Publications Details

On the generation of exact solutions using the method of nearby problems

The Method of Nearby Problems is employed to generate exact solutions to equations 'nearby' the steady and unsteady Burgers equation. Burgers equation is chosen because of the existence of exact solutions, and these exact solutions are discussed. Legendre polynomials are used to derive the exact solutions to the nearby problems, and the application of Legendre polynomials for both 1D and 2D problems is also discussed. Results are presented for the steady-state Burgers equation corresponding to a viscous shock wave for Reynolds numbers of 8, 16, and 512. The low Reynolds number cases are well approximated by 10th order Legendre polynomial fits, while the high Reynolds number case is not. The unsteady Burgers equation corresponding to coalescence of two viscous shock waves at a Reynolds number of 8 is also examined. Preliminary results indicate that further investigation is required to accurately capture this 2D solution.