MODIFIED METHOD OF SEPARATION OF VARIABLES FOR SOLVING DIFFRACTION PROBLEMS ON MULTILAYER DIELECTRIC GRATINGS

2021 ◽  
Vol 9 (4) ◽  
pp. 76-88
Author(s):  
Yu. G. Smirnov ◽  
◽  
V. Yu. Martynova ◽  
M. A. Moskaleva ◽  
A. V. Tikhonravov ◽  
...  
Keyword(s):  
Direct Problem ◽  
Order Differential Equation ◽  
Separation Of Variables ◽  
Grating Period ◽  
Piecewise Constant ◽  
One Dimensional ◽  
Modified Method ◽  
Dielectric Gratings ◽  
Constant Coefficients ◽  
Multilayer Dielectric

A modified method of separation of variables is proposed for solving the direct problem of diffraction of electromagnetic wave by multilayer dielectric gratings (MDG). To apply this method, it is necessary to solve a one-dimensional eigenvalue problem for a 2nd- order differential equation on a segment with piecewise constant coefficients. The accuracy of the method is verified by comparison with the results obtained by the commercially available RCWA method. It is demonstrated that the method can be applied not only to commonly used MDG elements with one line in a grating period but also to potentially promising MDG elements with several different lines in a grating period.

scholarly journals Immersed Interface CIP for One Dimensional Hyperbolic Equations

2014 ◽  
Vol 16 (1) ◽  
pp. 96-114
Author(s):  
Kazufumi Ito ◽  
Tomoya Takeuchi
Keyword(s):  
Hyperbolic Equations ◽  
Third Order ◽  
Immersed Interface ◽  
Order Of Accuracy ◽  
Piecewise Constant ◽  
One Dimensional ◽  
The Third ◽  
Numerical Tests ◽  
Jump Discontinuities ◽  
Constant Coefficients

AbstractThe immersed interface technique is incorporated into CIP method to solve one-dimensional hyperbolic equations with piecewise constant coefficients. The proposed method achieves the third order of accuracy in time and space in the vicinity of the interface where the coefficients have jump discontinuities, which is the same order of accuracy of the standard CIP scheme. Some numerical tests are given to verify the accuracy of the proposed method.

scholarly journals Fractional stochastic heat equation with piecewise constant coefficients

2020 ◽  
Vol 21 (01) ◽  
pp. 2150002
Author(s):  
Yuliya Mishura ◽  
Kostiantyn Ralchenko ◽  
Mounir Zili ◽  
Eya Zougar
Keyword(s):  
Diffusion Coefficient ◽  
Heat Equation ◽  
Divergence Form ◽  
Stochastic Heat Equation ◽  
Piecewise Constant ◽  
Constant Diffusion Coefficient ◽  
Infinite Dimensional ◽  
Order Elliptic Operator ◽  
Constant Diffusion ◽  
Constant Coefficients

We introduce a fractional stochastic heat equation with second-order elliptic operator in divergence form, having a piecewise constant diffusion coefficient, and driven by an infinite-dimensional fractional Brownian motion. We characterize the fundamental solution of its deterministic part, and prove the existence and the uniqueness of its solution.

scholarly journals A weak convergence approach to hybrid LQG problems with infinite control weights

2002 ◽  
Vol 15 (1) ◽  
pp. 1-21
Author(s):  
G. George Yin ◽  
Jiongmin Yong
Keyword(s):  
Markov Chain ◽  
Weak Convergence ◽  
Small Parameter ◽  
Limit Problem ◽  
Discrete Events ◽  
Linear Quadratic ◽  
Piecewise Constant ◽  
Constant Coefficients ◽  
The Cost ◽  
Random Times

This work is concerned with a class of hybrid LQG (linear quadratic Gaussian) regulator problems modulated by continuous-time Markov chains. In contrast to the traditional LQG models, the systems have both continuous dynamics and discrete events. In lieu of a model with constant coefficients, these coefficients vary with time and exhibit piecewise constant behavior. At any time t, the system follows a stochastic differential equation in which the coefficients take one of the m possible configurations where m is usually large. The system may jump to any of the possible configurations at random times. Further, the control weight in the cost functional is allowed to be indefinite. To reduce the complexity, the Markov chain is formulated as singularly perturbed with a small parameter. Our effort is devoted to solving the limit problem when the small parameter tends to zero via the framework of weak convergence. Although the limit system is still modulated by a Markov chain, it has a much smaller state space and thus, much reduced complexity.

scholarly journals Explicit Numerical Solution of High - Dimensional Advection - Diffusion

2013 ◽  
Vol 2 (3) ◽  
pp. 17-22
Author(s):  
Elham Bayatmanesh
Keyword(s):  
High Dimensional ◽  
Numerical Techniques ◽  
Science And Engineering ◽  
Numerical Examples ◽  
One Dimensional ◽  
Advection Diffusion Equation ◽  
Advection Diffusion ◽  
Constant Coefficients ◽  
The Subject ◽  
The One

The Several numerical techniques have been developed and compared for solving the one-dimensional and three-dimentional advection-diffusion equation with constant coefficients. the subject has played very important roles to fluid dynamics as well as many other field of science and engineering. In this article, we will be presenting the of n-dimentional and we neglect the numerical examples.

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