Lyapunov type operators for numerical solutions of PDEs

In the present paper, numerical methods are developed to approximate the solutions of some evolutionary nonlinear problems. The continuous problems are transformed into some Lyapunov type equations and then analysed for existence, uniqueness, convergence, stability and error estimates. The main idea consists in applying Fourier analysis and Von Neumann criterion acting translation and scaling parameter methods to obtain contractive operators leading to fixed point theory.