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Full Description
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 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 ∩ ℝ3 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.
Contents
Preface.- Introduction, PDE, and IE Formulations.- Spaces of Analytic Functions.- Spaces of Solution of the N-S Equations.- Proof of Convergence of Iteration 1.6.3.- Numerical Methods for Solving N-S Equations.- Sinc Convolution Examples.- Implementation Notes.- Result Notes.
