1. Formulation of Elementary Boundary Value Problems.- 1. The Concept of the Classical Formulation of a Boundary Value Problem for Equations with Discontinuous Coefficients.- 2. The Concept of Generalized Solution.- 3. 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.- 1. Asymptotic expansion. A Formal Asymptotic Solution.- 2. Asymptotic Expansion of a Solution of the Equation u = 1 + ?u3.- 3. Asymptotic Expansion of a Solution of the Equation (K(x/?)u?)?= f(x) by the Averaging Method.- 4. Generalization of the Averaging Method in the Case of a Piecewise Smooth Coefficient.- 5. Averaging the System of Differential Equations.- 3. Averaging Processes in Layered Media.- 1. Problem of Small Longitudinal Vibrations of a Rod.- 2. Nonstationary Problem of Heat Conduction.- 3. Averaging Maxwell Equations.- 4. Averaging Equations of a Viscoelastic Medium.- 5. Media with Slowly Changing Geometric Characteristics.- 6. Heat Transfer Through a System of Screens.- 7. Averaging a Nonlinear Problem of the Elasticity Theory in an Inhomogeneous Rod.- 8. The System of Equations of Elasticity Theory in a Layered Medium.- 9. Considerations Permitting Reduction of Calculations in Constructing Averaged Equations.- 10. Nonstationary Nonlinear Problems.- 11. Averaging Equations with Rapidly Oscillating Nonperiodic Coefficients.- 12. Problems of Plasticity and Dynamics of Viscous Fluid as Described by Functions Depending on Fast Variables.- 4. Averaging Basic Equations of Mathematical Physics.- 1. Averaging Stationary Thermal Fields in a Composite.- 2. Asymptotic Expansion of Solution of the Stationary Heat Conduction Problem.- 3. Stationary Thermal Field in a Porous Medium.- 4. Averaging a Stationary System of Equations of Elasticity Theory in Composite and Porous Materials.- 5. Nonstationary Systems of Equations of Elasticity and Diffusion Theory.- 6. Averaging Nonstationary Nonlinear System of Equations of Elasticity Theory.- 7. Averaging Stokes and Navier-Stokes Equations. The Derivation of the Percolation Law for a Porous Medium (Darcy's Law).- 8. Averaging in case of Short-Wave Propagation.- 9. Averaging the Transition Equation for a Periodic Medium.- 10. Eigenvalue Problems.- 5. General Formal Averaging Procedure.- 1. Averaging Nonlinear Equations.- 2. Averaged Equations of Infinite Order for a Linear Periodic Medium and for the Equation of Moment Theory.- 3. 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.- 1. Maintaining the Properties of Convexity and Symmetry of the Minimized Functional in Averaging.- 2. On the Principle of Equivalent Homogeneity.- 3. The Symmetry Properties of Effective Coefficients and Reduction of Periodic Problems to Boundary Value Problems.- 4. Agreement Between Theoretically Predicted Values of Effective Coefficients and Those Determined by an Ideal Experiment.- 7. Composite Materials Containing High-Modulus Reinforcement.- 1. The Stationary Field in a Layered Material.- 2. Composite Materials with Grains for Reinforcement.- 3. Dissipation of Waves in Layered Media.- 4. High-Modulus 3D Composite Materials.- 5. The Splitting Principle for the Averaged Operator for 3D High-Modulus Composites.- 8. Averaging of Processes in Skeletal Structures.- 1. An Example of Averaging a Problem on the Simplest Framework.- 2. A Geometric Model of a Framework.- 3. The Splitting Principle for the Averaged Operator for a Periodic Framework.- 4. The Splitting Principle for the Averaged Operator for Trusses and Thin-walled Structures.- 5. On Refining the Splitting Principle for the Averaged Operator.- 6 Asymptotic Expansion of a Solution of a Linear Equation in Partial Derivatives for a Rectangular Framework.- 7 Skeletal Structures with Random Properties.- 9. Mathematics of Boundary-Layer Theory in Composite Materials.- 1. Problem on the Contact of Two Layered Media.- 2. The Boundary Layer for an Elliptic Equation Defined on a Half-Plane.- 3. The Boundary Layer Near the Interface of Two Periodic Structures.- 4. Problem on the Contact of Two Media Divided by a Thin Interlayer.- 5. The Boundary Layer for the Nonstationary System of Equations of Elasticity Theory.- 6. On the Ultimate Strength of a Composite.- 7. Boundary Conditions of Other Types.- 8. On the Averaging of Fields in Layer Media with Layers of Composite Materials.- 9. The Time Boundary Layer for the Cauchy Parabolic Problem.- Supplement: Existence and Uniqueness Theorems for the Problem on a Cell.