Boundary value problems and dichotomic stability

Since the conditioning of a boundary value problem (BVP) is closely related to the existence of a dichotomic fundamental solution (i.e., where one set of modes is increasing and a complementary set is decreasing), it is important to have discretization methods that conserve this dichotomy property. The conditions this imposes on such a method are investigated in this paper. They are worked out in more detail for scalar second-order equations (the central difference scheme), and for linear first-order systems as well; for the latter type both one-step methods (including collocation) and multistep methods (those that may be used in multiple shooting) are examin

A Guide to Special Functions in Fractional Calculus

Dedicated to the memory of Professor Richard Askey (1933-2019) and to pay tribute to the Bateman Project. Harry Bateman planned his project (accomplished after his death as Higher Transcendental Functions, Vols. 1-3, 1953-1955, under the editorship by A. Erdelyi) as a "Guide to the Functions". This inspired the author to use the modified title of the present survey. Most of the standard (classical) Special Functions are representable in terms of the Meijer G-function and, specially, of the generalized hypergeometric functions pFq. These appeared as solutions of differential equations in mathematical physics and other applied sciences that are of integer order, usually of second order. However, recently, mathematical models of fractional order are preferred because they reflect more adequately the nature and various social events, and these needs attracted attention to "new" classes of special functions as their solutions, the so-called Special Functions of Fractional Calculus (SF of FC). Generally, under this notion, we have in mind the Fox H-functions, their most widely used cases of the Wright generalized hypergeometric functions pΨq and, in particular, the Mittag-Leffler type functions, among them the "Queen function of fractional calculus", the Mittag-Leffler function. These fractional indices/parameters extensions of the classical special functions became an unavoidable tool when fractalized models of phenomena and events are treated. Here, we try to review some of the basic results on the theory of the SF of FC, obtained in the author's works for more than 30 years, and support the wide spreading and important role of these functions by several examples

One-dimensional and multi-dimensional integral transforms of Buschman-Erdélyi type with Legendre functions in kernels

This paper consists of two parts. In the first part we give a brief survey of results on Buschman-Erdélyi operators, which are transmutations for the Bessel singular operator. Main properties and applications of Buschman-Erdélyi operators are outlined. In the second part of the paper we consider multi-dimensional integral transforms of Buschman-Erdélyi type with Legendre functions in kernels. Complete proofs are given in this part, main tools are based onMellin transformproperties and usage of Fox H-function

Summability of formal solutions for a family of generalized moment integro-differential equations

Generalized summability results are obtained regarding formal solutions of certain families of linear moment integro-differential equations with time variable coefficients. The main result leans on the knowledge of the behavior of the moment derivatives of the elements involved in the problem. A refinement of the main result is also provided giving rise to more accurate results which remain valid in wide families of problems of high interest in practice, such as fractional integro-differential equations.Agencia Estatal de InvestigaciónUniversidad de Alcal

The stability of laminar incompressible boundary layers in the presence of compliant boundaries

Thesis (Sc. D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 1964.Microfiche copy available in Barker.Vita.Includes bibliographical references (leaves 118-122).by Richard E. Kaplan.Sc.D

Finite element solutions to boundary value problems

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.This thesis consists of two distinct parts which deal with two-point boundary value problems and parabolic problems, respectively. In Section 1 we examine the numerical solution of a two-point boundary value problem by a collocation method based on the consistency relationship of regular splines. An existence and convergence result is established which generalises the 0(h^2) convergence result of the cubic spline collocation scheme for the problem in question. Contrary to most previously documented finite element schemes this method employs splines that may be non-linear in structure. Consequently, by a judicious choice of regular spline, the dominant terms of the true solution may be imitated more accurately than by the conventional polynomial based splines. The scheme is implemented by an algorithm that examines the suitability of various classes of regular splines and determines the subsequent deployment of them. The second section investigates semi-discrete finite element schemes for approximating the linear parabolic equation. A standard finite element discretization is employed for the space variable whilst an A0-stable, linear multistep, multiderivative discretization scheme, (L.M.S.D.) is used in time. We consider both the homogeneous and the nonhomogeneous linear parabolic equations and derive optimal convergence results for the above schemes. The convergence results achieved with a k-step L.M.S.D. scheme, incorporating the first m derivatives, generalise and extend the studies of several authors who concentrate on the particular cases of linear multistep formulae, m-l, and one-step schemes, k=1. Ao-stable L.M.S.D. 's are constructed and their implementation procedures examined. The suitability of selecting a L.M.S.D. method, with m, k>1, in a semi-discrete Galerkin scheme is discussed, and its superiority over semi-discrete Galerkin schemes, that incorporate linear multistep or one-step formulae, is confirmed in several aspects. Finally, a class of quasi-linear parabolic equations is solved by a semi-discrete Galerkin scheme that is third order accurate in time. This method is based on a particular third order L.M.S.D. scheme and requires the solution of linearly algebraic systems of equations at each time level. Thus, we improve on all the previously documented linearised schemes as they are only second order accurate in time. All the schemes described in Section 2 are unconditionally stable

High Order Multistep Methods with Improved Phase-Lag Characteristics for the Integration of the Schr\"odinger Equation

In this work we introduce a new family of twelve-step linear multistep methods for the integration of the Schr\"odinger equation. The new methods are constructed by adopting a new methodology which improves the phase lag characteristics by vanishing both the phase lag function and its first derivatives at a specific frequency. This results in decreasing the sensitivity of the integration method on the estimated frequency of the problem. The efficiency of the new family of methods is proved via error analysis and numerical applications.Comment: 36 pages, 6 figure

Department of Applied Mathematics Academic Program Review, Self Study / June 2010

The Department of Applied Mathematics has a multi-faceted mission to provide an exceptional mathematical education focused on the unique needs of NPS students, to conduct relevant research, and to provide service to the broader community. A strong and vibrant Department of Applied Mathematics is essential to the university's goal of becoming a premiere research university. Because research in mathematics often impacts science and engineering in surprising ways, the department encourages mathematical explorations in a broad range of areas in applied mathematics with specific thrust areas that support the mission of the school