An implicit-explicit local method for stiff differential equations

Abstract

In this thesis, a new implicit-explicit local differential transform method (IELDTM) with an arbitrary order is produced for stiff differential equations. In the produced method, which is a stability-preserved numerical algorithm that accepts the idea of differential transformation as a starting point, all information about the numerical process is taken directly from the corresponding differential equations. The currently proposed method is derived for stiff initial value problems (IVPs), stiff boundary value problems (BVPs), and stiff parabolic initial-boundary value problems. All theoretical analyses including priori error estimations and stability analysis are provided and the theoretical order expectations are verified through various numerical experiments. The IELDTM is found to be a high order, stability preserved, and versatile numerical approach for both ordinary differential equations (ODEs) and parabolic partial differential equations (PDEs) up to three spatial dimensions. The IELDTM is proven to eliminate both some existing drawbacks of the differential transform-based methods and other numerical techniques such as the finite element method (FEM) and finite difference method (FDM) by providing optimized degrees of freedom.