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) ∈ ℝ³ × [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 ∩ ℝ³ × [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.