WELL-POSEDNESS OF THE REACTION DIFFUSION SYSTEM WITH TIME DELAY AND IMPULSES - KUSTIAWAN (NIM:30112006), CECE
Well-posedness of a system of reaction-diffusion with time delay and impulses is discussed in this dissertation. The system is as follows : (∂u(x,t))/∂t=(∂^2 u(x,t))/(∂x^2 )+u(x,t)[1-u(x,t-τ)],τ>0. (1) Many differential equations are well posed although the solution can not be expressed explicitly. For example in the case of Equation (1), although the explicit solution is not found but the existence of the solution can be determined by the semigroup approach. To find the solution we turn our attention to numerical method. In the first part of this dissertation, the system (1) with single time delay and impulses are investigated. The boundary condition is homogeneous Neumann boundary condition. The impulses occur at fixed times. Semigroup approach is used to obtain the well-posedness of the problem. First, the problem is considered on each time interval between two consecutive impulses. Some theorems developed in Pazy (1983), Goldstein (1985), Ruess and Summers (1994), Batkai and Piazzera (2001) that are related to the strongly continuous semigroup due to some m-accretive operators, are used. Some sufficient conditions are obtained to prove the well-posedness of the problem. Integrated semigroup is used to obtained the well-posedness of the problem. Some sufficient conditions are imposed to the operators involving in the problem, in order to achieve the well-posedness. In the second part of this dissertation, a numerical scheme is developed along with its stability condition. Forward Time Centered Space (FTCS) is used to find the approximate solution of (1). To obtain how effective the scheme is, the following equation: (∂u(x,t))/∂t=(∂^2 u(x,t))/(∂x^2 )-u(x,t)+u(x,t-τ),τ>0. (2) whose explicit solution can be used as a benchmark. The explicit solution is compared to the numerical solution by finding the error, that is the difference between the two solutions. The numerical scheme used is finite difference method based on Taylor expansions. The FTCS scheme is applied to both temporal and spatial variables. Some simulations are presented to verify the scheme. In a certain problem, the simulations are compared to the corresponding explicit solutions.