Symmetry group classification of some problems in mathematical physics

In this thesis, some problems in physics and engineering sciences are examined by symmetry methods. In the literature, there are a lot of methods to solve nonlinear differential equations and these methods play an important role. One of these methods is to use symmetry groups. We consider some symmetry group related methods to solve problems in mathematical physics. Firstly, we deal with the Noether symmetry classification of the nonlinear fin equation, in which thermal conductivity and heat transfer coefficient are assumed to be functions of the temperature. This classification includes Noether symmetries, first integrals and some invariant solutions with respect to different choices of thermal conductivity and heat transfer coefficient functions. In this thesis, Noether symmetries of the fin equation are investigated using the partial Lagrangian approach. Secondly, we consider Lienard II-type harmonic nonlinear oscillator equation as a nonlinear dynamical system. Firstly, we examine the first integrals in the form A(t; x)x˙+B(t; x) and corresponding exact solutions, the integrating factors. In addition, we analyze other types of the first integrals via λ-symmetry approach. It is shown that the equation can be linearized by means of nonlocal transformation, which is called Sundman transformation. Using the modified Prelle-Singer approach, time independent first integrals are derived for the Lienard II-type harmonic nonlinear oscillator equation. The modified Prelle-Singer procedure is used for a class of second order nonlinear ordinary differential equations and several physically interesting nonlinear systems are solved. Prelle and Singer have proposed an algorithmic procedure to find the integrating factor for the system of first order ordinary differential equation. Once the integrating factor for the equation is determined then it leads to a time independent integral of motion for the first order ordinary differential equation. The Prelle-Singer method guarantees that if the first order ordinary differential equation has a first integral in terms of elementary functions then this first integral can be found. This method has been generalized to incorporate the integrals with non-elementary functions. Recently, this theory is generalized to obtain general solutions for second order and higher order ordinary differential equations without any integration. Moreover, it is possible to consider some feasible algorithm to obtain first integral, integrating factor and invariant solution and one can apply this algorithm to nonlinear equation. The another method for application to nonlinear differential equation is the transformation method. Considering this transformation procedure, a nonlinear equation can be converted to a linear differential equation whose solutions are known. It is well-known that Lie proves the general algorithm that all second order nonlinear differential equations can be converted to second order linear differential equations by the method of change of variables, which is called Lie linearization test. In fact, the mathematical procedure of linearizing transformation is quite diffucult work and this can be applied to only second order ordinary differential equations that have a eight-dimensional Lie algebra. Therefore, it is necessary to consider other type of transformation techniques of nonlinear differential equations for linearization of larger classes of equations. One of nonlocal transformations is of the form X = F(t;x); dT = G(t;x)dt, which is called the generalized Sundman transformation. This transformation is also called S-transformation and the equations that can be linearized by means of S-transformations are called S-linearizable. In the second problem, λ-symmetries via Lie symmetries, integrating factors, first integrals and invariant solutions of Lienard II-type harmonic nonlinear oscillator equation are obtained. In third problem, we examine first integrals, transformation pair and invariant solutions of fin equation by linearization methods. And we apply nonlocal transformation to fin equation. The important relations λ-symmetry with Lie point symmetry, Prelle Singer method with λ-symmetry and Lie symmetry are examined. The first integrals, integrating factors, Sundman transformation pair and invariant solutions of fin equation are found.