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Traditional Functional-Discrete Methods for the Problems of Mathematical Physics

E-BookEPUB2 - DRM Adobe / EPUBE-Book
352 Seiten
Englisch
Polityerschienen am23.02.20241. Auflage
This book is devoted to the construction and study of approximate methods for solving mathematical physics problems in canonical domains. It focuses on obtaining weighted a priori estimates of the accuracy of these methods while also considering the influence of boundary and initial conditions. This influence is quantified by means of suitable weight functions that characterize the distance of an inner point to the boundary of the domain.

New results are presented on boundary and initial effects for the finite difference method for elliptic and parabolic equations, mesh schemes for equations with fractional derivatives, and the Cayley transform method for abstract differential equations in Hilbert and Banach spaces. Due to their universality and convenient implementation, the algorithms discussed throughout can be used to solve a wide range of actual problems in science and technology. The book is intended for scientists, university teachers, and graduate and postgraduate students who specialize in the field of numerical analysis.



Volodymyr Makarov is Doctor of Physical and Mathematical Sciences, Professor, and Academician at the National Academy of Sciences of Ukraine, Kyiv, where he is also the founder and head of their Computational Mathematics Department.

Nataliya Mayko is Doctor of Physical and Mathematical Sciences and Professor in the Department of Computational Mathematics of the Faculty of Computer Science and Cybernetics at Taras Shevchenko National University of Kyiv, Ukraine.mehr
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Produkt

KlappentextThis book is devoted to the construction and study of approximate methods for solving mathematical physics problems in canonical domains. It focuses on obtaining weighted a priori estimates of the accuracy of these methods while also considering the influence of boundary and initial conditions. This influence is quantified by means of suitable weight functions that characterize the distance of an inner point to the boundary of the domain.

New results are presented on boundary and initial effects for the finite difference method for elliptic and parabolic equations, mesh schemes for equations with fractional derivatives, and the Cayley transform method for abstract differential equations in Hilbert and Banach spaces. Due to their universality and convenient implementation, the algorithms discussed throughout can be used to solve a wide range of actual problems in science and technology. The book is intended for scientists, university teachers, and graduate and postgraduate students who specialize in the field of numerical analysis.



Volodymyr Makarov is Doctor of Physical and Mathematical Sciences, Professor, and Academician at the National Academy of Sciences of Ukraine, Kyiv, where he is also the founder and head of their Computational Mathematics Department.

Nataliya Mayko is Doctor of Physical and Mathematical Sciences and Professor in the Department of Computational Mathematics of the Faculty of Computer Science and Cybernetics at Taras Shevchenko National University of Kyiv, Ukraine.
Details
Weitere ISBN/GTIN9781394276653
ProduktartE-Book
EinbandartE-Book
FormatEPUB
Format Hinweis2 - DRM Adobe / EPUB
FormatFormat mit automatischem Seitenumbruch (reflowable)
Verlag
Erscheinungsjahr2024
Erscheinungsdatum23.02.2024
Auflage1. Auflage
Seiten352 Seiten
SpracheEnglisch
Dateigrösse25707 Kbytes
Artikel-Nr.13992100
Rubriken
Genre9201

Inhalt/Kritik

Inhaltsverzeichnis
Preface ix

Introduction xi

Chapter 1 Elliptic Equations in Canonical Domains with the Dirichlet Condition on the Boundary or its Part 1

1.1 A standard finite-difference scheme for Poisson's equation with mixed boundary conditions 1

1.2 A nine-point finite-difference scheme for Poisson's equation with the Dirichlet boundary condition 18

1.3 A finite-difference scheme of the higher order of approximation for Poisson's equation with the Dirichlet boundary condition 31

1.4 A finite-difference scheme for the equation with mixed derivatives 46

Chapter 2 Parabolic Equations in Canonical Domains with the Dirichlet Condition on the Boundary or its Part 69

2.1 A standard finite-difference scheme for the one-dimensional heat equation with mixed boundary conditions 69

2.2 A standard finite-difference scheme for the two-dimensional heat equation with mixed boundary conditions 82

2.3 A standard finite-difference scheme for the two-dimensional heat equation with the Dirichlet boundary condition 102

Chapter 3 Differential Equations with Fractional Derivatives 115

3.1 BVP for a differential equation with constant coefficients and a fractional derivative of order ¿ 115

3.2 BVP for a differential equation with constant coefficients and a fractional derivative of order alpha element of (0,1) 124

3.3 BVP for a differential equation with variable coefficients and a fractional derivative of order alpha element of (0,1) 145

3.4 Two-dimensional differential equation with a fractional derivative 166

3.5 The Goursat problem with fractional derivatives 181

Chapter 4 The Abstract Cauchy Problem 213

4.1 The approximation of the operator exponential function in a Hilbert space 213

4.2 Inverse theorems for the operator sine and cosine functions 230

4.3 The approximation of the operator exponential function in a Banach space 236

4.4 Conclusion 247

Chapter 5 The Cayley Transform Method for Abstract Differential Equations 249

5.1 Exact and approximate solutions of the BVP in a Hilbert space 249

5.2 Exact and approximate solutions of the BVP in a Banach space 282

References 307

Index 315mehr
Leseprobe

Introduction

It is well known that the vast majority of boundary value and initial value problems cannot be solved exactly and require the use of appropriate approximate methods. An important characteristic of any approximate method is its accuracy. To estimate the accuracy, we traditionally use a certain discretization parameter: a mesh step, the number of terms of the partial sum of the series, etc.

However, for both theoretical and practical reasons, it is also important to take into account the influence of other factors, for example, the so-called boundary and initial effects. Precisely, the boundary effect means that due to the Dirichlet boundary condition for a differential equation in the canonical domain, the accuracy of the approximate solution near the boundary of the domain is higher compared to the accuracy further from the boundary. A similar situation is observed for non-stationary equations near those mesh nodes where the initial condition is set.

For the quantitative characteristics of the boundary or initial effect, we can take an a priori error estimate (in a certain mesh norm) with a certain weight function, which characterizes the distance of a point inside the domain to the boundary of the domain. The idea of such estimates was first announced by Volodymyr Makarov in Makarov (1987) for an elliptic equation in case of generalized solutions from Sobolev spaces and developed further in publications for quasilinear stationary and non-stationary equations. Since the concept was quite new, there were (and still are) very few publications on this subject. In some respects, the same issue is studied in the works of Galba (1985) and Molchanov and Galba (1990). However, they assume only the classical smoothness of solutions and do not consider time-dependent problems.

In this book, we develop our previous studies and present some new results on the impact of initial and boundary conditions on the accuracy of the following methods: the finite-difference method for elliptic and parabolic equations, the discrete method for solving equations with fractional derivatives, and the Cayley transform method for abstract differential equations in Hilbert and Banach spaces. Regardless of the type of problem or method, our main focus is always on obtaining weighted estimates with a proper weight function.

For a better understanding of the reasoning and easier navigation through the computation, some information is assumed to be known to the reader from classical mathematics courses, while the rest is provided directly in the text. Some of the formulas may seem a bit long and cumbersome, but this is partly because we are trying to be as detailed as possible and help the reader follow the calculations with ease.

This book consists of five chapters. Chapters 1 and 2 are devoted to the study of the accuracy of finite-difference schemes for stationary and non-stationary equations respectively, taking into account the influence of boundary and initial conditions (in the sense of Makarov as mentioned above).

The finite-difference method is historically one of the first and most recognized numerical methods for solving problems of mathematical physics, mainly due to its universality and convenience in practical implementation. In recent decades, it has gained considerable popularity due to growing interest in the study of nonlinear processes in various fields of physics, chemistry, seismology, ecology, etc. Mathematical models of such phenomena involve nonlinear partial differential equations. For example, in aerodynamics and hydrodynamics, the one-dimensional quasilinear Burgers parabolic equation arises as an adequate mathematical model of turbulence. A special case of the Burgers equation is the quasilinear transport equation (the Hopf equation), which is the simplest equation describing discontinuous flows or flows with shock waves. In biology, ecology, physiology, combustion theory, crystallization theory, plasma physics, etc., the Fisher-Kolmogorov-Petrovsky-Piskunov equation (the Fisher-KPP equation) plays an important role as the simplest semi-linear parabolic equation. The propagation of shallow water waves that weakly and nonlinearly interact, ion acoustic waves in plasma, acoustic waves on crystal lattices, etc. are often modeled by the Korteweg-de Vries equation (the KdV equation). Many publications are devoted to finite-difference schemes for solving problems for elliptic and parabolic equations with dynamic conjugation conditions at the contact boundary (which is associated with the presence of concentrated heat capacities in a heat-conducting medium) and/or dynamic boundary conditions (which model heat conduction in a solid body in contact with fluid, as well as processes in semiconductor devices). In the mathematical modeling of some processes in ecology, physics and technology, when it is impossible to set the exact values of the desired solution at the boundary of a domain, problems with non-local boundary conditions usually arise.

These and many other examples demonstrate that the finite-difference method is actively developing and is widely used to solve current scientific and technical problems. At the same time, there are very few publications dedicated to the study of the initial and boundary effects in the above sense, and our book is a certain step towards filling this gap. One of the first such works is the announcement (Makarov 1989) that deals with the problem

where

and Ω = {(x1,x2) : 0 xα α = 1,2} is a unit square. The problem is discretized by the finite-difference scheme

[I.1]

where Ï = Ï1×Ï2, Ïα = {xα = iαhα : iα = 1,2,â¦, Nα â1, hα = 1/Nα}, α = 1,2; γ is a boundary of the mesh Ï. Some traditional notations for finite-difference schemes from Samarskii (2001) are used here, for example: ,

The main result was presented in the following statement.

THEOREM.- Let and . Then, there exists h0 > 0 such that for all h â (0, h0] the accuracy of the finite-difference scheme [I.1] is characterized by the weighted estimate

with the weight function Ï(x) = min{x1x2, x1(1 â x2), (1 â x1)x2, (1 â x1)(1 â x2)}.

This idea is further developed in the present book for other types of boundary conditions for elliptic and parabolic equations. It is worth mentioning that the important stages in obtaining such weighted estimates are the evaluation of discrete Green s functions and the analysis of approximation errors. Each time, when it is necessary to estimate discrete Green s functions, we apply the following proposition, which is formulated and proved in Samarskii et al. (1987, p. 54).

MAIN LEMMA.- Let the following assumptions be fulfilled: 1) A : H â H is a self-adjoint operator acting in a Hilbert space H; 2) B : H* â H is a linear operator; 3) the inverse operator Aâ1 exists; 4) âB*vâ* ⤠γâAvâ for all v â H, where B* : H â H* is the adjoint operator of B, (y, v)* and are an inner product and an associate norm in H* respectively. Then, âAâ1Bvâ ⤠γâvâ* for all v â H*.

Similarly, when it comes to estimating an approximation error for a generalized solution from Sobolev spaces, we refer to the Bramble-Hilbert lemma (e.g. Samarskii et al. (1987, p. 29)). We recall it here for convenience.

LEMMA (BRAMBLE-HILBERT).- Let be an open convex bounded set of the diameter d > 0, let l(u) be a bounded linear functional in the space with , where is a positive non-negative number and 0 λ ⤠1, namely:

and let l(u) turn into zero on polynomials of degree of variables x1, x2,â¦,xn . Then, there exists a positive constant , which is dependent on Ω and independent of u(x), such that the following inequality holds true:

The study of the boundary and initial effects is also of great interest for new classes of problems, for example, related to the application of fractional integro-differentiation. In Chapter 3, we address the accuracy of the mesh methods for solving boundary value problems for differential equations with fractional derivatives.

For almost 300 years (from 1695 until recently) this branch of classical analysis was no more than an abstract mathematical theory. However, over the past several decades, fractional analysis has found wide applications in the construction of adequate mathematical models of many natural and social phenomena, as evidenced by a considerable number of publications (e.g. Kilbas et al. (2006); Sabatier et al. (2007); Nakagawa et al. (2010), to mention a few). Due to the ability to model...
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