scholarly journals On the Analytical and Numerical Solutions of the Linear Damped NLSE for Modeling Dissipative Freak Waves and Breathers in Nonlinear and Dispersive Mediums: An Application to a Pair-Ion Plasma

2021 ◽  
Vol 9 ◽  
Author(s):  
S. A. El-Tantawy ◽  
Alvaro H. Salas ◽  
M. R. Alharthi
Keyword(s):  
Analytical Solution ◽  
Numerical Solution ◽  
Numerical Solutions ◽  
Chebyshev Approximation ◽  
Rogue Waves ◽  
Approximation Technique ◽  
Damping Term ◽  
Analytical And Numerical Solutions ◽  
Ion Plasma ◽  
Finite Differences Method

In this work, two approaches are introduced to solve a linear damped nonlinear Schrödinger equation (NLSE) for modeling the dissipative rogue waves (DRWs) and dissipative breathers (DBs). The linear damped NLSE is considered a non-integrable differential equation. Thus, it does not support an explicit analytic solution until now, due to the presence of the linear damping term. Consequently, two accurate solutions will be derived and obtained in detail. The first solution is called a semi-analytical solution while the second is an approximate numerical solution. In the two solutions, the analytical solution of the standard NLSE (i.e., in the absence of the damping term) will be used as the initial solution to solve the linear damped NLSE. With respect to the approximate numerical solution, the moving boundary method (MBM) with the help of the finite differences method (FDM) will be devoted to achieve this purpose. The maximum residual (local and global) errors formula for the semi-analytical solution will be derived and obtained. The numerical values of both maximum residual local and global errors of the semi-analytical solution will be estimated using some physical data. Moreover, the error functions related to the local and global errors of the semi-analytical solution will be evaluated using the nonlinear polynomial based on the Chebyshev approximation technique. Furthermore, a comparison between the approximate analytical and numerical solutions will be carried out to check the accuracy of the two solutions. As a realistic application to some physical results; the obtained solutions will be used to investigate the characteristics of the dissipative rogue waves (DRWs) and dissipative breathers (DBs) in a collisional unmagnetized pair-ion plasma. Finally, this study helps us to interpret and understand the dynamic behavior of modulated structures in various plasma models, fluid mechanics, optical fiber, Bose-Einstein condensate, etc.

scholarly journals Analytical and Numerical Solution for a Rigid Liquid-Column Moving in a Pipe With Fluctuating Reservoir-Head and Venting Entrapped-Gas

2016 ◽  
Author(s):  
Arris S. Tijsseling ◽  
Qingzhi Hou ◽  
Zafer Bozkuş
Keyword(s):  
Numerical Solution ◽  
Numerical Solutions ◽  
Parameter Variation ◽  
Driving Pressure ◽  
Liquid Column ◽  
Analytical And Numerical Solutions ◽  
Trapped Gas

The motion of liquid filling a pipeline is impeded when the gas ahead of it cannot escape freely. Trapped gas will lead to a significant pressure build-up in front of the liquid column, which slows down the column and eventually bounces it back. This paper is an extension of previous work by the authors in the sense that the trapped gas can escape through a vent. Another addition is that the driving pressure is not kept constant but fluctuating. The obtained analytical and numerical solutions are utilized in parameter variation studies that give deeper insight in the system’s behavior.

NUMERICAL SOLUTION OF SINGLE MODE GYROTRON EQUATION

2005 ◽  
Vol 9 (1) ◽  
pp. 25-38 ◽  
Author(s):  
O. Dumbrajs ◽  
H. Kalis ◽  
A. Reinfelds
Keyword(s):  
Numerical Solution ◽  
Finite Differences ◽  
Numerical Solutions ◽  
Single Mode ◽  
Eigenvalues And Eigenfunctions ◽  
Analytical And Numerical Solutions ◽  
Quasistationary Solutions

In this paper we study numerical problems arising in solving the single mode gyrotron equation. Using the method of finite differences analytical and numerical solutions are obtained. Quasistationary solutions and corresponding eigenvalues and eigenfunctions of this problem are investigated.

Multiple interaction between a radiating dipole and parallel conducting cylinder

1987 ◽  
Vol 65 (2) ◽  
pp. 101-108
Author(s):  
H. A. Ragheb ◽  
M. Hamid
Keyword(s):  
Analytical Solution ◽  
Multiple Scattering ◽  
Radiation Field ◽  
Excellent Agreement ◽  
Numerical Solutions ◽  
Numerical Technique ◽  
Analytical And Numerical Solutions ◽  
Zone Field ◽  
Multiple Interaction ◽  
Approximate Procedure

The radiation from a long dipole in the presence of an infinitely long, circular conducting parallel cylinder is examined analytically. The expression for the far-zone field, which includes all successive multiple-scattering terms between the dipole and the cylinder, is derived using an iterative approximate procedure. The current induced on the dipole due to backscattering from the cylinder is calculated for each iteration. The derivation includes integral expressions, which are solved asymptotically using the far-zone field approximations. Also, a numerical technique is employed for solving the problem to verify the analytical solution. The results due to both analytical and numerical solutions are in excellent agreement. It is also found that the interaction between the dipole and the cylinder has a significant effect on the radiation field.

scholarly journals Approximate Solutions to the Forced Damped Parametric Driven Pendulum Oscillator: Chebyshev Collocation Numerical Solution

2021 ◽  
Author(s):  
Wedad Albalawi ◽  
Alvaro H. Salas ◽  
S. A. El-Tantawy
Keyword(s):  
Analytical Solution ◽  
Numerical Solution ◽  
Numerical Solutions ◽  
Classical Mechanics ◽  
Elliptic Functions ◽  
Approximate Solutions ◽  
Electronic Circuits ◽  
Chebyshev Collocation ◽  
Pendulum Equation ◽  
First Time

Abstract In this work, novel semi-analytical and numerical solutions to the forced damped driven nonlinear (FDDN) pendulum equation on the pivot vertically for arbitrary angles are obtained for the first time. The semi-analytical solution is derived in terms of the Jacobi elliptic functions with arbitrary elliptic modulus. For the numerical analysis, the Chebyshev collocation numerical method is introduced for analyzing tthe forced damped parametric driven pendulum equation. Moreover, the semi-analytical solution and Chebyshev collocation numerical solution are compared with the Runge-Kutta (RK) numerical solution. Also, the maximum distance error to the obtained approximate solutions is estimated with respect to the RK numerical solution. The obtained results help many authors to understand the mechanism of many phenomena related to the plasma physics, classical mechanics, quantum mechanics, optical fiber, and electronic circuits.

scholarly journals Corrigendum

1985 ◽  
Vol 34 (3) ◽  
pp. 481-483
Author(s):  
T. G. Forbes ◽  
E. R. Priest ◽  
A. W. Hood
Keyword(s):  
Analytical Solution ◽  
Numerical Solution ◽  
Current Sheet ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Linear Solution ◽  
Significant Change ◽  
Parameter Values ◽  
A Current

Numerical solutions were obtained by Forbes, Priest & Hood (1982) for the resistive decay of a current sheet in an MHD fluid. To check the accuracy of the numerical solutions, a linear, analytical solution was also deived for the regime where diffusion is dominant. In a subsequent reinvestigation of this problem an error in the linear, analytical solution has been discovered. For the parameter values used in the numerical solution this error is too small (≲ 2%) to produce any significant change in the previous test comparison between the numerical and analytical solutions. However, for parameter values much different from those used in the numerical solution, the error in the linear solution can be significant.

scholarly journals A fractional order epidemic model for the simulation of outbreaks of Ebola

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Weiqiu Pan ◽  
Tianzeng Li ◽  
Safdar Ali
Keyword(s):  
Numerical Solution ◽  
Root Mean Square ◽  
Fractional Order ◽  
Relative Error ◽  
Numerical Solutions ◽  
Real Data ◽  
Mean Square ◽  
Order Model ◽  
The Real ◽  
Fractional Orders

AbstractThe Ebola outbreak in 2014 caused many infections and deaths. Some literature works have proposed some models to study Ebola virus, such as SIR, SIS, SEIR, etc. It is proved that the fractional order model can describe epidemic dynamics better than the integer order model. In this paper, we propose a fractional order Ebola system and analyze the nonnegative solution, the basic reproduction number $R_{0}$ R 0 , and the stabilities of equilibrium points for the system firstly. In many studies, the numerical solutions of some models cannot fit very well with the real data. Thus, to show the dynamics of the Ebola epidemic, the Gorenflo–Mainardi–Moretti–Paradisi scheme (GMMP) is taken to get the numerical solution of the SEIR fractional order Ebola system and the modified grid approximation method (MGAM) is used to acquire the parameters of the SEIR fractional order Ebola system. We consider that the GMMP method may lead to absurd numerical solutions, so its stability and convergence are given. Then, the new fractional orders, parameters, and the root-mean-square relative error $g(U^{*})=0.4146$ g ( U ∗ ) = 0.4146 are obtained. With the new fractional orders and parameters, the numerical solution of the SEIR fractional order Ebola system is closer to the real data than those models in other literature works. Meanwhile, we find that most of the fractional order Ebola systems have the same order. Hence, the fractional order Ebola system with different orders using the Caputo derivatives is also studied. We also adopt the MGAM algorithm to obtain the new orders, parameters, and the root-mean-square relative error which is $g(U^{*})=0.2744$ g ( U ∗ ) = 0.2744 . With the new parameters and orders, the fractional order Ebola systems with different orders fit very well with the real data.

The physical problem of neutron slowing down: analytical and numerical solutions in finite media

1992 ◽  
Vol 13 (1) ◽  
pp. 38-46 ◽  
Author(s):  
V Colombo ◽  
G G M Coppa ◽  
S E Corno ◽  
P Ravetto
Keyword(s):  
Numerical Solutions ◽  
Physical Problem ◽  
Slowing Down ◽  
Analytical And Numerical Solutions

Analytical and numerical solutions of elastoplastic problems for spherical shells with circular cuts

1993 ◽  
Vol 29 (1) ◽  
pp. 22-27 ◽  
Author(s):  
I. A. Guseinov ◽  
R. Yu. Kerimov ◽  
I. S. Chernyshenko
Keyword(s):  
Numerical Solutions ◽  
Spherical Shells ◽  
Analytical And Numerical Solutions
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