The Exact Solutions Of Two Types Nonlinear Equations Of Evolution And Boundary Value Problems Of Plain Elastic Mechanics Equations

In this paper, several exact solitary wave solutions for combinatorial KdV equation u_t + au^pu_x + bu^2pu_x + δu_xxx = 0 and generalized Boussinesq equation u_tt + (?)/((?)_x)(u_x + au^pu_x + bu^2pu_x + ru_xx + δu_xxx) = 0 are solved. For overcome nonlinear terms of any degree in the equation, we made a proper transformation according to the structure of the equation at first, make the equations into ordinary differential equation. Then solve the ordinary differential equation exactly, by auxiliary functions method and undetermined the coefficient method and algebraic computer system Mathematica. As a result some exact bell and kink solitary wave solutions for the original equations are obtained. The method used here is useful for further investigating and solving mathematics physics equation with nonlinear terms of any degree.We also have investigated applications of the symmetry method in the boundary value problem of the mechanics equations. By symmetry method, we successful solved (1) the problem of the half infinite plane received the concentrated strength in vertical direction ; (2) the problem of the wedge received the strength in its top. the applications of symmetry method can avoid ambiguity of the inverse method . Furthermore the obtained results have proved that the symmetry method can be used in more extensive mechanics problems. It is known that the symmetry method is not widely used to solve the boundary value problems of mechanics equation. So,it is still need further investigating that how to utilize the advantage of symmetry method effectively and design suitable algorithm to deal with such problem. The results of this paper are only a preliminary exploring of this question.