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.

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 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 315