B‐spline collocation methods for numerical solutions of the Burgers′ equation

The Burgers' equation first appeared in the paper by Bateman (3), who mentioned two of the essentially steady solutions. Due to extensive works of Burgers (4) involving the Burgers' equation especially as a mathematical model for the turbulence, it is known as Burgers' equation. The equation is used as a model in fields as wide as heat conduction (5), gas dynamics (13), shock waves (4), longitudinal elastic waves in an isotropic solid (15), number theory (18), continues stochastic processes (5), and so forth. Hopf (8 )a nd Cole (5) solved the Burgers' equation analytically and independently for arbitrary ini- tial conditions. In many cases, these solutions involve infinite series which may converge very slowly for small values of viscosity coefficients ν, which correspond to steep wave fronts in the propagation of the dynamic wave forms. Burgers' equation shows a similar features with Navier-Stokes equation due to the form of the nonlinear convection term and the occurrence of the viscosity term. Before concentrating on the numerical solution of the Navier-Stokes equation, it seems reasonable to first study a simple model of the Burgers' equation. Therefore, the Burgers' equation has been used as a model equation to test the numerical methods in terms of accuracy and stability for the Navier-Stokes equa- tion. Many authors have used a variety of numerical techniques for getting the numerical solution of the Burgers' equation. Numerical difficultieshavebeencomeacrossinthenu- merical solution of the Burgers' equation with a very small viscosity. Various numerical techniques accompanied with spline functions have been set up for computing the solu- tionsoftheBurgers'equation.RubinandGraveshaveusedthecubicsplinefunctiontech- nique and quasilinearisation for the numerical solutions of the Burgers' equation in one space variable at low Reynolds numbers (16). A cubic spline collocation procedure has been developed for the numerical solution of the Burgers' equation (17). A combination