The correction terms in approximate mean value analysis

This paper shows that the correction term in Schweitzer's approximate mean value analysis of closed queueing network satisfies three properties: invariance under class-splitting, no self-queueing, local dependence. In addition, it is uniquely specified by these three properties.     

The existence of approximate solutions for two-dimensional potential flow problems

A proof is given of the existence of an approximate Complex Variable Boundary Element Method solution for a Birichlet problem. This constructive proof can be used as a basis for numerical calculations. © 1996 John Wiley & Sons, Inc.     

A note on Euler approximations for SDEs with Hölder continuous diffusion coefficients

We provide a rate for the strong convergence of Euler approximations for stochastic differential equations (SDEs) whose diffusion coefficient is not Lipschitz but only (1/2+α)-Hölder continuous for some α≥0.     

A direct numerical method for approximate solution of inverse reaction diffusion equation via two-dimensional Legendre hybrid functions

Numerical Algorithms - In this paper, we propose an efficient numerical method based on two-dimensional hybrid of block-pulse functions and Legendre polynomials for numerically solving an inverse...     

Finite element approximation of diffusion equations with convolution terms

Approximation of solutions to diffusion equations with memory represented by convolution integral terms is considered. Such problems arise from modeling of flows in fissured media. Convergence of the method is proved and results of numerical experiments confirming the theoretical results are presented. The advantages of implementation of the algorithm in a multiprocessing environment are discussed.

     

On two-dimensional magnetohydrodynamic equations with fractional diffusion

We consider the existence of the smooth solution for two-dimensional magnetohydrodynamic equations with fractional diffusion in the fluid equations. In particular, we prove the global-in-time existence of fractional diffusion with power greater than 1/2.     

A two-dimensional additive problem with an increasing number of terms

In this paper there is established an asymptotic formula for the number of simultaneous representations of two numbers as sums of an increasing number of terms involving a power function, i.e., an asymptotic (as n→∞) formula is found for the number of solutions in integers x_i, 0 ≤ x_i ≤ p, of the following system of diophantine equations: $$\left\{ {\begin{array}{*{20}c} {x_1 + x_2 + \ldots x_n = N_{1,} } \\ {x_1^2 + x_{_2 }^{_2 } + \ldots x_n = N_{2.} } \\ \end{array} } \right.$$ . The analysis is carried out as in the proof of a local limit theorem of probability theory and involves estimates of Weyl sums.     

Parallel implementation of a multiblock method with approximate subdomain solution

Solution of large linear systems encountered in computational fluid dynamics often naturally leads to some form of domain decomposition, especially when it is desired to use parallel machines. It has been proposed to use approximate solvers to obtain fast but rough solutions on the separate subdomains. In this paper approximate solutions via (1) an inner preconditioned GMRES iteration to fixed tolerance, and (2) incomplete factorization (RILU, restricted to the diagonal) are considered. Numerical experiments for a fundamental test problem are included which show speedups obtained on a cluster of workstations as well as on a distributed memory parallel computer. Additionally, the parallel implementation of GCR is addressed, with particular focus on communication costs associated with orthogonalization processes. This consideration brings up questions concerning the use of Householder reflections with GCR.     

An approximate solution of two-dimensional convex piston problem

A new theory of shock dynamics (NTSD) has been used to study the propagation of curved shocks originating from the motion of two-dimensional convex piston of various shapes. The results have been compared with those obtained by Whitham's classical shock dynamics and by TVD version of MacCormack's finite difference scheme.     

Elliptic operators with unbounded diffusion,drift and potential terms

We prove that the realization $A_p$ in $L^p({\mathbb{R}}^N),1<p<\infty$, of the elliptic operator $A=(1+|x{|}^{\alpha })\mathrm{\Delta }+b|x{|}^{\alpha ?1}\frac{x}{|x|}?\mathrm{?}?c|x{|}^{\beta }$ with domain $D(A_p)=\{u\in W^{2,p}({\mathbb{R}}^N)|Au\in L^p({\mathbb{R}}^N)\}$ generates a strongly continuous analytic semigroup $T(?)$ provided that $\alpha >2,\beta >\alpha ?2$ and any constants $b\in \mathbb{R}$ and $c>0$. This generalizes the recent results in [4] and in [16]. Moreover we show that $T(?)$ is consistent, immediately compact and ultracontractive.     

Global solutions of the fast diffusion equation with gradient absorption terms

We investigate the existence of nonnegative weak solutions to the problem ut=Δ(um)−p|∇u| in Rn×(0,∞) with ⁺(1−2/n)<m<1. It will be proved that: (i) When 1<p<2, if the initial datum u₀∈D(Rn) then there exists a solution; (ii) When 1<p<(2+mn)/(n+1), if the initial datum u₀(x) is a bounded and nonnegative measure then the solution exists; (iii) When (2+mn)/(n+1)?p<2, if the initial datum is a Dirac mass then the solution does not exist. We also study the large time behavior of the L¹-norm of solutions for 1<p?(2+mn)/(n+1), and the large time behavior of t1/β‖u(⋅,t)−Ec(⋅,t)L^∞_‖ for (2+mn)/(n+1)<p<2.     

Global dynamics of approximate solutions to an age-structured epidemic model with diffusion

We consider an age-dependent s-i-s epidemic model with diffusion whose mortality is unbounded. We approximate the solution using Galerkin methods in the space variable combined with backward Euler along the characteristic direction in the age and time variables. It is proven that the scheme is stable and convergent in optimal rate in l ^∞,2 (L ²) norm. To investigate the global behavior of the discrete solution resulting from the algorithm, we reformulate the resulting system into a monotone form. Positivity of the nonlocal birth process is proved using the positivity of the first eigenvalue of the resulting matrix system and using the fact that the positivity is preserved along the characteristics. The difference equation of the steady state coupled with nonlocal birth process is solved by developing monotone iterative schemes. The stability of the discrete solution of the steady state is then analyzed by constructing suitable positive subsolutions. Mathematics subject classifications (2000) 65M12, 65M25, 65M60, 92D25 M.-Y. Kim: This work was supported by Korea Research Foundation Grant (KRF-2001-041-D00037).     

The drift–diffusion system in two-dimensional critical Hardy space

We establish the time local well-posedness for large data of a solution to two-dimensional drift–diffusion system in the critical Hardy space . This solution is obtained in the Hardy class, for which the natural free energy is well defined from the initial time. The time global well-posedness result for small data is also obtained in a similar class. The crucial parts of the proof are to employ the end-point type of maximal regularity for the homogeneous heat equation and some new bilinear estimates in the Hardy space.