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A Spectral Method for the Eigenvalue Problem for Elliptic Equations
Kendall Atkinson,Olaf Hansen +1 more
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In this article, the eigenvalue problem Lu = u for an elliptic partial differential operator L over with zero values for either Dirichlet or Neumann boundary conditions is considered.Abstract:
Let be an open, simply connected, and bounded region in R d , d � 2, and assume its boundary @ is smooth. Consider solving the eigenvalue problem Lu = �u for an elliptic partial differential operator L over with zero values for either Dirichlet or Neumann boundary conditions. We propose, analyze, and illustrate a ‘spectral method’ for solving numerically such an eigenvalue problem. This is an extension of the methods presented earlier in [5], [6].read more
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Computation of Special Functions
TL;DR: In this article, a set of subroutines uses rational approximations to compute Bessel functions of integral order, and empirical formulae have been developed to express the limiting boundaries of the modes of computation.
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Reproducing kernels of Sobolev spaces via a green kernel approach with differential operators and boundary operators
Gregory E. Fasshauer,Qi Ye +1 more
TL;DR: The theoretical results provide a more intuitive way of understanding what kind of functions are well approximated by the reproducing kernel-based interpolant to a given multivariate data sample.
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Reproducing Kernels of Sobolev Spaces via a Green Kernel Approach with Differential Operators and Boundary Operators
Gregory E. Fasshauer,Qi Ye +1 more
TL;DR: In this paper, the reproducing kernel along with its associated Hilbert space is shown to be embedded in a classical Sobolev space, and the inner product of reproducing-kernel Hilbert space in terms of the operators $\mathbf{P}$ and $\mathBF{B}$.
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Hierarchical model reduction for incompressible fluids in pipes
TL;DR: An extensive investigation of hierarchical model reduction in polar coordinates is performed to discuss different possible choices for the transverse basis, pointing out pros and cons of the polar coordinate system.
Hierarchical model reduction for incompressible flows in cylindrical domains
TL;DR: The HiMod approximation of Advection-Diffusion-Reaction as well as Incompressible Navier-Stokes equations in axisymmetric domains are addressed, having computational hemodynamics as reference application.
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Handbook of Mathematical Functions
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Partial Differential Equations
TL;DR: In this paper, the authors present a theory for linear PDEs: Sobolev spaces Second-order elliptic equations Linear evolution equations, Hamilton-Jacobi equations and systems of conservation laws.
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The Mathematical Theory of Finite Element Methods
TL;DR: In this article, the construction of a finite element of space in Sobolev spaces has been studied in the context of operator-interpolation theory in n-dimensional variational problems.
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Orthogonal Polynomials
TL;DR: In this paper, different aspects of the theory of orthogonal polynomials of one (real or complex) variable are reviewed and orthogonality on the unit circle is not discussed.
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