The Finite Element Method For Solving The Burgers' Equation Based On The Hope-Cole Transformation

Burgers'equation is a model example for several physical phenomena such as traffic, shock waves, turbulence problems and continuous stochastic processes. It can also be used to test various numerical algorithms. Due to its wide range of applicability, several researchers have constructed various numerical schemes for finding its approximate solution, such as Adomian's decomposition method, a mixed finite difference and boundary element approach, spline finite element method, the exact-explicit finite difference method, Douglas finite difference scheme, the direct variational method and variational iteration method.Hopf-Cole transformation is a powerful analytical tool. It has been applied to derive some exact solutions for the Burgers'equation. Recently, it has been recognized as a promising numerical tool to obtain some successful results.In this paper, Hopf-Cole transformation is used to transform the Burgers' equation into the linear diffusion equation, then the existence and uniqueness of the corresponding variation problem are verified. The convergence analysis and error estimate of the semi-discrete and full-discrete finite element schemes are given.