Monotone iterative methods for nonlinear partial diffrential equations

Abstract

A uniform approach is adopted throughout this thesis by appropriately approximating the solutions of nonlinear di erential equations by sequences of linear ones that monotonically converge to the unique solution of the problem. The existence and uniqueness of the solutions of di erent nonlinear partial di erential equations with initial and/or boundary conditions arising from mathematical models are obtained for both continuous and/or discretized domains. All the proposed methods supply lower and upper bounds for the solutions of the given nonlinear di erential equations. The e cacy of the proposed iterative schemes in terms of their faster convergence and/or higher exibility in choosing the initial guess are demonstrated through numerical simulations. In Chapter 1 provides an outline of the historic development of the method of monotone iterations as a powerful tool for nonlinear di erential equations of various types. Few basic results and de nitions that are relevant to the rest of the chapters are also given in this chapter.Chapter 2 deals with an accelerated monotone iterative procedure for a coupled system of partial di erential equations arising from a catalytic converter model. The monotone property as well as the convergence analysis and the error estimate of the proposed iterative schemes for continuous domain as well as discretized domain based on nite di erence approximations are proved theoretically. The e ciency of the proposed scheme is illustrated by providing a comparative numerical study with the existing method. In Chapter 3, an alternative approach to the one provided in Chapter 2 is proposed in which one has to evaluate the derivative only once throughout the procedure. The proposed scheme also accelerates the procedure studied in the literature. An interesting theoretical study on the monotone convergence as well as error estimate of the proposediterative procedure are provided for continuous as well as nite di erence based discretized problems. Chapter 4 proposes an accelerated iterative procedure for a nonlinear fourth order elliptic equation with nonlocal boundary conditions. Theoretically, the monotone property as well as the convergence analysis are proved for both the continuous and nite di erence discretized cases. The proposed scheme not only accelerates the scheme in the literature but also provides a greater exibility in choosing the initial guess. The e cacy of the proposed scheme is demonstrated through a comparative numerical study with the recent literature. In Chapter 5, a nite di erence method based monotone iterative technique is employed to solve an important class of Volterra type parabolic partial integro-di erential equations. The monotone property, convergence analysis and an error estimate in termsof the stopping criteria are proved theoretically. The e ectiveness of the proposed scheme is demonstrated by applying it to nonlinear integro-partial di erential equations arising in population models and nuclear reactor models.