Spectral Solution of GKdVB and its variant equations using different basis functions

Abstract

In the present study, we present the spectral collocation analysis to solve all variants of Generalised Korteweg-deVries Burger (GKdVB) equations. The different type of equations studied are Burger, Modified Burger, KdV, Modified KdV, KdV Burger and Modified KdV Burger equations. Representative solutions are obtained with respect to a set of initial and boundary conditions available in the literature and the results agree very well with the present solutions. Error analysis is also done for several cases. The main advantage of the present analysis is the simplified mathematical model for a single equation termed as GKdVB and its elegant and simple mathematical analysis to reframe into a set of nonlinear ODEs. After the mathematical procedures are incorporated, we need to adjust the parametric values for $p$, $\nu$, $\mu$ and $\epsilon$ to obtain the solutions of all its variant equations. These set of computations can be applied for different basis functions. It is observed that, in the present analysis, mathematical procedure and computations are very efficient and robust to obtain accurate solutions with least computing time and better numerical accuracy. This analysis particularly favours in capturing the shocking up effect of the solutions for smaller values of kinematic viscosity very well.