Legendre-Petrov-Galerkin Chebyshev spectral collocation method for second-order nonlinear differential equations

We propose a Legendre-Petrov-Galerkin Chebyshev spectral collocation method for initial value problems (IVPs) of second-order nonlinear ordinary differential equations (ODEs). The is applied to time discretization and the term dealt with method. scheme results in simple algebraic system by choosing appropriate basis functions. Optimal error estimates $ L^2 $-norm single multiple interval methods are given. As an application method, we construct space-time schemes solving some time-dependent partial (PDEs). Numerical experiments suggest efficiency methods.