Numerical study of nonlinear mckendrick Von foerster type equations

Abstract

ABSTRACT newlineThis thesis consists of four chapters. Chapter 1 is dedicated to the introduction newlineand the literature survey of the McKendrick Von Foerster type equations. newlineIn Chapter 2, an implicit finite difference scheme is presented to approximate newlinethe solution to the McKendrick Von Foerster equation with diffusion (M-VD) newlinewith Robin condition at both the end points. The notion of upper solution is newlineintroduced and used effectively with aid of discrete maximum principle to study newlinethe wellposedness and stability of the numerical scheme. A relation between the newlinenumerical solutions to the M-V-D and the steady state problem is established. newlineIn Chapter 3, a numerical scheme to find approximate solutions to the M-V-D newlinewith Robin condition at the left end point and Dirichlet boundary condition at newlineright point is presented. The main difficulty in employing the standard analysis newlineto study the properties of this scheme is due to presence of nonlinear and nonlocal newlineterm in the Robin boundary condition in the M-V-D. To overcome this, we use newlinethe abstract theory of discretizations based on the notion of stability threshold to newlineanalyze the scheme. Stability, and convergence of the proposed numerical scheme newlineare established. newlineIn Chapter 4, higher order numerical schemes to the McKendrick Von Foerster newlineequation are presented when the death rate has singularity at the maximum age. newlineThe third, fourth order schemes that are proposed are based on the characteristics newline(non intersecting lines in this case), and are multi-step methods with appropriate newlinecorrections at each step. In fact, the convergence analysis of the schemes are newlinediscussed in detail. Moreover, numerical experiments are provided to validate newlinethe orders of convergence of the proposed third order and fourth order schemes. newline