Partial Differential Equations: New Methods for Their Treatment and Solution / Edition 1

by N.D. Bellman, G. Adomian

Overview

The purpose of this book is to present some new methods in the treatment of partial differential equations. Some of these methods lead to effective numerical algorithms when combined with the digital computer. Also presented is a useful chapter on Green's functions which generalizes, after an introduction, to new methods of obtaining Green's functions for partial differential operators. Finally some very new material is presented on solving partial differential equations by Adomian's decomposition methodology. This method can yield realistic computable solutions for linear or non­ linear cases even for strong nonlinearities, and also for deterministic or shastic cases - again even if strong shasticity is involved. Some interesting examples are discussed here and are to be followed by a book dealing with frontier applications in physics and engineering. In Chapter I, it is shown that a use of positive operators can lead to monotone convergence for various classes of nonlinear partial differential equations. In Chapter II, the utility of conservation technique is shown. These techniques are suggested by physical principles. In Chapter III, it is shown that dynamic programming applied to variational problems leads to interesting classes of nonlinear partial differential equations. In Chapter IV, this is investigated in greater detail. In Chapter V, we show. that the use of a transformation suggested by dynamic programming leads to a new method of successive approximations.

Product Details

ISBN-13:	9789027716811
Publisher:	Springer Netherlands
Publication date:	12/31/1984
Series:	Mathematics and Its Applications , #15
Edition description:	1985
Pages:	290
Product dimensions:	8.27(w) x 11.69(h) x 0.24(d)

Table of Contents

I/Monotone Convergence and Positive Operators.- II/Conservation.- III / Dynamic Programming and Partial Differential Equations.- IV / The Euler-Lagrange Equations and Characteristics.- V / Quasilinearization and a New Method of Successive Approximations.- VI / The Variation of Characteristic Values and Functions.- VII / The Hadamard Variational Formula.- VIII / The Two-Dimensional Potential Equation.- IX / The Three-Dimensional Potential Equation.- X / The Heat Equation.- XI / Nonlinear Parabolic Equations.- XII / Differential Quadrature.- XIII / Adaptive Grids and Nonlinear Equations.- XIV / Infinite Systems of Differential Equations.- XV / Green’s Functions.- XVI / Approximate Calculation of Green’s Functions.- XVII / Green’s Functions for Partial Differential Equations.- XVIII / The Itô Equation and a General Shastic Model for Dynamical Systems.- XIX / Nonlinear Partial Differential Equations and the Decomposition Method.

Editorial Reviews

‘As a rich source of ideas the text should be highly recommended to mature students in appropriate fields and to research workers.'
Zentralblatt für Mathematik (1986)

From the Publisher