Mathematical Analysis and Numerical Methods for Science and Technology: Volume 4 Integral Equations and Numerical Methods

The advent of high-speed computers has made it possible for the first time to calculate values from models accurately and rapidly. Researchers and engineers thus have a crucial means of using numerical results to modify and adapt arguments and experiments along the way. Every facet of technical and industrial activity has been affected by these developments. The objective of the present work is to compile the mathematical knowledge required by researchers in mechanics, physics, engineering, chemistry and other branches of application of mathematics for the theoretical and numerical resolution of physical models on computers. Since the publication in 1924 of the "Methoden der mathematischen Physik" by Courant and Hilbert, there has been no other comprehensive and up-to-date publication presenting the mathematical tools needed in applications of mathematics in directly implementable form.

Table of Contents:
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.