Existence And Computation Of Partial Functional Differential Equations On Unbounded Domains

Abstract

This thesis studies numerical approximation of neutral differential newlineequations with infinite delay asymptotic stability of infinite delay differential newlineequations semidiscretization of partial differential equations with delay and newlinenumerical solution of first order hyperbolic equations newlineIn the available literature existence of infinite delay equations is usually newlineposed in Banach spaces which are defined as newlinewhere is a continuous function with newlinelim newlineand the norm is given by As the numerical schemes do newlinenot converge in these spaces theorems on numerical approximations are posed in newlineThe present thesis remedies this situation by the study of infinite newlinedelay equations on Banach spaces C which are defined below newlineLet J be a sequence of compact intervals such that newlineis sequence of positive reals such that Then is defined as newlineExistence as well as results on numerical approximation are obtained using newlinethese spaces The presence of a sequence of projections converging to identity newlinesuch that each projection can be computed with values of functions on a newlinefinite interval facilitates the study of numerical approximations Denoting the newlinesequence of projections by existence of a sequence of pair of operators newlineand where are finite dimensional newlineSpaces the identity operator on is used to construct newlinediscretizations newlineIn Chapter 2 the existence of a mild solution and the convergence of a newlinenumerical scheme for the neutral differential equation with infinite delay newlineare obtained Here and both belong to 0 and lim newlineThe phase space used here is newline newlinewhere max newlineIn Chapter 3 a theorem on asymptotic stability is obtained for the newlinedifferential equation with an infinite delay given by newline newline