Some aspects of monotone iterative methods for classical and fractional differential equations

Abstract

The present thesis, in six chapters, proposes existence and uniqueness theorems for classical and fractional order di erential equations with initial conditions and/or boundary conditions, via monotone iterative methods. The proposed iterative schemes are easy to apply and computationally inexpensive. Based on the iterative schemes, numerical methods are developed to solve the problems numerically. To show the e ciency of the proposed schemes, they are compared with existing methods in the literature, wherever necessary.Chapter 1 presents a short literature survey of monotone iterative methods as well as fractional di erential equations. This chapter also provides basic de nitions, properties and formulas which are useful in later chapters. In Chapter 2, an existence and uniqueness theorem for solving a nonlinear fractional order initial value problem of Caputo type of order q 2 (0; 1] is proposed using the method of modi ed quasilinearization. The main theorem has been illustrated numerically using appropriate examples which show that the proposed quasilinearization method is robust and easy to apply. Chapter 3 proposed an existence and uniqueness theorem for fractional order Volterra population model via an e cient monotone iterative scheme. By coupling spectral method with the proposed iterative scheme, the fractional order integro di erential equation is solved numerically. The numerical experiments support the fact that the proposed iterative scheme is e cient than the existing iterative scheme in the literature. In Chapter 4 a proof, via monotone quasilinearization method, for the existence and uniqueness of the solution for a two-point nonlinear boundary value problem of fractional order 1 < q < 2 is proposed. Using the lower and upper solution, two sequences are constructed that converge uniformly, monotonically and quadratically to the unique solution of the problem. An interesting numerical study is also provided to support the proposed theory.Chapter 5 provides an existence and uniqueness solution of a class of parabolic partial integro-di erential equations via a monotone iterative scheme. A bivariate spectral collocation method is also proposed to solve these problems numerically. Finally, the robustness and e ciency of the numerical scheme is illustrated using partial integro di erential equation that arises in population dynamics. Chapter 6 provides a short note on monotone iterative methods available for handling a coupled system of partial di erential equation that arises in catalytic converter. Chapter primarily focuses on providing a theoretical justi cation for the better performance of some monotone methods over other existing monotone methods in the literature.