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Navier-Stokes Equations on R3 × [0, T]

von: Frank Stenger, Don Tucker, Gerd Baumann
96,29 €
Verlag:	Springer
Format:	PDF
Veröffentl.:	23.09.2016
ISBN/EAN:	9783319275260
Sprache:	englisch

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Beschreibungen

Titelbeschreibung

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:<ul>
<li>The functions of S are
nearly always conceptual rather than explicit</li>
<li>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</li>
<li>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</li>
<li>Enables sharper bounds on
the solution enabling easier existence proofs, and a more accurate and
more efficient method of solution, including accurate error bounds</li>
</ul>

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.

Inhaltsverzeichnis

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.

Back cover copy

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:<ul>
<li>The
functions of S are nearly always conceptual rather than explicit</li>
<li>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</li>
<li>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</li>
<li>Enables
sharper bounds on the solution enabling easier existence proofs, and a
more accurate and more efficient method of solution, including accurate
error bounds</li>
</ul>

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.

Sonderbeitrag

Studies the properties of solutionsof the Navier–Stokes partial differential equations on (x , y, z , t) ? R3 × [0, T]Demonstrates a new method fordetermining solutions of the Navier–Stokes equations by converting partialdifferential equations to a system of integral equations describing spaces ofanalytic functions containing solutionsEnables sharper bounds on solutions
to Navier–Stokes equations, easier existence proofs, and a more accurate,
efficient method of determining a solution with accurate error boundsIncludes an custom-written
Mathematica package for computing solutions to the Navier–Stokes equations
based on the author's approximation methodIncludes supplementary material: sn.pub/extras