The Method of Summary Representation for Numerical Solution of Problems of Mathematical Physics

Author's Preface to the English EditionPrefaceIntroductionChapter 1. General Theory of the One-Dimensional Problem of Eigenvalues and Eigenfunctions of Discrete Argument. Matrices of Type II § 1. Ordinary Finite-Difference Equations § 2. General Problem of Eigenvalues and Eigenfunctions of Discrete Argument. Matrices of Type II § 2.1 Formulae of Multiple Summation by Parts § 2.2 The Space II and the Space II' of Functions of Discrete Argument. Self-Adjoint Finite-Difference Operators § 2.3 Self-Adjoint Finite-Difference Boundary-Value Problems. Matrices of Type II § 2.4 Eigenvalues and Eigenfunctions of Discrete Argument § 2.5 Matrices of Simple Structure. Fundamental Properties of Matrices of Type II § 2.6 General Problem of Eigenvalues and Eigenfunctions for Second-Order Finite-Difference Equations § 3. Solution of Particular Boundary-Value Problems, and the Construction of Fundamental Matrices in Explicit Form § 4. On Special Functions of Discrete Argument, and Special Matrices of Type II § 4.1 Functions of Discrete Argument, Connected with Second-Order Finite-Difference Operators § 4.2 Special Functions of Discrete Argument of the First and Second Types § 4.3 Special CasesChapter 2. Numerical Solution of Two-Dimensional and Three-Dimensional Boundary-Value Problems of Mathematical Physics § 1. Solution of Boundary-Value Problems for Second-Order Elliptic Differential Equations with Constant Coefficients § 1.1 Solution of Two-Dimensional Boundary-Value Problems § 1.2 Extension of the Method to the Solution of Three-Dimensional Boundary-Value Problems § 1.3 Numerical Example § 1.4 Generalization of the Basic Formulae of Summary Representation for Two-Dimensional Boundary-Value Problems § 1.5 Generalization of the Basic Formulae of Summary Representation for Three-Dimensional Boundary-Value Problems § 2. Solution of Boundary-Value Problems for Fourth-Order Elliptic Differential Equations with Constant Coefficients § 2.1 Formula of Summary Representation for the Finite-Difference Biharmonic Operator § 2.2 Solution of Biharmonic Boundary-Value Problems § 2.3 Generalization of the Basic Formula of Summary Representation § 3. Formulae of Summary Representation for Finite-Difference Equations, Corresponding to Second-Order Parabolic Differential Equations with Constant Coefficients § 3.1 Equations with Two Independent Variables § 3.2 Equations with Three Independent Variables § 4. Solution of Finite-Difference Boundary-Value Problems, Connected with Boundary-Value Problems for Second-Order Hyperbolic Differential Equations with Constant Coefficients § 4.1 Equations of Hyperbolic Type with Two Independent Variables § 4.2 Hyperbolic Equations with Three Independent Variables § 5. Differential Equation for the Transverse Vibrations of Beams § 6. On the Numerical Solution of Two-Dimensional and Three-Dimensional Boundary-Value Problems for Differential Equations with Variable CoefficientsReferencesSupplement to the English Edition § 1. Alternating Iterative Method for the Numerical Solution of Connection Equations § 2. On the Approach to the Limit in Certain Formulae of Summary Representation § 3. On the Application of the Method of Summary Representation to the Solution of Problems of Filtration Under Pressure § 4. On the Solution of Bending and Torsion Problems for Prismatic Beams by the Method of Summary Representation § 4.1. Problems of the Bending of Prismatic Beams § 4.2. Problems of the Torsion of Prismatic Beams § 5. On the Application of the Method of Summary Representation to Biharmonic Problems of the Bending of Plates § 6. On Certain Formulae of Summary Representation § 6.1.