Focusing on advanced numerical problem-solving, this handbook offers innovative methods akin to finite elements, specifically tailored for analytic issues with singularities in complex regions. It simplifies sinc methods for users by minimizing technical explanations, making it practical for a range of mathematical applications, from calculus to differential equations. With over 470 MATLAB programs included, along with a CD-ROM for user convenience, it serves as a comprehensive resource for students and professionals in mathematics, computer science, and engineering.
Sinc methods, rooted in the Sinc function, offer powerful solutions to a variety of computational challenges, including those involving singularities and boundary layers. This book, authored by a leading expert, introduces these methods, highlighting their significance beyond the well-known Fast Fourier Transform. It serves both as a comprehensive research sourcebook and a textbook for advanced numerical analysis and applied approximation theory. The inclusion of problem sections and historical notes enriches the learning experience for students and researchers alike.
In this monograph, leading researchers in numerical analysis, partial differential equations, and complex computational problems explore the properties of solutions to the Navier–Stokes partial differential equations in the domain (x, y, z, t) ∈ ℝ³ × [0, T]. The authors begin by transforming the PDE into a system of integral equations and identifying spaces A of analytic functions that contain the solutions. They demonstrate that these spaces are dense within the spaces S of rapidly decreasing and infinitely differentiable functions. This approach offers several advantages: functions in S are often conceptual rather than explicit, initial and boundary conditions from applied sciences are typically piece-wise analytic, allowing solutions to retain similar properties. Furthermore, approximation methods applied to functions in A converge exponentially, while those for functions in S converge at a polynomial rate. This leads to sharper bounds on solutions, facilitating existence proofs and providing a more accurate, efficient solution method with precise error bounds. After establishing denseness, the authors prove the existence of solutions in the space A ∩ ℝ³ × [0, T] and introduce a novel algorithm based on Sinc approximation and Picard-like iteration for computing these solutions. Appendices include a custom Mathematica program for implementing the algorithm and visualizing the computed solutions.
