Lie Group Method for Solving the Generalized Burgers', Burgers'-KdV and KdV Equations with Time-Dependent Variable Coefficients

In this study, the Lie group method for constructing exact and numerical solutions of the generalized time-dependent variable coefficients Burgers', Burgers'-KdV, and KdV equations with initial and boundary conditions is presented. Lie group theory is applied to determine symmetry reductions which reduce the nonlinear partial differential equations to ordinary differential equations. The obtained ordinary differential equations were solved analytically and the solutions are obtained in closed form for some specific choices of parameters, while others are solved numerically. In the obtained results we studied effects of both the time t and the index of nonlinearity on the behavior of the velocity, and the solutions are graphically presented.