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-Stokespartial differential equations on (x, y, z,
t) ? ?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 ? ?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.
