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Navier–stokes Equations on R3 X 0, T (Hardcover) (Frank Stenger)
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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 (N-S) partial differential equations (PDE) on (x, y, z, t) ? R3 × [0, T].
Initially converting the PDE to a system of integral equations (IE), 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 IE in the space of functions A n R3 × [0, T], and we 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.