Traditional Functional-Discrete Methods for the Problems of Mathematical Physics: New Aspects [Hardcover]

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.

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-dimensionaló´