Numerical and synchronizational behaviors of some evolution equations

Abstract

This study provides several new combined methods to capture the numerical behaviour of nature, governed by the nonlinear advection-diffusion-reaction equation, in one and two dimensions. To achieve this, the implicit backward differentiation formula-spline (BDFS), the optimal five-stage and fourth-order strong stability preserving Runge-Kutta (SSPRK54)-spline and the modified cubic B-spline-SSPRK54 methods are proposed. Without any linearization, the given problems through the proposed schemes are converted to a system of nonlinear and linear differential equations. The current methods are seen to be very reliable alternatives in solving the problem by conserving the physical properties of nature. In addition, the generalized synchronization behaviours of nonlinear advection-diffusion-reaction processes, without losing their natural properties, are investigated to demonstrate the effectiveness of the proposed technique and to reduce computational difficulties in capturing numerical solutions for advection dominant cases. Within the framework of this thesis, a new version of the synchronization methods, based on the design of response systems, is also proposed to solve the synchronization problem discussed here. This technique utilizes the master configuration to monitor the synchronized motions. To show the effectiveness and feasibility of those approaches, various numerical simulations are carried out.