Teoman Özer

Symmetry group classification for two-dimensional elastodynamics problems in nonlocal elasticity

In this study the symmetry groups of two-dimensional elastodynamics problems in nonlocal elasticity are identified and classified. The determining equations are found, and then the differential equations are obtained that include the kernel function and the independent term. The symmetry group classification is determined by using these differential equations and solutions of the determining equations.

Symmetry group classification for one-dimensional elastodynamics problems in nonlocal elasticity

Abstract The symmetry groups of one-dimensional elastodynamics problem of nonlocal elasticity are investigated and we get a classification for the problem. The determining equations of the system of Fredholm integro-differential equations corresponding to one-dimensional nonlocal elasticity equation are found and solved. We get the differential equations that include the kernel function and the independent term. The symmetry groups are determined using these functions. We compare the results of one-dimensional nonlocal elasticity with the results of the Voltera integro-differential equation corresponding to one-dimensional visco-elasticity equation in the conclusion section of the manuscript.

The solution of Navier equations of classical elasticity using Lie symmetry groups

Abstract The analytical solutions of axially-symmetric Navier equations in classical elasticity are found by applying Lie group theory. We investigate two different systems of partial differential equations corresponding elastostatics and elastodynamics problems, and find similarity solutions of both cases by solving the reduced system of ordinary differential equations which have fewer independent variables. As an example of the elastostatics case, the displacements and stress components are obtained for porous, polymeric foam material by using similarity solutions.

Invariant solutions and conservation laws to nonconservative FP equation

We generate conservation laws for the one dimensional nonconservative Fokker-Planck (FP) equation, also known as the Kolmogorov forward equation, which describes the time evolution of the probability density function of position and velocity of a particle, and associate these, where possible, with Lie symmetry group generators. We determine the conserved vectors by a composite variational principle and then check if the condition for which symmetries associate with the conservation law is satisfied. As the Fokker-Planck equation is evolution type, no recourse to a Lagrangian formulation is made. Moreover, we obtain invariant solutions for the FP equation via potential symmetries.

Group properties and conservation laws for nonlocal shallow water wave equation

Abstract Symmetry groups, symmetry reductions, optimal system, conservation laws and invariant solutions of the shallow water wave equation with nonlocal term are studied. First, Lie symmetries based on the invariance criterion for nonlocal equations and the solution approach for nonlocal determining equations are found and then the reduced equations and optimal system are obtained. Finally, new conservation laws are generated and some similarity solutions for symmetry reduction forms are discussed.

An application of symmetry groups to nonlocal continuum mechanics

In this study the symmetry group properties of the one-dimensional elastodynamics problem in nonlocal continuum mechanics is discussed by using an approach developed for symmetry group analysis of integro-differential equations with general form. This approach is based on the modification of the invariance criterion of the differential equations, which include nonlocal variables and integro-differential operators. Lie point symmetries of the nonlocal elasticity equation are obtained based on solving nonlocal determining equations by using a new approach. The symmetry groups for different types of kernel function and the free term including the classical linear elasticity case are presented.

Invariant solutions of integro-differential Vlasov-Maxwell equations in Lagrangian variables by Lie group analysis

The Lie point symmetries of the Vlasov-Maxwell system in Lagrangian variables are investigated by using a direct method for symmetry group analysis of integro-differential equations, with emphasis on solving nonlocal determining equations. All similarity reduction forms for the system are obtained by using different approaches and some analytical and numerical solutions are presented.

The group-theoretical analysis of nonlocal Benney equation

This study deals with the symmetry group analysis of the nonlinear and nonlocal Benney equation in hydrodynamics. The generalization of the invariance criterion for the integro- differential equations is used to calculate the Lie point symmetries. Furthermore, the solution technique for nonlocal determining equations is introduced. The optimal system, reduced equations and similarity solutions are discussed.

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.

Linearization properties, first integrals, nonlocal transformation for heat transfer equation

We examine first integrals and linearization methods of the second-order ordinary differential equation which is called fin equation in this study. Fin is heat exchange surfaces which are used widely in industry. We analyze symmetry classification with respect to different choices of thermal conductivity and heat transfer coefficient functions of fin equation. Finally, we apply nonlocal transformation to fin equation and examine the results for different functions.