This paper is devoted to a review of the prolate spheroidal wave functions
(PSWFs) and their variants from the viewpoint of spectral ⁄ spectral-element approximations
using such functions as basis functions. We demonstrate the pros and cons
over their polynomial counterparts, and put the emphasis on the construction of essential
building blocks for efficient spectral algorithms.
In this paper, we use Pacard-Xu's methods to discuss the complex deformation
of constant scalar curvature metrics in the case of fixed and varying complex
structures. Moreover, we also discuss the complex deformation of Kähler-Ricci solitons.
In this paper we use the maximum principle and the Hopf lemma to prove
symmetry results to some overdetermined boundary value problems for domains in
the hemisphere or star-shaped domains in $S^n$. Our method is based on finding suitable $P$-functions as Weinberger ().
Numerical simulations by high order methods for the blood flow model in
arteries have wide applications in medical engineering. This blood flow model admits
the steady state solutions, for which the flux gradient is non-zero, and is exactly balanced
by the source term. In this paper,we design a high order discontinuous Galerkin
method to this model by means of a novel source term approximation as well as well-balanced
numerical fluxes. Rigorous theoretical analysis as well as extensive numerical
results all suggests that the resulting method maintains the well-balanced property,
enjoys high order accuracy and keeps good resolutions for smooth and discontinuous
We prove the concavity of the power of a solution is preserved for a class of
doubly nonlinear parabolic equation, which is a well-known feature in some particular
cases such as the porous medium equation or the parabolic $p$-Laplace equation.
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