Details

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


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



von: Frank Stenger, Don Tucker, Gerd Baumann

85,59 €

Verlag: Springer
Format: PDF
Veröffentl.: 23.09.2016
ISBN/EAN: 9783319275260
Sprache: englisch

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Beschreibungen

<p>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<b> </b>partial differential equations on (x, y, z,
t) ∈ ℝ<sup>3</sup> × [0, <i>T</i>]. Initially converting the PDE to a
system of integral equations, the authors then describe spaces <b>A</b> of analytic functions that house
solutions of this equation, and show that these spaces of analytic functions
are dense in the spaces <i>S</i> of rapidly
decreasing and infinitely differentiable functions. This method benefits from
the following advantages:</p><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 <b>A</b> they converge at an exponential rate, whereas methods of
approximation applied to the functions of <b>S</b> 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><p>



</p><p>Following the proofs of denseness, the authors prove the
existence of a solution of the integral equations in the space of functions <b>A</b> ∩ ℝ<sup>3</sup> × [0, <i>T</i>], 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.</p>
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.
<p>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<b> </b>partial
differential equations on (x, y, z, t) ∈ ℝ<sup>3</sup> × [0, <i>T</i>]. Initially converting the PDE to a
system of integral equations, the authors then describe spaces <b>A</b> of analytic functions that house
solutions of this equation, and show that these spaces of analytic functions
are dense in the spaces <i>S</i> of rapidly
decreasing and infinitely differentiable functions. This method benefits from
the following advantages:</p><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 <b>A</b> they converge at an exponential rate, whereas methods of
approximation applied to the functions of <b>S</b> 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><p>



</p><p>Following the proofs of denseness, the
authors prove the existence of a solution of the integral equations in the
space of functions <b>A</b> ∩ ℝ<sup>3</sup> × [0, <i>T</i>], 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.</p>
<p>Studies the properties of solutions</p><p>of the Navier–Stokes partial differential equations on (x , y, z , t) ? R3 × [0, T]</p><p>Demonstrates a new method for</p><p>determining solutions of the Navier–Stokes equations by converting partial</p><p>differential equations to a system of integral equations describing spaces of</p><p>analytic functions containing solutions</p><p>Enables sharper bounds on solutions
to Navier–Stokes equations, easier existence proofs, and a more accurate,
efficient method of determining a solution with accurate error bounds</p><p>Includes an custom-written
Mathematica package for computing solutions to the Navier–Stokes equations
based on the author's approximation method</p>

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