In this monograph leading researchers in the world ofnumerical analysis partial differential
equations and hard computationalproblems study the properties of solutions of the
Navier-Stokes partial differential equations on (x y z t) 3 × [0 T]. Initially converting
the PDE to asystem of integral equations the authors then describe spaces A of analytic
functions that housesolutions of this equation and show that these spaces of analytic
functionsare dense in the spaces S of rapidlydecreasing and infinitely differentiable
functions. This method benefits fromthe 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 boundsFollowing the proofs of
denseness the authors prove theexistence of a solution of the integral equations in the space
of functions A 3 × [0 T] and provide an explicit novelalgorithm based on Sinc approximation
and Picard-like iteration for computingthe solution. Additionally the authors include
appendices that provide acustom Mathematica program for computing solutions based on the
explicitalgorithmic approximation procedure and which supply explicit illustrations ofthese
computed solutions.
