اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدOn the method of reflections
172
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
This paper aims at reviewing and analysing the method of reflections. The latter is an iterative procedure designed to linear boundary value problems set in multiply connected domains. Being based on a decomposition of the domain boundary, this method is particularly well-suited to numerical solvers relying on integral representation formulas. For the parallel and sequential forms of the method appearing in the literature, we propose a general abstract formulation in a given Hilbert setting and interpret the procedure in terms of subspace corrections. We then prove the unconditional convergence of the sequential form and propose a modification of the parallel one that makes it unconditionally converging. An alternative proof of convergence is provided in a case which does not fit into the previous framework. We finally present some numerical tests.
قيم البحث
اقرأ أيضاً
The theory of specular X-ray reflectivity from a rough interface based upon the reflection function method (RFM) is proposed. The RFM transforms the second order differential equation for the wave amplitude into the non-linear first order differentia
l equation of Riccati type for the reflection function. This equation is solved in the approximation of the abruptly changing potential, which is justified for the typical angles of X-ray reflectometry. The reflectivity is represented as a series. The first term of this series reproduces the Nevot-Croce approximation and second one gives the phase correction for greater angles. It is shown that the phase correction can be used to obtain the degree of interface asymmetry. The X-ray reflectometry model profiles for Fe/Cr superlattice are used to illustrate the method.
In this paper, we first prove that the cubic, defocusing nonlinear Schrodinger equation on the two dimensional hyperbolic space with radial initial data in $H^s(mathbb{H}^2)$ is globally well-posed and scatters when $s > frac{3}{4}$. Then we extend t
he result to nonlineraities of order $p>3$. The result is proved by extending the high-low method of Bourgain in the hyperbolic setting and by using a Morawetz type estimate proved by the first author and Ionescu.
We use modular invariance and crossing symmetry of conformal field theory to reveal approximate reflection symmetries in the spectral decompositions of the partition function in two dimensions in the limit of large central charge and of the four-poin
t function in any dimension in the limit of large scaling dimensions $Delta_0$ of external operators. We use these symmetries to motivate universal upper bounds on the spectrum and the operator product expansion coefficients, which we then derive by independent techniques. Some of the bounds for four-point functions are valid for finite $Delta_0$ as well as for large $Delta_0$. We discuss a similar symmetry in a large spacetime dimension limit. Finally, we comment on the analogue of the Cardy formula and sparse light spectrum condition for the four-point function.
Boundary value problems for integrable nonlinear evolution PDEs formulated on the finite interval can be analyzed by the unified method introduced by one of the authors and used extensively in the literature. The implementation of this general method
to this particular class of problems yields the solution in terms of the unique solution of a matrix Riemann-Hilbert problem formulated in the complex $k$-plane (the Fourier plane), which has a jump matrix with explicit $(x,t)$-dependence involving six scalar functions of $k$, called spectral functions. Two of these functions depend on the initial data, whereas the other four depend on all boundary values. The most difficult step of the new method is the characterization of the latter four spectral functions in terms of the given initial and boundary data, i.e. the elimination of the unknown boundary values. Here, we present an effective characterization of the spectral functions in terms of the given initial and boundary data. We present two different characterizations of this problem. One is based on the analysis of the so-called global relation, on the analysis of the equations obtained from the global relation via certain transformations leaving the dispersion relation of the associated linearized PDE invariant, and on the computation of the large $k$ asymptotics of the eigenfunctions defining the relevant spectral functions. The other is based on the analysis of the global relation and on the introduction of the so-called Gelfand-Levitan-Marchenko representations of the eigenfunctions defining the relevant spectral functions. We also show that these two different characterizations are equivalent and that in the limit when the length of the interval tends to infinity, the relevant formulas reduce to the analogous formulas obtained recently for the case of boundary value problems formulated on the half-line.
Boundary value problems for integrable nonlinear evolution PDEs formulated on the half-line can be analyzed by the unified method introduced by one of the authors and used extensively in the literature. The implementation of this general method to th
is particular class of problems yields the solution in terms of the unique solution of a matrix Riemann-Hilbert problem formulated in the complex $k$-plane (the Fourier plane), which has a jump matrix with explicit $(x,t)$-dependence involving four scalar functions of $k$, called spectral functions. Two of these functions depend on the initial data, whereas the other two depend on all boundary values. The most difficult step of the new method is the characterization of the latter two spectral functions in terms of the given initial and boundary data, i.e. the elimination of the unknown boundary values. For certain boundary conditions, called linearizable, this can be achieved simply using algebraic manipulations. Here, we present an effective characterization of the spectral functions in terms of the given initial and boundary data for the general case of non-linearizable boundary conditions. This characterization is based on the analysis of the so-called global relation, on the analysis of the equations obtained from the global relation via certain transformations leaving the dispersion relation of the associated linearized PDE invariant, and on the computation of the large $k$ asymptotics of the eigenfunctions defining the relevant spectral functions.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
