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Neumann Problem (neumann + problem)

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Mathematics and Statistics100%

Selected Abstracts

Stability of global and exponential attractors for a three-dimensional conserved phase-field system with memory

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 18 2009
Gianluca Mola
Abstract We consider a conserved phase-field system on a tri-dimensional bounded domain. The heat conduction is characterized by memory effects depending on the past history of the (relative) temperature ,, which is represented through a convolution integral whose relaxation kernel k is a summable and decreasing function. Therefore, the system consists of a linear integrodifferential equation for ,, which is coupled with a viscous Cahn,Hilliard type equation governing the order parameter ,. The latter equation contains a nonmonotone nonlinearity , and the viscosity effects are taken into account by a term ,,,,t,, for some ,,0. Rescaling the kernel k with a relaxation time ,>0, we formulate a Cauchy,Neumann problem depending on , and ,. Assuming a suitable decay of k, we prove the existence of a family of exponential attractors {,,,,} for our problem, whose basin of attraction can be extended to the whole phase,space in the viscous case (i.e. when ,>0). Moreover, we prove that the symmetric Hausdorff distance of ,,,, from a proper lifting of ,,,0 tends to 0 in an explicitly controlled way, for any fixed ,,0. In addition, the upper semicontinuity of the family of global attractors {,,,,,} as ,,0 is achieved for any fixed ,>0. Copyright © 2009 John Wiley & Sons, Ltd. [source]

A new integral equation approach to the Neumann problem in acoustic scattering

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 16 2001
P. A. Krutitskii
We suggest a new approach of reduction of the Neumann problem in acoustic scattering to a uniquely solvable Fredholm integral equation of the second kind with weakly singular kernel. To derive this equation we placed an additional boundary with an appropriate boundary condition inside the scatterer. The solution of the problem is obtained in the form of a single layer potential on the whole boundary. The density in the potential satisfies a uniquely solvable Fredholm integral equation of the second kind and can be computed by standard codes. Copyright © 2001 John Wiley & Sons, Ltd. [source]

Iterated Neumann problem for the higher order Poisson equation

MATHEMATISCHE NACHRICHTEN, Issue 1-2 2006
H. Begehr
Abstract Rewriting the higher order Poisson equation ,nu = f in a plane domain as a system of Poisson equations it is immediately clear what boundary conditions may be prescribed in order to get (unique) solutions. Neumann conditions for the Poisson equation lead to higher-order Neumann (Neumann- n ) problems for ,nu = f . Extending the concept of Neumann functions for the Laplacian to Neumann functions for powers of the Laplacian leads to an explicit representation of the solution to the Neumann- n problem for ,nu = f . The representation formula provides the tool to treat more general partial differential equations with leading term ,nu in reducing them into some singular integral equations. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

On the number of interior peak solutions for a singularly perturbed Neumann problem

COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 2 2007
Fang-Hua Lin
We consider the following singularly perturbed Neumann problem: where , = , ,2/,x is the Laplace operator, , > 0 is a constant, , is a bounded, smooth domain in ,N with its unit outward normal ,, and f is superlinear and subcritical. A typical f is f(u) = up where 1 < p < +, when N = 2 and 1 < p < (N + 2)/(N , 2) when N , 3. We show that there exists an ,0 > 0 such that for 0 < , < ,0 and for each integer K bounded by where ,N, ,, f is a constant depending on N, ,, and f only, there exists a solution with K interior peaks. (An explicit formula for ,N, ,, f is also given.) As a consequence, we obtain that for , sufficiently small, there exists at least [,N, ,f/,N (|ln ,|)N] number of solutions. Moreover, for each m , (0, N) there exist solutions with energies in the order of ,N,m. © 2006 Wiley Periodicals, Inc. [source]

Fast direct solver for Poisson equation in a 2D elliptical domain

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 1 2004
Ming-Chih Lai
Abstract In this article, we extend our previous work M.-C. Lai and W.-C. Wang, Numer Methods Partial Differential Eq 18:56,68, 2002 for developing some fast Poisson solvers on 2D polar and spherical geometries to an elliptical domain. Instead of solving the equation in an irregular Cartesian geometry, we formulate the equation in elliptical coordinates. The solver relies on representing the solution as a truncated Fourier series, then solving the differential equations of Fourier coefficients by finite difference discretizations. Using a grid by shifting half mesh away from the pole and incorporating the derived numerical boundary value, the difficulty of coordinate singularity can be elevated easily. Unlike the case of 2D disk domain, the present difference equation for each Fourier mode is coupled with its conjugate mode through the numerical boundary value near the pole; thus, those two modes are solved simultaneously. Both second- and fourth-order accurate schemes for Dirichlet and Neumann problems are presented. In particular, the fourth-order accuracy can be achieved by a three-point compact stencil which is in contrast to a five-point long stencil for the disk case. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 72,81, 2004 [source]