Partial Differential Equations, Softcover reprint of the original 1st ed. 1985 New Methods for Their Treatment and Solution Mathematics and Its Applications Series, Vol. 15

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 stochastic cases - again even if strong stochasticity 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 dyn~mic 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.

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 Stochastic Model for Dynamical Systems.- XIX / Nonlinear Partial Differential Equations and the Decomposition Method.