Abstract: Einstein vacuum equations, that is a system of nonlinear
partial differential equations (PDEs) are derived from Weyl metric
by using relation between Einstein tensor and metric tensor. The
symmetries of Einstein vacuum equations for static axisymmetric
gravitational fields are obtained using the Lie classical method. We
have examined the optimal system of vector fields which is further
used to reduce nonlinear PDE to nonlinear ordinary differential
equation (ODE). Some exact solutions of Einstein vacuum equations
in general relativity are also obtained.
Abstract: In this paper, we study a new modified Novikov equation for its classical and nonclassical symmetries and use the symmetries to reduce it to a nonlinear ordinary differential equation (ODE). With the aid of solutions of the nonlinear ODE by using the modified (G/G)-expansion method proposed recently, multiple exact traveling wave solutions are obtained and the traveling wave solutions are expressed by the hyperbolic functions, trigonometric functions and rational functions.
Abstract: We seek exact solutions of the coupled Klein-Gordon-Schrödinger equation with variable coefficients with the aid of Lie classical approach. By using the Lie classical method, we are able to derive symmetries that are used for reducing the coupled system of partial differential equations into ordinary differential equations. From reduced differential equations we have derived some new exact solutions of coupled Klein-Gordon-Schrödinger equations involving some special functions such as Airy wave functions, Bessel functions, Mathieu functions etc.