## Mathematical Analysis and Numerical Methods for Science and Technology: Volume 2 Functional and Variational MethodsThese 6 volumes - the result of a 10 year collaboration between the authors, two of France's leading scientists and both distinguished international figures - 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. The advent of large computers has in the meantime revolutionised methods of computation and made this gap in the literature intolerable: the objective of the present work is to fill just this gap. Many phenomena in physical mathematics may be modeled by a system of partial differential equations in distributed systems: a model here means a set of equations, which together with given boundary data and, if the phenomenon is evolving in time, initial data, defines the system. 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. Modeling by distributed systems now also supports work in many areas of physics (plasmas, new materials, astrophysics, geophysics), chemistry and mechanics and is finding increasing use in the life sciences. |

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### Contents

1 | |

2 The Mellin Transform | 24 |

3 The Hankel Transform | 40 |

Review of Chapter IIIA | 57 |

Sobolev Spaces | 92 |

5 The Spaces H Q for all me N | 120 |

8 Supplementary Remarks | 138 |

Linear Differential Operators | 148 |

Closed Operators | 334 |

3 Linear Operators in Hilbert Spaces | 348 |

Review of Chapter VI | 374 |

2 Examples of Second Order Elliptic Problems | 393 |

3 Regularity of the Solutions of Variational Problems | 425 |

Review of Chapter VII | 456 |

of Two Functions | 458 |

84 | 480 |

2 Linear Differential Operators with Constant Coefficients | 170 |

Cauchy Problem for Differential Operators | 204 |

WellPosed Cauchy Problem | 214 |

Parabolic and Weakly Parabolic Operators | 221 |

WellPosed Cauchy Problem in Q Hyperbolic Operators | 230 |

Comparison of Operators | 241 |

Construction of an Elementary Solution | 247 |

5 The Maximum Principle | 250 |

Review of Chapter V | 268 |

57 | 292 |

71 | 298 |

Ideas About Functions of a Real or Complex Variable | 304 |

on the Real Line | 485 |

Convolution of Distributions | 492 |

Fourier Transforms | 500 |

Fourier Transform in L | 506 |

Partial Fourier Transform | 513 |

533 | |

551 | |

562 | |

569 | |

Contents of Volumes 1 36 | 584 |

### Other editions - View all

Mathematical Analysis and Numerical Methods for Science and Technology ... Robert Dautray,Jacques-Louis Lions No preview available - 1988 |

Mathematical Analysis and Numerical Methods for Science and Technology ... Robert Dautray,Jacques-Louis Lions No preview available - 1988 |

### Common terms and phrases

analytic Banach space bounded open set bounded set Cauchy problem Chap compact support constant coefficients continuous linear continuous linear form converges convex convolution Corollary deduce defined Definition denote dense differential operator Dirichlet problem domain D(A elementary solution elliptic operator equation Example exists a constant finite Fourier transform given graph Green's formula Hahn–Banach theorem Hankel transform hence Hilbert space Hö(Q hyperbolic operators hyperbolic with respect hypo-elliptic hypothesis inequality injection inverse isomorphism kernel Laplacian Lemma Let Q linear form linear mapping linear operator maximum principle neighbourhood normed space notation open set open set Q orthogonal parabolic properties prove regular relatively compact Remark resp result satisfies scalar product sequence sesquilinear form Sobolev spaces Theorem topology transpose u e H'(Q unbounded unique vector space verify weakly x e Q