Chapman and Hall/CRC
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.
CONSTRAINED NUMBERS. Constrained Coordinate System. Generation of the Coordinate System. Natural Coordinates. Computation of the Number of Elements. An Ordering Relation. Application to Symbolic Computation of Derivatives. HERMITE INTERPOLATING POLYNOMIALS. Multivariate Hermite Interpolating Polynomials. Generation of the Hermite Interpolating Polynomials. Hermite Interpolating Polynomials: The Classical and Present Approaches. Normalized Symmetric Square Domain. Rectangular Non-Symmetric Domain. Generic Domains. Extensions of the Constrained Numbers. Field of the Complex Numbers. Analysis of the Behavior of the Hermite Interpolating Polynomials. SELECTED APPLICATIONS. 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.