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3 results

  • 26 - Spectral and Pseudo-Spectral Methods for Periodic Elliptic Equations

  • from Part V - Boundary and Initial Boundary Value Problems
  • Abner J. Salgado, University of Tennessee, Knoxville, Steven M. Wise, University of Tennessee, Knoxville
  • Book:
    Classical Numerical Analysis
    Published online:
    29 September 2022
    Print publication:
    20 October 2022, pp 721-741

  • Summary

    Periodic differential equations and their approximation are the topic of this chapter. We discuss the application of classical finite difference schemes and their analysis in this setting. The spectral Galerkin method, that is, using trigonometric polynomials as basis is then discussed and we show its spectral accuracy. Finally the pseudo-spectral method is presented, its implementation via the DFT is discussed.

  • Existence and Uniqueness of Solutions to Singular p-Laplace Equations of Kirchhoff Type

  • Part of
  • Qingwei Li
  • Journal:
    Canadian Mathematical Bulletin / Volume 63 / Issue 3 / September 2020
    Published online by Cambridge University Press:
    16 December 2019, pp. 677-691
    Print publication:
    September 2020

  • In this paper, we study both the existence and uniqueness of nonnegative solutions for the nonlocal $p$-Laplace equation with singular term

    $$\begin{eqnarray}\left\{\begin{array}{@{}ll@{}}-B\Bigl(\frac{1}{p}\int _{\unicode[STIX]{x1D6FA}}|\unicode[STIX]{x1D6FB}u|^{p}\text{d}x\Bigr)\unicode[STIX]{x1D6E5}_{p}u=\frac{h(x)}{u^{\unicode[STIX]{x1D6FE}}}+k(x)u^{q},\quad & x\in \unicode[STIX]{x1D6FA},\\ u>0,\quad & x\in \unicode[STIX]{x1D6FA},\\ u=0,\quad & x\in \unicode[STIX]{x2202}\unicode[STIX]{x1D6FA},\end{array}\right.\end{eqnarray}$$
    where $\unicode[STIX]{x1D6FA}\subset \mathbb{R}^{N}(N\geqslant 1)$ is a bounded domain with smooth boundary $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$, $h\in L^{1}(\unicode[STIX]{x1D6FA})$, $h>0$ almost everywhere in $\unicode[STIX]{x1D6FA}$, $k\in L^{\infty }(\unicode[STIX]{x1D6FA})$ is a non-negative function, $B:[0,+\infty )\rightarrow [m,+\infty )$ is continuous for some positive constant $m$, $p>1$, $0\leqslant q\leqslant p-1$, and $\unicode[STIX]{x1D6FE}>1$. A “compatibility condition” on the couple $(h(x),\unicode[STIX]{x1D6FE})$ will be given for the problem to admit at least one solution. To be a little more precise, it is shown that the problem admits at least one solution if and only if there exists a $u_{0}\in W_{0}^{1,p}(\unicode[STIX]{x1D6FA})$ such that $\int _{\unicode[STIX]{x1D6FA}}h(x)u_{0}^{1-\unicode[STIX]{x1D6FE}}\text{d}x<\infty$. When $k(x)\equiv 0$, the weak solution is unique.

  • SYMMETRY RESULT FOR SOME OVERDETERMINED VALUE PROBLEMS

  • Part of
  • MOHAMMED BARKATOU, SAMIRA KHATMI
  • Journal:
    The ANZIAM Journal / Volume 49 / Issue 4 / April 2008
    Published online by Cambridge University Press:
    01 April 2008, pp. 479-494
    • Article
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  • The aim of this article is to prove a symmetry result for several overdetermined boundary value problems. For the two first problems, our method combines the maximum principle with the monotonicity of the mean curvature. For the others, we use essentially the compatibility condition of the Neumann problem.