Homogenisation

1. Formulation of Elementary Boundary Value Problems.- x1. The Concept of the Classical Formulation of a Boundary Value Problem for Equations with Discontinuous Coefficients.- x2. The Concept of Generalized Solution.- x3. Generalized Formulations of Problems for the Basic Equations of Mathematical Physics.- 2. The Concept of Asymptotic Expansion. A Model Example to Illustrate the Averaging Method.- x1. Asymptotic expansion. A Formal Asymptotic Solution.- x2. Asymptotic Expansion of a Solution of the Equation u = 1 + ?u3.- x3. Asymptotic Expansion of a Solution of the Equation (K(x/?)u?)?= f(x) by the Averaging Method.- x4. Generalization of the Averaging Method in the Case of a Piecewise Smooth Coefficient.- x5. Averaging the System of Differential Equations.- 3. Averaging Processes in Layered Media.- x1. Problem of Small Longitudinal Vibrations of a Rod.- x2. Nonstationary Problem of Heat Conduction.- x3. Averaging Maxwell Equations.- x4. Averaging Equations of a Viscoelastic Medium.- x5. Media with Slowly Changing Geometric Characteristics.- x6. Heat Transfer Through a System of Screens.- x7. Averaging a Nonlinear Problem of the Elasticity Theory in an Inhomogeneous Rod.- x8. The System of Equations of Elasticity Theory in a Layered Medium.- x9. Considerations Permitting Reduction of Calculations in Constructing Averaged Equations.- x10. Nonstationary Nonlinear Problems.- x11. Averaging Equations with Rapidly Oscillating Nonperiodic Coefficients.- x12. Problems of Plasticity and Dynamics of Viscous Fluid as Described by Functions Depending on Fast Variables.- 4. Averaging Basic Equations of Mathematical Physics.- x1. Averaging Stationary Thermal Fields in a Composite.- x2. Asymptotic Expansion of Solution of the Stationary Heat Conduction Problem.- x3. Stationary Thermal Field in a Porous Medium.- x4. Averaging a Stationary System of Equations of Elasticity Theory in Composite and Porous Materials.- x5. Nonstationary Systems of Equations of Elasticity and Diffusion Theory.- x6. Averaging Nonstationary Nonlinear System of Equations of Elasticity Theory.- x7. Averaging Stokes and Navier-Stokes Equations. The Derivation of the Percolation Law for a Porous Medium (Darcy's Law).- x8. Averaging in case of Short-Wave Propagation.- x9. Averaging the Transition Equation for a Periodic Medium.- x10. Eigenvalue Problems.- 5. General Formal Averaging Procedure.- x1. Averaging Nonlinear Equations.- x2. Averaged Equations of Infinite Order for a Linear Periodic Medium and for the Equation of Moment Theory.- x3. A Method of Describing Multi-Dimensional Periodic Media that does not Involve Separating Fast and Slow Variables.- 6. Properties of Effective Coefficients. Relationship Among Local and Averaged Characteristics of a Solution.- x1. Maintaining the Properties of Convexity and Symmetry of the Minimized Functional in Averaging.- x2. On the Principle of Equivalent Homogeneity.- x3. The Symmetry Properties of Effective Coefficients and Reduction of Periodic Problems to Boundary Value Problems.- x4. Agreement Between Theoretically Predicted Values of Effective Coefficients and Those Determined by an Ideal Experiment.- 7. Composite Materials Containing High-Modulus Reinforcement.- x1. The Stationary Field in a Layered Material.- x2. Composite Materials with Grains for Reinforcement.- x3. Dissipation of Waves in Layered Media.- x4. High-Modulus 3D Composite Materials.- x5. The Splitting Principle for the Averaged Operator for 3D High-Modulus Composites.- 8. Averaging of Processes in Skeletal Structures.- x1. An Example of Averaging a Problem on the Simplest Framework.- x2. A Geometric Model of a Framework.- x3. The Splitting Principle for the Averaged Operator for a Periodic Framework.- x4. The Splitting Principle for the Averaged Operator for Trusses and Thin-walled Structures.- x5. On Refining the Splitting Principle for the Averaged Operator.- x6 Asymptotic Expansion of a Solution of a Linear Equation in Partial Derivatives for a Rectangular Framework.- x7 Skeletal Structures with Random Properties.- 9. Mathematics of Boundary-Layer Theory in Composite Materials.- x1. Problem on the Contact of Two Layered Media.- x2. The Boundary Layer for an Elliptic Equation Defined on a Half-Plane.- x3. The Boundary Layer Near the Interface of Two Periodic Structures.- x4. Problem on the Contact of Two Media Divided by a Thin Interlayer.- x5. The Boundary Layer for the Nonstationary System of Equations of Elasticity Theory.- x6. On the Ultimate Strength of a Composite.- x7. Boundary Conditions of Other Types.- x8. On the Averaging of Fields in Layer Media with Layers of Composite Materials.- x9. The Time Boundary Layer for the Cauchy Parabolic Problem.- Supplement: Existence and Uniqueness Theorems for the Problem on a Cell.