduction of the equation. As a result, several new exact solutions are obtained. A similar analysis is performed for a system of variant Boussinesq (VB) equations.

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.

Generalization of approximate partial Noether approach in phase space

The approximate partial Noether theorem proposed earlier for the ordinary differential equations (ODEs) (Naeem and Mahomed in Nonlinear Dyn 57(1–2):303–311, 2009) is generalized in phase space for the perturbed Hamiltonian-type systems. The notion of approximate partial Hamiltonian is developed. An approximate partial Hamiltonian gives rise to an approximate Hamiltonian-type perturbed dynamical system of first-order ODEs. An approximate Legendre-type transformation connects the approximate partial Lagrangian and what we term as approximate partial Hamiltonian. The formulas for approximate partial Hamiltonian operators determining equations and first integrals are provided explicitly. We have explained our approach with the help of simple illustrative example. Then, it is applied to establish the approximate first integrals, reductions and exact solutions of two perturbed cubically coupled Duffing–Van der Pol oscillators. Both resonant and nonresonant cases are considered.

First integrals and analytical solutions of the nonlinear fin problem with temperature-dependent thermal conductivity and heat transfer coefficient

Fin materials can be observed in a variety of engineering applications. They are used to ease the dissipation of heat from a heated wall to the surrounding environment. In this work, we consider a nonlinear fin problem with temperature-dependent thermal conductivity and heat transfer coefficient. The equation(s) under study are highly nonlinear. Both the thermal conductivity and the heat transfer coefficient are given as arbitrary functions of temperature. Firstly, we consider the Lie group analysis for different cases of thermal conductivity and the heat transfer coefficients. These classifications are obtained from the Lie group analysis. Then, the first integrals of the nonlinear straight fin problem are constructed by three methods, namely, Noether’s classical method, partial Noether approach and Ibragimov’s nonlocal conservation method. Some exact analytical solutions are also constructed. The obtained result is also compared with the result obtained by other methods.

On Conservation Forms and Invariant Solutions for Classical Mechanics Problems of Liénard Type

In this study we apply partial Noether andλ-symmetry approaches to a second-order nonlinear autonomous equation of the formy′′+fyy′+g(y)=0, called Liénard equation corresponding to some important problems in classical mechanics field with respect tof(y)andg(y)functions. As a first approach we utilize partial Lagrangians and partial Noether operators to obtain conserved forms of Liénard equation. Then, as a second approach, based on theλ-symmetry method, we analyzeλ-symmetries for the case thatλ-function is in the form ofλ(x,y,y′)=λ1(x,y)y′+λ2(x,y). Finally, a classification problem for the conservation forms and invariant solutions are considered.

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.

Conservation Laws For the (1+2)-Dimensional Wave Equation in Biological Environment

The derivation of conservation laws for the wave equation on sphere, cone and flat space is considered. The partial Noether approach is applied for wave equation on curved surfaces in terms of the coefficients of the first fundamental form (FFF) and the partial Noether operator's determining equations are derived. These determining equations are then used to construct the partial Noether operators and conserved vectors for the wave equation on different surfaces. The conserved vectors for the wave equation on the sphere, cone and flat space are simplified using the Lie point symmetry generators of the equation and conserved vectors with the help of the symmetry conservation laws relation.

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 and exact solutions of a class of non linear regularized long wave equations via double reduction theory and Lie symmetries

The conservation laws for (1+1)-dimensional non-linear generalized regularized long wave (GRLW) equation are derived via partial Noether approach after increasing its order. The GRLW equation is a third-order partial differential equation. We convert GRLW equation to fourth order equation by assuming new dependent variable v to be the derivative of original dependent variable u by setting either u=vx or u=vt. The partial Noether’s approach is then used to derive the conservation laws. The derived conserved vectors are adjusted to satisfy the divergence relationship. Finally, the conservation laws are expressed in the variable u and they constitute the conservation laws for the third-order GRLW equation. The Lie point symmetries for GRLW equation are computed. The double reduction theory based on symmetry and its associated conserved vector is utilized and two independent exact solutions are obtained. Moreover, the Lie symmetry method is used to derive an invariant solution. One of the solutions obtained by the double reduction method is the same as derived by Lie symmetry method. The second solution constructed by the double reduction theory is not obtained by the Lie symmetry method. A similar analysis is performed for regularized long wave (RLW) and modified Benjamin–Bona–Mahoney (MBBM) equations.

Conserved quantities and group classification of wave equation on hyperbolic space

This paper discusses the nonlinear wave equation on hyperbolic space. We extend some results reported in Nadjafikhah and Zaeim (2011) [17]. Partial Noether theorem is applied to the equation under consideration and partial Noether operators are computed for different cases of f(u). Furthermore, these partial Noether operators are utilized to calculate corresponding conserved quantities. A complete group classification is discussed for the considered equation. Maximal solvable algebra of Lie point symmetries is used to reduce the class of considered equation. Then algebras of translational symmetries are associated with its infinite conservation laws.

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.

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.

On the conservation laws for a certain class of nonlinear wave equation via a new conservation theorem

Bokhari et al. [14] applied the partial Noether approach to find the Noether-type operators and conservation laws for a certain class of the nonlinear (2 + 1) wave equation u tt = ( f( u) u x ) x + ( g( u) u y ) y describing waves in two dimension involving arbitrary velocity functions. These arbitrary functions generally arise when transmitting a signal on a transmission plane with material properties that are changing along the plane. Recently, Ibragimov in [15] presented nonlocal conservation theorem method. In this study, we investigate conservation laws via the new conservation theorem. We show that Y = β ( x ) ∂ ∂ v is a Noether symmetry for the system consisting of the PDE and the its adjoint equation with respect to the formal Lagrangian, where v = β( x) is a solution for the adjoint equation. Then we show that the basic set of conservation laws that is estimated by the partial Noether approach can be given by the new conservation theorem if we consider the Noether symmetry β ( x ) ∂ ∂ v .We note that the classification of the variable v in the adjoint equation in the new conservation theorem approach is equivalent to the classification of the characteristics of the Noether type operator in the partial Noether approach. New four sets of conservation laws corresponding to translations and scaling symmetries are constructed that did not appear before. We show that each one of these new four sets is linearly independent set and form a subgroup of conservation laws.

Approximate partial Noether operators and first integrals for cubically coupled nonlinear Duffing oscillators subject to a periodically driven force

The approximate first integrals for a system of two cubically coupled nonlinear Duffing oscillators subject to a periodically driven force are constructed with the help of approximate partial Noether approach. Firstly, approximate partial Noether operators associated with a partial Lagrangian are derived. Then approximate first integrals are obtained for both the resonant and non-resonant cases.

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.

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.