X. Mixed Problems and the Tricomi Equation.- § 1. Description and Formulation of the Problem.- 1. Stationary Plane Flow of a Compressible Fluid.- 2. Solution in the Hodograph Plane; The Frankl' Equation.- § 2. Methods for Solving Problems of Mixed Type.- 1. An Example of a Well-Posed Boundary Value Problem for the Frankl' Equation.- 2. Particular Solutions.- 3. Existence and Uniqueness Results.- Bibliographic Commentary.- XI. Integral Equations.- A. Solution Methods Using Analytic Functions and Sectionally Analytic Functions.- § 1. The Wiener-Hopf Method.- Wiener-Hopf Equations.- 1. The Wiener-Hopf Method.- 2. Decomposition of an Analytic Function Defined in a Strip in the Complex Plane.- 3. Factorisation of an Analytic Function Defined in a Strip in the Complex Plane.- 4. Application to the Wiener-Hopf Integral Equation of the Second Kind.- 5. Application to the Milne Problem.- 6. Application to the Dock Problem.- § 2. Sectionally Analytic Functions.- 1. S. Analytic Functions.- 2. Cauchy Integrals and Plemelj Formulas.- 3. The Poincaré-Bertrand Formula and the Hilbert Inversion Formula.- § 3. The Hilbert Problem.- 1. The Hilbert Problem in the Case where L is a Contour.- 2. The Hilbert Problem in the Case where L is an Arc.- 3. The Hilbert Problem in the Case of a Straight Line.- 4. Some Problems Reducible to a Hilbert Problem.- § 4. Application to Some Problems in Physics.- 1. Simple Layer and Double Layer Problems.- 2. Determination of the Charge Density on the Surface of a Cylindrical Body at Potential V.- 3. The Problem of the Thin Aerofoil Profile.- 4. Plane Elasticity and the Biharmonic Equation.- B. Integral Equations Associated with Elliptic Boundary Value Problems in Domains in ?3.- § 1. Study of Certain Weighted Sobolev Spaces.- § 2. Integral Equations Associated with the Boundary Value Problems of Electrostatics.- 1. Integral Representations.- 2. Dirichlet Problems Relative to the Operator ?.- 3. Neumann Problems Relative to the Operator ?.- § 3. Integral Equations Associated with the Helmholtz Equation.- § 4. Integral Equations Associated with Problems of Linear Elasticity.- § 5. Integral Equations Associated with the Stokes System.- XII. Numerical Methods for Stationary Problems.- 1. The Basic Ideas of Finite Difference Methods and Finite Element Methods.- 2. Comparison of the Two Methods. Field of Applications of the Finite Element Method.- 3. The Different Topics Treated in this Chapter XII.- 4. The Lax-Milgram Theorem and Sobolev Spaces.- § 1. Principal Aspects of the Finite Element Method Applied to the Problem of Linear Elasticity.- 1. Variational Formulation of the Continuous Problem.- 2. Construction of Approximation Function Spaces.- 3. The First Approximation Problem (Ph1).- 4. Numerical Quadrature Schemes and the Definition of the Second Approximation Problem (Ph2).- 5. Error Estimates.- 6. Numerical Implementation.- § 2. Treatment of Domains with Curved Boundaries.- 1. "Exact" Triangulation of the Domain ?.- 2. Construction of an "Approximate" Triangulation of the Domain ?.- 3. Examples of the Construction of the Mappings FK.- 4. Definition of Curved Finite Elements of Class ?0.- 5. Estimation of the Interpolation Error.- 6. Application to the Solution of the Problem of Plane Linear Elasticity.- § 3. A Non Conforming Method of Finite Elements.- 1. The Wilson Finite Element.- 2. Estimation of the Interpolation Error.- 3. The Space Xh of Finite Elements.- 4. The Discrete Problem. Abstract Error Estimate.- 5. The Bilinear Lemma.- 6. Estimation of the Error $$ \parallel \vec{u} - {\vec{u}_h}{\parallel_h} $$.- § 4. Applications to the Problems of Plates and Shells.- 1. Approximation of the Problems of Plates.- 2. Approximation of the Problems of Shells.- § 5 Approximation of Eigenvalues and Eigenvectors.- 1. Some Results from the Spectral Theory of Differential Operators.- 2. The Approximate Problem.- 3. Estimation of the Errors $$ \left| {{{\vec{\lambda }}_j} - {{\vec{\lambda }}_{{hj}}}} \right|,1 \leqslant j \leqslant 3{M_h} $$.- 4. Estimation of the Errors $$ \parallel {\vec{u}_j} - {\vec{u}_{{hj}}}\parallel, 1 \leqslant j \leqslant 3{M_h} $$.- 5. Numerical Solutions.- § 6. An Example of the Approximate Calculation for a Problem of the Eigenvalues of a Non Self-Adjoint Operator.- 1. The Neutron Diffusion Equations Recalled.- 2. The Critical Problem with Two Energy Groups.- 3. Determination of the Positive Solution.- 4. Extension to the Case Where the Number of Neutron (Kinetic) Energy Groups is Greater than Two.- 5. The Eigenvalue Problem Connected with the Evolution Problem of Neutron Diffusion.- 6. Some Comments.- Review of Chapter XII.- XIII. Approximation of Integral Equations by Finite Elements. Error Analysis.- § 1. The Case of a Polyhedral Surface.- 1. The Simple Layer Potential Case for the Dirichlet Problem.- 2. Study of the Potential of a Double Layer for the Neumann Problem in ?3.- 3. Study of the Exterior Neumann Problem Represented by a Simple Layer.- § 2. The Case of a Regular Closed Surface.- 1. The Approximation of Surfaces.- 2. Notions on the Error Generated by the Approximation of the Surface.- Appendix. "Singular Integrals".- § 1. Operator, Convolution Operator, Integral Operator.- § 2. The Hilbert Transformation.- § 3. Generalities on Singular Integral Operators.- § 5. The Calderon-Zygmund Theorem.- § 6. Marcinkiewicz Spaces.- 1. Definitions.- 2. Application to the Homogeneous Convolution Kernel.- 4. Operators of Weak Type. The Marcinkiewicz Theorem.- 5. The Maximal Hardy-Littlewood Operator..- Proof of Lemma 1 in § 2.- Table of Notations.- of Volumes1-3, 5, 6.
