scholarly journals Comparison of Iteration Schemes for the Solution of the Multidimensional Schrödinger-Poisson Equations

VLSI Design ◽  
1998 ◽  
Vol 8 (1-4) ◽  
pp. 105-109 ◽  
Author(s):  
A. Trellakis ◽  
A. T. Galick ◽  
A. Pacelli ◽  
U. Ravaioli
Keyword(s):  
Coupled System ◽  
Iteration Scheme ◽  
Simple Expression ◽  
Software Implementation ◽  
Quantum Structures ◽  
System Of Differential Equations ◽  
Poisson Equations ◽  
Quantum Electron ◽  
Predictor Corrector ◽  
Bound Electron

We present a fast and robust iterative method for obtaining self-consistent solutions to the coupled system of Schrödinger's and Poisson's equations in quantum structures. A simple expression describing the dependence of the quantum electron density on the electrostatic potential is used to implement a predictor – corrector type iteration scheme for the solution of the coupled system of differential equations. This approach simplifies the software implementation of the nonlinear problem, and provides excellent convergence speed and stability. We demonstrate the algorithm by presenting an example for the calculation ofthe two-dimensional bound electron states within the cross-section of a GaAs-AlGaAs based quantum wire. For this example, six times fewer iterations are needed when our predictor – corrector approach is applied, compared to a corresponding underrelaxation algorithm.

Iteration scheme for the solution of the two-dimensional Schrödinger-Poisson equations in quantum structures

1997 ◽  
Vol 81 (12) ◽  
pp. 7880-7884 ◽  
Author(s):  
A. Trellakis ◽  
A. T. Galick ◽  
A. Pacelli ◽  
U. Ravaioli
Keyword(s):  
Iteration Scheme ◽  
Quantum Structures ◽  
Two Dimensional ◽  
Poisson Equations

scholarly journals Analysis of a Coupled System of Nonlinear Fractional Langevin Equations with Certain Nonlocal and Nonseparated Boundary Conditions

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Zaid Laadjal ◽  
Qasem M. Al-Mdallal ◽  
Fahd Jarad
Keyword(s):  
Boundary Conditions ◽  
Coupled System ◽  
Fractional Integrals ◽  
Langevin Equations ◽  
Uniqueness Of Solutions ◽  
Caputo Fractional Derivatives ◽  
Nonseparated Boundary Conditions ◽  
New Type ◽  
Predictor Corrector ◽  
Derivatives Of

In this article, we use some fixed point theorems to discuss the existence and uniqueness of solutions to a coupled system of a nonlinear Langevin differential equation which involves Caputo fractional derivatives of different orders and is governed by new type of nonlocal and nonseparated boundary conditions consisting of fractional integrals and derivatives. The considered boundary conditions are totally dissimilar than the ones already handled in the literature. Additionally, we modify the Adams-type predictor-corrector method by implicitly implementing the Gauss–Seidel method in order to solve some specific particular cases of the system.

Numerical Solution of the Equations Governing the Steady State of a Thin Cylindrical Web Supported by an Air Cushion

1999 ◽  
Author(s):  
Sinan Müftü
Keyword(s):  
Steady State ◽  
Nonlinear Partial Differential Equations ◽  
Coupled System ◽  
Steady State Solution ◽  
Iteration Scheme ◽  
Two Dimensions ◽  
Navier Stokes ◽  
Air Cushion ◽  
Raphson Method ◽  
The Web

Abstract The numerical method used to solve the coupled nonlinear, partial differential equations (PDEs) representing the interaction between a thin, flexible, cylindrical web and an air cushion at steady state is analyzed. The web deflections are modeled by a cylindrical shell theory that allows moderately large deflections. The airflow is modeled in two-dimensions with a modified form of the Navier-Stokes and mass balance equations that have non-linear source terms. The coupled fluid/structure system is solved numerically in a stacked iteration scheme: The fluid equations are solved using pseudo-compressibility method with artificial viscosity; and the web equations are solved with a modified Newton-Raphson method. The convergence characteristics of the coupled system and the effects of the numerical parameters on the steady state solution are studied by numerical experiments.

scholarly journals Existence Results for a Nonlocal Coupled System of Differential Equations Involving Mixed Right and Left Fractional Derivatives and Integrals

Symmetry ◽  
2020 ◽  
Vol 12 (4) ◽  
pp. 578
Author(s):  
Sotiris K. Ntouyas ◽  
Abrar Broom ◽  
Ahmed Alsaedi ◽  
Tareq Saeed ◽  
Bashir Ahmad
Keyword(s):  
Existence And Uniqueness ◽  
Fractional Derivatives ◽  
Fixed Point Theorems ◽  
Coupled System ◽  
Uniqueness Result ◽  
Existence Results ◽  
System Of Differential Equations ◽  
Liouville Type ◽  
Uniqueness Of Solutions ◽  
Fractional Derivatives And Integrals

In this paper, we study the existence and uniqueness of solutions for a new kind of nonlocal four-point fractional integro-differential system involving both left Caputo and right Riemann–Liouville fractional derivatives, and Riemann–Liouville type mixed integrals. The Banach and Schaefer fixed point theorems are used to obtain the desired results. An example illustrating the existence and uniqueness result is presented.

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