Lie Symmetry Classification of the Generalized Nonlinear Beam Equation

Abstract

Abstract:In this paper we make a Lie symmetry analysis of a generalized nonlinear beam equation with both second-order and fourth-order wave terms, which is extended from the classical beam equation arising in the historical events of travelling wave behavior in the Golden Gate Bridge in San Francisco. We perform a complete Lie symmetry group classification by using the equivalence transformation group theory for the equation under consideration. Lie symmetry reductions of a nonlinear beam-like equation which are singled out from the classification results are investigated. Some classes of exact solutions, including solitary wave solutions, triangular periodic wave solutions and rational solutions of the nonlinear beam-like equations are constructed by means of the reductions and symbolic computation.

Similarity solutions and conservation laws for the Beam Equations: a complete study

“…In this work we study the algebraic properties of the Euler-Bernoulli, the Rayleigh and of the Timoshenko-Prescott according to the admitted Lie point symmetries, for the source-free equation as also in the case where a homogeneous source term exists. The application of symmetry analysis for the Euler-Bernoulli equation is not new, there are various studies in the literature [6,12,16,29,32], however in this paper, we obtained some new results, as the reduction of Euler-Bernoulli form to perturbed form of Painlevé-Ince [20] equation, which is integrable and the third-order ode which falls into the category of equations studied by Chazy, Bureau and Cosgrove. Also, we show that the three beam equations of our study admit the same travelling-wave solution.…”

Similarity Solutions and Conservation Laws for the Beam Equations: A Complete Study

“…In this work, we study the algebraic properties of the Euler-Bernoulli, the Rayleigh and of the Timoshenko-Prescott according to the admitted Lie point symmetries, for the source-free equation as also in the case where a homogeneous source term exists. The application of the symmetry analysis for the Euler-Bernoulli equation is not new, there are various studies in the literature [5][6][7][8][9], however in this paper, we obtained some new results, as the reduction of the Euler-Bernoulli form to a perturbed form of Painlevé-Ince [10] equation, which is integrable and the third-order ode, which falls into the category of equations studied by Chazy, Bureau and Cosgrove. Also, we show that the three beam equations of our study admit the same travelling-wave solution.…”

Symmetry analysis for the $2+1$ generalized quantum Zakharov-Kuznetsov equation

“…In [22], Ovsiannikov classified all forms of the nonlinear heat equation u t = (f (u) u x ) x according to the admitted Lie algebra. Since then, the classification problem has been widely studied in the literature [23][24][25][26][27][28][29][30][31].…”