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.
Table of Contents
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 ½ 115
3.2 BVP for a differential equation with constant coefficients and a fractional derivative of order α ∈ (0,1) 124
3.3 BVP for a differential equation with variable coefficients and a fractional derivative of order α ∈ (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 315
