An integral equation method for the numerical solution of the Burgers equation

We consider an initial–boundary value problem for the two-dimensional Burgers equation on the plane. This problem is reformulated by an equivalent integral equation on the Fourier transform space. For the solution of this integral equation, two numerical methods are proposed. One of these two methods is based on the properties of the Gaussian function, whereas the other one is based on the FFT algorithm. Finally, the Galerkin method with Gaussian basis functions is applied to the original initial–boundary value problem, in order to compare the performances of the proposed methods with a standard numerical procedure. Some numerical examples are given to evaluate the efficiency of the proposed methods. The performances obtained from these numerical experiments promise that these methods can be applied effectively to more complex problems, such as the Navier–Stokes equation and the turbulence flows.