Remark on Algorithm 982: Explicit Solutions of Triangular Systems of First-order Linear Initial-value Ordinary Differential Equations with Constant Coefficients

Algorithm 982: Explicit solutions of triangular systems of first-order linear initial-value ordinary differential equations with constant coefficients provides an explicit solution for an homogeneous system, and a brief description of how to compute a solution for the inhomogeneous case. The method described is not directly useful if the coefficient matrix is singular. This remark explains more completely how to compute the solution for the inhomogeneous case and for the singular coefficient matrix case.

Physics-based control of neoclassical tearing modes on TCV

This paper presents recent progress on the studies of neoclassical tearing modes (NTMs) on TCV, concerning the new physics learned and how this physics contributes to a better real-time (RT) control of NTMs. A simple technique that adds a small (sinusoidal) sweeping to the target electron cyclotron (EC) beam deposition location has proven effective both for the stabilization and prevention of 2⁄1 NTMs. This relaxes the strict requirement on beam-mode alignment for NTM control, which is difficult to ensure in RT. In terms of the EC power for NTM stabilization, a control scheme making use of RT island width measurements has been tested on TCV. NTM seeding through sawtooth (ST) crashes or unstable current density profiles (triggerless NTMs) has been studied in detail. A new NTM prevention strategy utilizing only transient EC beams near the relevant rational surface has been developed and proven effective for preventing ST-seeded NTMs. With a comprehensive modified Rutherford equation (co-MRE) that considers the classical stability both at zero and finite island width, the prevention of triggerless NTMs with EC beams has been simulated for the first time. The prevention effects are found to result from the local effects of the EC beams (as opposed to global current profile changes), as observed in a group of TCV experiments scanning the deposition location of the preemptive EC beam. The co-MRE has also proven able to reproduce well the island width evolution in distinct plasma scenarios on TCV, ASDEX Upgrade and MAST, with very similar constant coefficients. The co-MRE has the potential of being applied in RT to provide valuable information such as the EC power required for NTM control with RT-adapted coefficients, contributing to both NTM control and integrated control with a limited set of actuators.

Effect of Doping different Percentages of Lithium Fluoride on Some Optical Characteristics of Poly-Methyl Methacrylate Polymer

This study implements the optical characteristics of Poly-Methyl methacrylate (PMMA) polymer before and after doping different percentages of Lithium Fluoride (LiF). Where the specimens were formulated as disk shape with diameter of (2.5 cm) and thickness of (0.148 cm) using Thermal pressing technology. The absorbance and reflectivity spectra were recorded in addition to their coefficients at range (300-1100) nm. Also, the study has included the determination of refraction and real and imaginary part of dielectric constant coefficients.

Unique continuation property of solutions to general second order elliptic systems

This paper concerns the weak unique continuation property of solutions of a general system of differential equation/inequality with a second order strongly elliptic system as its leading part. We put not only some natural assumptions which we call basic assumptions, but also some technical assumptions which we call further assumptions. It is shown as usual by first applying the Holmgren transform to this equation/inequality and then establishing a Carleman estimate for the leading part of the transformed inequality. The Carleman estimate is given via a partition of unity and the Carleman estimate for the operator with constant coefficients obtained by freezing the coefficients of the transformed leading part at a point. A little more details about this are as follows. Factorize this operator with constant coefficients into two first order differential operators. Conjugate each factor by a Carleman weight, and derive an estimate which is uniform with respect to the point at which we froze the coefficients for each conjugated factor by constructing a parametrix for its adjoint operator.

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

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.

Twisted kink dynamics in multiflavor chiral Gross-Neveu model

The Gross-Neveu model with UL(Nf)xUR(Nf) chiral symmetry is reconsidered in the large Nc limit. The known analytical solution for the time dependent interaction of any number of twisted kinks and breathers is cast into a more revealing form. The (x,t)-dependent factors are isolated from constant coefficients and twist matrices. These latter generalize the twist phases of the single flavor model. The crucial tool is an identity for the inverse of a sum of two square matrices, derived from the known formula for the determinant of such a sum.

Dynamics of exact closed-form solutions to the Schamel Burgers and Schamel equations with constant coefficients using a novel analytical approach

In this paper, we investigate two different constant-coefficient nonlinear evolution equations, namely the Schamel Burgers equation and the Schamel equation. These models also have a great deal of potential for studying ion-acoustic waves in plasma physics and fluid dynamics. The primary goal of this paper is to establish closed-form solutions and dynamics of analytical solutions to the Schamel Burgers and the Schamel equations, which are special examples of the well-known Schamel–Korteweg-de Vries (S-KdV) equation. We derive completely novel solutions to the considered models using a variety of computation programmes and a newly proposed extended generalized [Formula: see text] expansion approach. The newly formed solutions, which include hyperbolic and trigonometric functions as well as rational function solutions, have been produced. The annihilation of three-dimensional shock waves, periodic waves, single soliton, singular soliton, and combo soliton, multisoliton as well as their three-dimensional and contour plots are used to show the dynamical representations of the acquired solutions. These results demonstrate that the proposed extended technique is efficient, reliable and simple.

Numerical Simulation for Unsteady Anisotropic-Diffusion Convection Equation of Spatially Variable Coefficients and Incompressible Flow

The anisotropic-diffusion convection equation of spatiallyvariable coefficients which is relevant for functionally graded mediais discussed in this paper to find numerical solutions by using acombined Laplace transform and boundary element method. The variablecoefficients equation is transformed to a constant coefficients equation.The constant coefficients equation is then Laplace-transformed sothat the time variable vanishes. The Laplace-transformed equationis consequently written in a pure boundary integral equation whichinvolves a time-free fundamental solution. The boundary integral equationis therefore employed to find numerical solutions using a standardboundary element method. Finally the results obtained are inverselytransformed numerically using the Stehfest formula to get solutionsin the time variable. The combined Laplace transform and boundaryelement method is easy to be implemented, efficient and accurate forsolving unsteady problems of anisotropic functionally graded mediagoverned by the diffusion convection equation.

Linear PDE with constant coefficients

We discuss practical methods for computing the space of solutions to an arbitrary homogeneous linear system of partial differential equations with constant coefficients. These rest on the Fundamental Principle of Ehrenpreis–Palamodov from the 1960s. We develop this further using recent advances in computational commutative algebra.