scholarly journals Solution to the Modified Helmholtz Equation for Arbitrary Periodic Charge Densities

2021 ◽  
Vol 8 ◽  
Author(s):  
Miriam Winkelmann ◽  
Edoardo Di Napoli ◽  
Daniel Wortmann ◽  
Stefan Blügel
Keyword(s):  
Helmholtz Equation ◽  
Charge Density ◽  
Poisson Equation ◽  
Yukawa Potential ◽  
Radial Solutions ◽  
Modified Helmholtz Equation ◽  
Poisson Solver ◽  
The Poisson Equation ◽  
Definition Of ◽  
Uniformly Convergent Fourier Series

We present a general method for solving the modified Helmholtz equation without shape approximation for an arbitrary periodic charge distribution, whose solution is known as the Yukawa potential or the screened Coulomb potential. The method is an extension of Weinert’s pseudo-charge method [Weinert M, J Math Phys, 1981, 22:2433–2439] for solving the Poisson equation for the same class of charge density distributions. The inherent differences between the Poisson and the modified Helmholtz equation are in their respective radial solutions. These are polynomial functions, for the Poisson equation, and modified spherical Bessel functions, for the modified Helmholtz equation. This leads to a definition of a modified pseudo-charge density and modified multipole moments. We have shown that Weinert’s convergence analysis of an absolutely and uniformly convergent Fourier series of the pseudo-charge density is transferred to the modified pseudo-charge density. We conclude by illustrating the algorithmic changes necessary to turn an available implementation of the Poisson solver into a solver for the modified Helmholtz equation.

scholarly journals Moving Heat Source Reconstruction from Cauchy Boundary Data: The Cartesian Coordinates Case

2011 ◽  
Vol 2011 ◽  
pp. 1-19 ◽  
Author(s):  
Nilson C. Roberty ◽  
Denis M. de Sousa ◽  
Marcelo L. S. Rainha
Keyword(s):  
Helmholtz Equation ◽  
Boundary Data ◽  
Representation Formula ◽  
Transient Heat Conduction ◽  
Thermal Source ◽  
Source Reconstruction ◽  
Modified Helmholtz Equation ◽  
Time Space ◽  
Definition Of ◽  
Characteristic Interval

We consider the problem of reconstruction of an unknown characteristic interval and block transient thermal source inside a domain. By exploring the definition of an Extended Dirichlet to Neumann map in the time space cylinder that has been introduced in Roberty and Rainha (2010a), we can treat the problem with methods similar to that used in the analysis of the stationary source reconstruction problem. Further, the finite differenceθ-scheme applied to the transient heat conduction equation leads to a model based on a sequence of modified Helmholtz equation solutions. For each modified Helmholtz equation the characteristic interval and parallelepiped source function may be reconstructed uniquely from the Cauchy boundary data. Using representation formula we establish reciprocity functional mapping functions that are solutions of the modified Helmholtz equation to their integral in the unknown characteristic support. Numerical experiment for capture of an interval and an rectangular parallelepiped characteristic source inside a cubic box domain from boundary data are presented in threedimensional and one-dimensional implementations. The problem of centroid determination is addressed and questions are discussed from an computational points of view.

Charge Based Current–Voltage Model for the Silicon on Insulator Junctionless Field-Effect Transistor

2020 ◽  
Vol 20 (8) ◽  
pp. 4920-4925
Author(s):  
Yongjin Jeong ◽  
In Man Kang ◽  
Seongjae Cho ◽  
Jisun Park ◽  
Hyungsoon Shin
Keyword(s):  
Charge Density ◽  
Poisson Equation ◽  
Oxide Thickness ◽  
Silicon On Insulator ◽  
Physical Conditions ◽  
The Poisson Equation ◽  
Proposed Model ◽  
Current Voltage ◽  
Gate Oxide Thickness ◽  
Channel Thickness

In this study, we propose an accurate and simple current–voltage model for an SOI-JLFET based on a solution of the Poisson equation. The model is divided into three regions: accumulation, accumulation–depletion, and depletion. The charge density in each region is calculated with the Poisson equation and region-specific boundary conditions, and then the current is obtained by integrating the charge density with consideration of the Vds effect. The proposed model, which was implemented in HSPICE using Verilog-A, was validated using TCAD simulation for various physical conditions such as SOI channel thickness, gate oxide thickness, and channel doping concentration type. According to simulation results by the error rate calculation, our model shows more than 90% accuracy.

scholarly journals Moving Heat Source Reconstruction from the Cauchy Boundary Data

2010 ◽  
Vol 2010 ◽  
pp. 1-22 ◽  
Author(s):  
Nilson C. Roberty ◽  
Marcelo L. S. Rainha
Keyword(s):  
Helmholtz Equation ◽  
Boundary Data ◽  
Representation Formula ◽  
Transient Heat Conduction ◽  
Thermal Source ◽  
Source Reconstruction ◽  
Modified Helmholtz Equation ◽  
Time Space ◽  
Definition Of ◽  
Transient Thermal

We consider the problem of reconstruction of an unknown characteristic transient thermal source inside a domain. By introducing the definition of an extended dirichlet-to-Neumann map in the time-space cylinder and the adoption of the anisotropic Sobolev-Hilbert spaces, we can treat the problem with methods similar to those used in the analysis of the stationary source reconstruction problem. Further, the finite differenceθscheme applied to the transient heat conduction equation leads to a model based on a sequence of modified Helmholtz equation solutions. For each modified Helmholtz equation the characteristic star-shape source function may be reconstructed uniquely from the Cauchy boundary data. Using representation formula, we establish reciprocity functional mapping functions that are solutions of the modified Helmholtz equation to their integral in the unknown characteristic support.

scholarly journals A Tikhonov-Type Regularization Method for Identifying the Unknown Source in the Modified Helmholtz Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Jinghuai Gao ◽  
Dehua Wang ◽  
Jigen Peng
Keyword(s):  
Helmholtz Equation ◽  
Regularization Method ◽  
Regularization Parameter ◽  
A Priori ◽  
Inverse Source Problem ◽  
A Posteriori ◽  
Modified Helmholtz Equation ◽  
Numerical Tests ◽  
Theoretical Frame ◽  
Set Up

An inverse source problem in the modified Helmholtz equation is considered. We give a Tikhonov-type regularization method and set up a theoretical frame to analyze the convergence of such method. A priori and a posteriori choice rules to find the regularization parameter are given. Numerical tests are presented to illustrate the effectiveness and stability of our proposed method.

Sign in / Sign up

Export Citation Format

Share Document