Iterative Operator Splitting Method For A Class Of Fisherâs Equation - Amir Hamzah, Dadang
The class of Fisher’s equations plays a significant role in the modeling of physical situations such as growth model in ecology, fluid dynamics, gas dynamics and heat transfer. In the study of partial differential equation, these equations are categorized as nonlinear parabolic type which describes the evolution of a density or concentration with respect to time. This thesis is devoted to the study of numerical solution of class of Fisher’s equation using operator splitting method. In particular, the iterative operator splitting method is discussed. This method splits a difficult problem into several sub-problems which may be easier to solve. Each sub-problem is solved using a combination of an iterative scheme with suitable integrators. We carried our analysis by considering the theoretical and numerical aspects. In the theoretical part, consistency, stability, and convergence of the numerical scheme are discussed. Consistency analysis is done by considering the boundedness of operators and the choices of initial solutions. Boundedness of operators plays the main role in deriving the local error by utilizing their nice properties. However, when the operator is unbounded such nice properties are no longer valid. In order to derive the local error for the unbounded case, the properties from semigroup theory are used. The stability analysis is done by considering the stability with respect to initial condition. We showed that the change of the solutions with respect to the change of initial conditions are bounded. The convergence analysis are done using the LadyWindermere’s Fan argument to relate the local and the global errors. We showed that the convergence of the scheme by showing that the global error is bounded. In the numerical part, we apply the scheme to Fisher’s equation. The maximum error is observed and then the errors are compared with non iterative operator splitting.