Each chapter is divided into sections. Usually the first sections of a chapter are of an introductory nature, explain numerical phenomena and exhibit numerical results. Investigations of a more theoretieal nature are presented in the later sections of each chapter. As in Volume I, the formulas, theorems, tables and figures are numbered consecutively in each section and indicate, in addition, the section num ber.
In cross references to other chapters the latin chapter number is put first. References to the bibliography are again by "author" plus "year" in parentheses. The bibliography again contains only those papers which are discussed in the text and is in no way meant to be complete. Author : Lawrence F. The book's approach not only explains the presentedmathematics, but also helps readers understand how these numericalmethods are used to solve real-world problems.
Unifying perspectives are provided throughout the text, bringingtogether and categorizing different types of problems in order tohelp readers comprehend the applications of ordinary differentialequations. Detailedreferences outline additional literature on both analytical andnumerical aspects of ordinary differential equations for furtherexploration of individual topics.
Numerical Solution of Ordinary Differential Equations isan excellent textbook for courses on the numerical solution ofdifferential equations at the upper-undergraduate and beginninggraduate levels. It also serves as a valuable reference forresearchers in the fields of mathematics and engineering. Such problems arise in a variety of applications, e. From a more theoretical viewpoint, the study of differential-algebraic problems gives insight into the behaviour of numerical methods for stiff ordinary differential equations.
These lecture notes provide a self-contained and comprehensive treatment of the numerical solution of differential-algebraic systems using Runge-Kutta methods, and also extrapolation methods. Readers are expected to have a background in the numerical treatment of ordinary differential equations.
Numerical Solution of Stochastic Differential Equations. The numerical analysis of stochastic differential equations SDEs differs significantly from that of ordinary differential equations.
This book provides an easily accessible introduction to SDEs, their applications and the numerical methods to solve such equations. From the reviews: "The authors draw upon their own research and experiences in obviously many. Either way analytical solutions are difficult to come by and generally are not in a readily usable form. An approximate algorithm will be very useful in the study of Ito equations as well as in solving stochastic control problems.
Discretization of the random integral equation. The functions a. Divide the interval T, over which the integral equation is defined, into smaller intervals of duration h, i.
Now one may write 2. The first strategy for the development of the algorithm consists in replacing the integrals in 2. Terms involving x x, which arise in the above expansions are successively replaced by the following expression analogous to 2. The number of successive substitutions would be governed by the desired order of error arising from truncation.
The present paper develops an algorithm for an error of order o h 2 and op h2. The development of higher order schemes involves evaluation of stochastic integrals of an increasingly complex nature and the difficulty in evaluating them does not appear to be commensurate with the gain in accuracy. The following notation is followed in this paper: 8b a, a t,, x, , bx. RAO, J. Now equation 2. The terms on the right-hand side of equation 2. We have 2. Now one may write: ax.
W,- W. All the integrals that are encountered in equations 2. So each one of them gives rise to a random variable. The properties of these random variables and their relationships are studied in detail in the Appendix. One may find from these relationships that ZI,, Z2,, Z16,, Z17,, Z s,, Z9, and Z:9 are dependent normal variables and Zs, is uncorrelated to Za, and Z2, and it may be approximated by a normal variable.
The random variables Z16,-Zs 1, are op h2. One may also observe that Zij i 1, 2, Z I, and Z2 are dependent normal variables and Z3 is approximately a normal variable. Now the algorithm given by 3.
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