Traditional Functional-Discrete Methods for the Problems of Mathematical Physics

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