Ebook: Partial Differential Equations: New Methods for Their Treatment and Solution

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.

---

---

Content:
Front Matter....Pages i-xvii
Monotone Convergence and Positive Operators....Pages 1-4
Conservation....Pages 5-27
Dynamic Programming and Partial Differential Equations....Pages 28-35
The Euler-Lagrange Equations and Characteristics....Pages 36-58
Quasilinearization and a New Method of Successive Approximations....Pages 59-61
The Variation of Characteristic Values and Functions....Pages 62-81
The Hadamard Variational Formula....Pages 82-87
The Two-Dimensional Potential Equation....Pages 88-102
The Three Dimensional Potential Equation....Pages 103-109
The Heat Equation....Pages 110-119
Nonlinear Parabolic Equations....Pages 120-128
Differential Quadrature....Pages 129-147
Adaptive Grids and Nonlinear Equations....Pages 148-152
Infinite Systems of Differential Equations....Pages 153-175
Green’s Functions....Pages 176-236
Approximate Calculation of Green’s Functions....Pages 237-242
Green’s Functions for Partial Differential Equations....Pages 243-247
The It? Equation and a General Stochastic Model for Dynamical Systems....Pages 248-253
Nonlinear Partial Differential Equations and the Decomposition Method....Pages 254-288
Back Matter....Pages 289-290

---

Content:
Front Matter....Pages i-xvii
Monotone Convergence and Positive Operators....Pages 1-4
Conservation....Pages 5-27
Dynamic Programming and Partial Differential Equations....Pages 28-35
The Euler-Lagrange Equations and Characteristics....Pages 36-58
Quasilinearization and a New Method of Successive Approximations....Pages 59-61
The Variation of Characteristic Values and Functions....Pages 62-81
The Hadamard Variational Formula....Pages 82-87
The Two-Dimensional Potential Equation....Pages 88-102
The Three Dimensional Potential Equation....Pages 103-109
The Heat Equation....Pages 110-119
Nonlinear Parabolic Equations....Pages 120-128
Differential Quadrature....Pages 129-147
Adaptive Grids and Nonlinear Equations....Pages 148-152
Infinite Systems of Differential Equations....Pages 153-175
Green’s Functions....Pages 176-236
Approximate Calculation of Green’s Functions....Pages 237-242
Green’s Functions for Partial Differential Equations....Pages 243-247
The It? Equation and a General Stochastic Model for Dynamical Systems....Pages 248-253
Nonlinear Partial Differential Equations and the Decomposition Method....Pages 254-288
Back Matter....Pages 289-290
....