Presents a fresh approach to the approximate solution of differential equationsSupplies algorithms and pseudocode for generating constrained numbers and Hermite interpolating polynomialsContains detailed examples that clarify the algorithms and aid in troubleshooting the development of computer codeDemonstrates several applications of the algorithms, with both one-dimensional and multivariate examples
Generation of Multivariate Hermite Interpolating Polynomials advances the study of approximate solutions to partial differential equations by presenting a novel approach that employs Hermite interpolating polynomials and bysupplying algorithms useful in applying this approach.
Organized into three sections, the book begins with a thorough examination of constrained numbers, which form the basis for constructing interpolating polynomials. The author develops their geometric representation in coordinate systems in several dimensions and presents generating algorithms for each level number. He then discusses their applications in computing the derivative of the product of functions of several variables and in the construction of expression for n-dimensional natural numbers. Section II focuses on the construction of Hermite interpolating polynomials, from their characterizing properties and generating algorithms to a graphical analysis of their behavior.
The final section of the book is dedicated to the application of Hermite interpolating polynomials to linear and nonlinear differential equations in one or several variables. Of particular interest is an example based on the author's thermal analysis of the space shuttle during reentry to the earth's atmosphere, wherein he uses the polynomials developed in the book to solve the heat transfer equations for the heating of the lower surface of the wing.
Table of Contents
Constrained Coordinate System
Generation of the Coordinate System
Computation of the Number of Elements
An Ordering Relation
Application to Symbolic Computation of Derivatives
HERMITE INTERPOLATING POLYNOMIALS
Multivariate Hermite Interpolating Polynomial
Generation of the Hermite Interpolating Polynomials
Hermite Interpolating Polynomials: The Classical and Present Approaches
Normalized Symmetric Square Domain
Rectangular Nonsymmetric Domain
Extensions of the Constrained Numbers
Field of the Complex Numbers
Analysis of the Behavior of the Hermite Interpolating Polynomials
Construction of the Approximate Solution
One-Dimensional Two-Point Boundary Value Problems
Application to Problems with Several Variables
Thermal Analysis of the Surface of the Space Shuttle
"…the book is timely, very useful, and quite unique. … [The book] contains many detailed constructions, interesting concepts, and practical discussions. … It could be used as a reference book and/or a supplement to a course on discretization methods for differential equations."
- Mathematics of Computation, April 2005
"…Overall this is a useful and well-written text. …I recommend the book for applied mathematicians and practitioners using the finite element method."
- SIAM Review