Özet:
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.
