Generation of Multivariate Hermite Interpolating Polynomials
This book describes approximate solutions of differential equations in the case when a solution can be expanded into a basis of polynomials with all derivatives up to a predefined order specified at the boundary. The book is divided into three sections. It begins with a thorough examination of constrained numbers, which form a basis for constructions of interpolating polynomials. The author develops their geometric representation in coordinate systems in several dimensions and he presents generating algorithms for each level number. As an application, it is possible to compute the derivative of a product of functions of several variables and to construct an expression for n-dimensional natural numbers.
Section II focuses on a construction of Hermite interpolating polynomials, from their characterizing properties and generating algorithms to a graphical analysis of their behaviour. The final section is devoted to applications of Hermite interpolating polynomials to linear and nonlinear differential equations of one or several variables. An example based on the author’s thermal analysis of the space shuttle during re-entry to the Earth’s atmosphere is particularly interesting. He uses here polynomials developed in the book to solve the heat transfer equations for heating of the lower surface of the wing. The author presents a lot of algorithms and pseudo-codes for generating constrained numbers and Hermite interpolating polynomials. The book can offer an inspiration for further research and will be of interest for graduate students, researchers and software developers in mathematics, physics and engineering.