Simulations of the one and two dimensional nonlinear evolutionary partial differential equations: A numerical study

In this work a hybrid scheme is proposed for the numerical study of various evolutionary partial differential equations (EPDEs). In proposed strategy, temporal derivatives are estimated via first and second order finite differences while the solution and spatial derivatives are approximated by Lucas and Fibonacci polynomials. Further, the collocation approach is applied which convert the EPDEs, to the system of coupled linear equations which are simple to solve. For non-linear problems, Quasilinearization is used to tackle nonlinearity. The scheme is implemented to solve different EPDEs which include, one dimensional non-linear Boussinesq, Hunter-Saxton, wave-like and two dimensional linear and non-linear wave like equations. Accuracy of the scheme is portrayed by computing L-infinity, L-2 error norms and the relative error. Moreover, the calculated outcomes are compared with previously existing results in literature. Simulation demonstrates that the scheme works well for the mentioned EPDEs.