Solution of Burgers equation using the characteristic function method

Abstract:
Burgers' equation is solved here using the Characteristic function method. This method is leads to an eight dimensional Lie Group. Four transformations are analyzed. An analytical solution is derived for each case.

In the present paper the method of characteristic function is presented to solve Burgers equation in one dimensional field

ut+u ux = su2x

where s is the kinematic viscosity, u(x,t) is function of x and t.

The method of characteristic function presented here reduces the formidable task of solution of a set of simultaneous partial differential equations, resulting from the application of the Lie group method. In 1971 Na and Hansen [2] simplify the non-classical method by expressing the infinitesimals generators in term of a single function W, called the "characteristic function". This function is the surface invariant condition introduced by Bluman and Cole . The procedure, well described in [1,2,3], consists in the determination of the characteristic function "W", from which the group generators are derived. We will illustrate this method through a similarity reduction of Burgers equation.

References:

1. M. Kassem, Similarity solution of a non linear diffusion equation using the characteristic function method, Int. Conf. Math. Nucl. Physics , 2003.
2. T. Y. Na, A.G. Hansen, Similarity Analysis of differential equation by Lie Group, J. of Franklin Institute, 1971, 6.
3. R. Seshadri, T.Y Na, Group Invariance in Engineering Boundary Value Problems, Springer Verlag, 1985.