Mathematical Analysis and Numerical Methods for Science and Technology: Volume 1

Description

In the first years of the 1970's Robert Dautray engaged in conversations with Jacques Yvon, High-Commissioner of Atomic energy, of the necessity of publish­ ing mathematical works of the highest level to put at the disposal of the scientific community a synthesis of the modern methods of calculating physical phe­ nomena. It is necessary to get away from the habit of treating mathematical concepts as elegant abstract entities little used in practice. We must develop a technique, but without falling into an impoverishing utilitarianism. The competence of the Commissariat a I'Energie Atomique in this matter can provide a support of exceptional value for such an enterprise. The work which I have the pleasure to present realises the synthesis ofmathemat­ ical methods, seen from the angle of their applications, and of use in designing computer programs. It should be seen as complete as possible for the present moment, with the present degree of development of each of the subjects. It is this specific approach which creates the richness of this work, at the same time a considerable achievement and a harbinger of the future. The encounter to which it gives rise among the originators of mathematical thought, the users of these concepts and computer scientists will be fruitful for the solution of the great problems which remain to be treated, should they arise from the mathematical structure itself (for example from non-linearities) or from the architecture of computers, such as parallel computers.

Description

These 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. The main physical examples examined in the 6 volumes are presented in Chapter I: Classical Fluids and the Navier-Stokes System; Linear Elasticity, Linear Viscoelasticity, Electromagnetism and Maxwell's Equation, Neutronics, and Quantum Physics. Then a first examination of the mathematical models is given. Chapter II is devoted to the study of the Laplacian operator by methods which only use classical tools.

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