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AuthorCollatz, Lothar. author
TitleThe Numerical Treatment of Differential Equations [electronic resource] / by Lothar Collatz
ImprintBerlin, Heidelberg : Springer Berlin Heidelberg, 1960
Edition Third Edition
Connect tohttp://dx.doi.org/10.1007/978-3-642-88434-4
Descript XV, 568 p. 6 illus., 1 illus. in color. online resource

SUMMARY

This book constitutes an attempt to present in a connected fashion some of the most important numerical methods for the solution of ordinary and partial differential equations. The field to be covered is extremely wide, and it is clear that the present treatment cannot be remotely exhaustive; in particular, for partial differential equations it has only been possible to present the basic ideas, and many of the methods developed extensively by workers in applied fields - hydroยญ dynamics, aerodynamics, etc. -, most of which have been developed for specific problems, have had to be dismissed with little more than a reference to the literature. However, the aim of the book is not so much to reproduce these special methods, their corresponding computing schemes, etc. , as to acquaint a wide circle of engineers, physicists and mathematicians with the general methods, and to show with the aid of numerous worked examples that an idea of the quantitative behaviour of the solution of a differential equation problem can be obtained by numerical means with nothing like the trouble and labour that widespread prejudice would suggest. This prejudice may be partly due to the kind of matheยญ matical instruction given in technical colleges and universities, in which, although the theory of differential equations is dealt with in detail, numerical methods are gone into only briefly


CONTENT

I Mathematical preliminaries and some general principles -- Some notes on the numerical examples -- ยง 1. Introduction to problems involving differential equations -- ยง 2. Finite differences and interpolation formulae -- ยง 3. Further useful formulae from analysis -- ยง 4. Some error distribution principles -- ยง 5. Some useful results from functional analysis -- II Initial-value problems in ordinary differential equations -- ยง 1. Introduction -- ยง 2. The Runge-Kutta method for differential equations of the n-th order -- ยง 3. Finite-difference methods for differential equations of the first order -- ยง 4. Theory of the finite-difference methods -- ยง 5. Finite-difference methods for differential equations of higher order -- III Boundary-value problems in ordinary differential equations -- ยง 1. The ordinary finite-difference method -- ยง 2. Refinements of the ordinary finite-difference method -- ยง 3. Some theoretical aspects of the finite-difference methods -- ยง 4. Some general methods -- ยง 5. Ritzโ{128}{153}s method for second-order boundary-value problems -- ยง 6. Ritzโ{128}{153}s method for boundary-value problems of higher order -- ยง 7. Series solutions -- ยง 8. Some special methods for eigenvalue problems -- IV Initial-and initial-/boundary-value problems in partial differential equations -- The need for a sound theoretical foundation -- ยง 1. The ordinary finite-difference method -- ยง 2. Refinements of the finite-difference method -- ยง 3. Some theoretical aspects of the finite-difference methods -- ยง 4. Partial differential equations of the first order in one dependent variable -- ยง 5. The method of characteristics for systems of two differential equations of the first order -- ยง 6. Supplements -- V Boundary-value problems in partial differential equations -- ยง 1. The ordinary finite-difference method -- ยง 2. Refinements of the finite-difference method -- ยง 3. The boundary-maximum theorem and the bracketing of solutions -- ยง 4. Some general methods -- ยง 5. The Ritz method -- ยง 6. The Trefftz method -- VI Integral and functional equations -- ยง 1. General methods for integral equations -- ยง 2. Some special methods for linear integral equations -- ยง 3. Singular integral equations -- ยง 4. Volterra integral equations -- ยง 5. Functional equations -- Table III. Finite-difference expressions for ordinary differential equations -- Table IV. Euler expressions for functions of one independent variable -- Table V. Euler expressions for functions of two independent variables -- Table VII. Catalogue of examples treated -- Author index


Mathematics Differential equations Partial differential equations Computer mathematics Mathematics Computational Mathematics and Numerical Analysis Ordinary Differential Equations Partial Differential Equations



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