## About this book

### Introduction

In this monograph, leading researchers in
the world of numerical analysis, partial differential equations, and hard
computational problems study the properties of solutions of the Navier–Stokes** **partial
differential equations on (x, y, z, t) ∈ ℝ^{3} × [0, *T*]. Initially converting the PDE to a
system of integral equations, the authors then describe spaces **A** of analytic functions that house
solutions of this equation, and show that these spaces of analytic functions
are dense in the spaces *S* of rapidly
decreasing and infinitely differentiable functions. This method benefits from
the following advantages:

- The functions of S are nearly always conceptual rather than explicit
- Initial and boundary conditions of solutions of PDE are usually drawn from the applied sciences, and as such, they are nearly always piece-wise analytic, and in this case, the solutions have the same properties
- When
methods of approximation are applied to functions of
**A**they converge at an exponential rate, whereas methods of approximation applied to the functions of**S**converge only at a polynomial rate - Enables sharper bounds on the solution enabling easier existence proofs, and a more accurate and more efficient method of solution, including accurate error bounds

Following the proofs of denseness, the
authors prove the existence of a solution of the integral equations in the
space of functions **A** ∩ ℝ^{3} × [0, *T*], and provide an explicit novel algorithm based on Sinc
approximation and Picard–like iteration for computing the solution.
Additionally, the authors include appendices that provide a custom Mathematica
program for computing solutions based on the explicit algorithmic approximation
procedure, and which supply explicit illustrations of these computed solutions.

### Keywords

### Bibliographic information

- DOI https://doi.org/10.1007/978-3-319-27526-0
- Copyright Information Springer International Publishing AG 2016
- Publisher Name Springer, Cham
- eBook Packages Mathematics and Statistics
- Print ISBN 978-3-319-27524-6
- Online ISBN 978-3-319-27526-0
- About this book