# American Institute of Mathematical Sciences

ISSN:
1534-0392

eISSN:
1553-5258

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## Communications on Pure & Applied Analysis

June 2006 , Volume 5 , Issue 2

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2006, 5(2): i-ii doi: 10.3934/cpaa.2006.5.2i +[Abstract](694) +[PDF](40.0KB)
Abstract:
Over the past decades there have been very rapid developments of analysis and numerical approximations for singular problems. To review the recent developments and to explore exciting new directions in this area, the International Workshop on Analysis and Numerical Approximation of Singular Problems was held at Instituto Superior Técnico, Lisbon, Portugal, from 10-12 November 2004. The aim of this workshop was to bring together active scientists working on singular problems in physics and engineering, and to provide a forum so that they would meet and exchange ideas in a stimulating environment. The conference was attended by more than forty participants from over ten countries, including 14 invited talks, 13 contributed talks and a poster session. The detailed information of the workshop can be found in http://www.math.ist.utl.pt/~plima/IWAN.

2006, 5(2): 241-250 doi: 10.3934/cpaa.2006.5.241 +[Abstract](1429) +[PDF](207.3KB)
Abstract:
We will consider the weakly singular Fredholm integral operator

$T:L^{1}([0,\tau^\star])\rightarrow L^{1}([0,\tau^\star]),\quad (T\varphi)(\tau)=\frac{\bar\omega}{2} \int_{0}^{\tau^\star}E_{1}(|\tau -\tau^'|)\varphi(\tau')\,d\tau',$

where $E_{1}$ denotes the first exponential integral function,

$E_{1}(\tau)=\int_{1}^{\infty}\frac{\exp(-\tau\mu)}{\mu}\mu,\quad\tau>0,$

and $\bar\omega$ is a constant. The spectral elements of a matrix operator representing the discretization of the integral operator $T$ by a projection method on a subspace of dimension $n$ will be computed. These spectral elements will be refined iteratively, by a defect correction type formula to yield an approximation to the spectral elements of $T$.

2006, 5(2): 251-259 doi: 10.3934/cpaa.2006.5.251 +[Abstract](1426) +[PDF](113.9KB)
Abstract:
We consider nonlinear boundary value problems with multiple solutions. A method is proposed for the computation of such solutions which depends crucially on known a priori qualitative information about the behavior of the solutions. The method is a two-stage method where the second stage is a shooting method and initial values of the shooting parameters are found in a first stage which approximates the boundary value problem with a discrete approximation. Both nonsingular and singular problems are considered.
2006, 5(2): 261-276 doi: 10.3934/cpaa.2006.5.261 +[Abstract](1169) +[PDF](216.0KB)
Abstract:
We analyze the attainable order of convergence of collocation solutions for linear and nonlinear Volterra functional integro-differential equations of neutral type containing weakly singular kernels and nonvanishing delays. The discretization of the initial-value problem is based on a reformulation as a sequence of ODEs with nonsmooth solutions. The paper concludes with a brief description of possible alternative numerical approaches for solving various classes of such functional integro-differential equations.
2006, 5(2): 277-288 doi: 10.3934/cpaa.2006.5.277 +[Abstract](1344) +[PDF](219.9KB)
Abstract:
We are concerned with the analytical and numerical analysis of a nonlinear weakly singular Volterra integral equation. Owing to the singularity of the solution at the origin, the global convergence order of Euler's method is less than one. The smoothness properties of the solution are investigated and, by a detailed error analysis, we prove that first order of convergence can be achieved away from the origin. Some numerical results are included confirming the theoretical estimates.
2006, 5(2): 289-307 doi: 10.3934/cpaa.2006.5.289 +[Abstract](1263) +[PDF](243.2KB)
Abstract:
In this paper we present a comparison of numerical methods for the solution of single term fractional differential equations. We review five available methods and use a graphical technique to compare their relative merits. We conclude by giving recommendations on the choice of efficient methods for any given single term fractional differential equation.
2006, 5(2): 309-319 doi: 10.3934/cpaa.2006.5.309 +[Abstract](972) +[PDF](280.6KB)
Abstract:
The purpose of the paper is to introduce a novel cell boundary element (CBE) method for the convection dominated diffusion equation. The CBE method can be viewed as a Petrov-Galerkin type method defined on the skeleton of a mesh. The proposed method utilizes continuity of normal flux on each inter-element boundary. By constructing a local basis (mesh-oriented element) that is dependent upon the orientation of the mesh we could obtain a stable non-oscillatory numerical scheme. We also consider a local basis (wind-oriented element) which incorporates the wind direction. Numerical examples are presented to compare various elements with the existing method such as the streamline diffusion method (SUPG).
2006, 5(2): 321-336 doi: 10.3934/cpaa.2006.5.321 +[Abstract](999) +[PDF](216.3KB)
Abstract:
In this work we are concerned about a second order nonlinear ordinary differential equation. Our main purpose is to describe one-parameter families of solutions of this equation which satisfy certain boundary conditions. These one-parameter families of solutions are obtained in the form of asymptotic or convergent series. The series expansions are then used to approximate the solutions of two boundary value problems. We are specially interested in the cases where these problems are degenerate with respect to the unknown function and/or to the independent variable. Lower and upper solutions for each of the considered boundary value problems are obtained and, in certain particular cases, a closed formula for the exact solution is derived. Numerical results are presented and discussed.
2006, 5(2): 337-347 doi: 10.3934/cpaa.2006.5.337 +[Abstract](911) +[PDF](195.7KB)
Abstract:
Applications of three-dimensional Galerkin boundary element methods require the numerical evaluation of many four-dimensional integrals. In this paper we explore the possibility of using extrapolation quadrature. To do so, one needs appropriate error functional expansions. The treatment here is limited to integration over a region $\mathcal T_1 \times \mathcal T_2$, where $\mathcal T_1$ and $\mathcal T_2$ are planar triangular elements in a hanging-chad configuration; that is, they have one vertex in common but are otherwise disjoint. We derive error expansions for product trapezoidal rules valid for integrands having an $|r_{12}|^{-1}$ factor. This factor gives rise to a weak singularity at the common vertex.
2006, 5(2): 349-365 doi: 10.3934/cpaa.2006.5.349 +[Abstract](909) +[PDF](220.0KB)
Abstract:
In this paper we describe and analyze the general framework of cascading multigrid preconditioning. In particular, we introduce two preconditioners based on the cascading multigrid approach. We then illustrate the application of the corresponding method to the heat equation with mild regularity in Besov spaces. Furthermore, we analyze a cascading multigrid preconditioner for the interior penalty discontinuous Galerkin method.
2006, 5(2): 367-382 doi: 10.3934/cpaa.2006.5.367 +[Abstract](983) +[PDF](505.5KB)
Abstract:
We consider numerical approximations to parameter-dependent linear and logistic stochastic delay differential equations with multiplicative noise. The aim of the investigation is to explore the parameter values at which there are changes in qualitative behaviour of the solutions. One may use a phenomenological approach but a more analytical approach would be attractive. A possible tool in this analysis is the calculation of the approximate local Lyapunov exponents. In this paper we show that the phenomenological approach can be used effectively to estimate bifurcation parameters for deterministic linear equations but one needs to use the dynamical approach for stochastic equations.
2006, 5(2): 383-393 doi: 10.3934/cpaa.2006.5.383 +[Abstract](940) +[PDF](4289.7KB)
Abstract:
Rayleigh-Bénard convection in a small rectangular domain is studied by the standard bifurcation analysis. The dynamics on the center manifold is calculated up to 3rd order. By this normal form, it is possible to determine the local bifurcation structures in principle. One can easily imagine that mixed mode solutions such as hexagonal, patchwork-quilt patterns are unstable from the knowledge of amplitude equation:Ginzburg-Landau equation. However they are not necessarily similar to each other in a small rectangular domain. Several non-trivial stable mixed mode patterns are introduced.
2006, 5(2): 395-413 doi: 10.3934/cpaa.2006.5.395 +[Abstract](1228) +[PDF](235.6KB)
Abstract:
We discuss a possibility to construct high order numerical algorithms on uniform or mildly graded grids for solving linear Fredholm integral equations of the second kind with weakly singular or other nonsmooth kernels. We first regularise the solution of the integral equation by introducing a suitable new independent variable and then solve the transformed equation by piecewise polynomial collocation and Galerkin methods on a mildly graded or uniform grid.

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