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)

3 x [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

3 x [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.