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

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 NavierStokes *partial 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 Picardlike 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.

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

  Inhalt
  Preface.- Introduction, PDE, and IE Formulations.- Spaces of Analytic Functions.- Spaces of Solution of the NS Equations.- Proof of Convergence of Iteration 1.6.3.- Numerical Methods for Solving NS Equations.- Sinc Convolution Examples.- Implementation Notes.- Result Notes.