| Preface | |
| Modelling Some Physical Phenomena In Terms of Partial Differential Equations | |
| Concept of partial differential equations | p. 1 |
| Partial differential equations as Euler-Lagrange equations for variational problems | p. 2 |
| Oscillations of a membrane | p. 5 |
| Propagation of heat | p. 7 |
| Equations of linear theory of elasticity | p. 8 |
| Basic equations of fluid mechanics | p. 13 |
| Equations of general theory of relativity | p. 16 |
| Model partial differential equations | p. 17 |
| Basic Types of Partial Differential Equations and Wellposedness of Statement of the Problems for Them | |
| Concept of characteristic form and type division of partial differential equations | p. 23 |
| Classification into types of linear second order partial differential equations | p. 25 |
| Canonical form of linear second order partial differential equations with two independent variables | p. 30 |
| Systems of linear partial differential equations | p. 34 |
| Concept of well-posedness in formulation of problems of partial differential equations | p. 41 |
| Wave equation | p. 43 |
| Laplace equation | p. 54 |
| Heat-conduction equation | p. 68 |
| Elliptic Type Equations | |
| Smoothness classes of functions and their domain of specification | p. 79 |
| Adjoint linear second order partial differential operators | p. 84 |
| Existence of solutions for second order linear elliptic equation. Elementary-solutions | p. 87 |
| Double layer and simple layer potentials | p. 98 |
| Dirichlet's problem for harmonic functions | p. 104 |
| Neumann's problem. Problem with directional derivative | p. 114 |
| Some other problems for elliptic equations | p. 122 |
| Extremum principle for equation (139) and the uniqueness of solution for problem (135)-(137) | p. 126 |
| Generalized Solutions of Partial Differential Equations | |
| A brief survey of methods for constructing classical solutions | p. 129 |
| Variational methods | p. 145 |
| Generalized solutions for classical problems of partial differential equations | p. 158 |
| A brief survey of methods for proving existence of generalized solutions | p. 177 |
| Some Classes of Non-linear Partial Differential Equations | |
| General observations | p. 185 |
| Cauchy-Kovalevski equations | p. 188 |
| Initial value and characteristic Cauchy's problems of some quasilinear hyperbolic equations | p. 199 |
| Dirichlet's problem for non-linear elliptic equations | p. 205 |
| Cauchy's problem for a class of first order quasilinear equations | p. 208 |
| Linearization of a class of quasilinear equations | p. 212 |
| Some other classes of non-linear partial differential equations | p. 217 |
| On the smoothness character of solutions for partial differential equations | p. 222 |
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