Symmetric Arc Solutions of ζ¨ = ζn

1968 ◽  
Vol 35 (3) ◽  
pp. 565-570
Author(s):  
C. P. Atkinson ◽  
B. L. Dhoopar
Keyword(s):  
Differential Equation ◽  
Time Domain ◽  
Complex Variable ◽  
Digital Computer ◽  
Nonlinear Differential Equations ◽  
Time Derivative ◽  
Approximate Solutions ◽  
Complex Variables ◽  
Complex Modes ◽  
The Time Domain

This paper, “Symmetric Arc Solutions of ζ¨ = ζn,” presents periodic solutions of this differential equation relating the complex variable ζ(t) = u(t) + iv(t) and its second time derivative ζ¨ The solutions are called symmetric arc solutions since they form such arcs on the ζ = u + iv-plane. The solutions, ζ(t), are “complex modes” of coupled nonlinear differential equations in the complex variables z1 and z2. Symmetric arc solutions are presented for a range of n from n = 3 to n = 101. Approximate solutions are presented and compared with solutions generated by digital computer. Solutions are presented on the ζ-plane and in the time domain as u(t) and v(t).

scholarly journals A New Approach to Approximate Solutions for Nonlinear Differential Equation

2018 ◽  
Vol 2018 ◽  
pp. 1-8
Author(s):  
Safia Meftah
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Small Parameter ◽  
Perturbation Method ◽  
Periodic Solutions ◽  
Asymptotic Expansions ◽  
Nonlinear Differential Equations ◽  
Approximate Solutions ◽  
Approximation Methods ◽  
New Approach

The question discussed in this study concerns one of the most helpful approximation methods, namely, the expansion of a solution of a differential equation in a series in powers of a small parameter. We used the Lindstedt-Poincaré perturbation method to construct a solution closer to uniformly valid asymptotic expansions for periodic solutions of second-order nonlinear differential equations.

Analytical approximate solutions of the time-domain diffusion equation in layered slabs

2002 ◽  
Vol 19 (1) ◽  
pp. 71 ◽  
Author(s):  
Fabrizio Martelli ◽  
Angelo Sassaroli ◽  
Yukio Yamada ◽  
Giovanni Zaccanti
Keyword(s):  
Diffusion Equation ◽  
Time Domain ◽  
Approximate Solutions ◽  
The Time Domain

Uncertain Circuit Equation

2021 ◽  
Vol 14 (03) ◽  
Author(s):  
Yang Liu
Keyword(s):  
Differential Equation ◽  
Method Of Moments ◽  
Time Domain ◽  
Electrical Circuit ◽  
Unknown Parameters ◽  
Uncertainty Distribution ◽  
Liu Process ◽  
The Time Domain ◽  
Steady State Solutions ◽  
Transient And Steady State

Differential equation is a powerful tool for investigating the transient and steady-state solutions of electrical circuit in the time domain. By considering the noise in actual circuit system, this paper first presents an uncertain circuit equation, which is a type of differential equation driven by Liu process. Then the solution of uncertain circuit equation and the inverse uncertainty distribution of solution are derived. Following that, two applications of solution are provided as well. Based on the observations, the method of moments is used to estimate the unknown parameters in uncertain circuit equation. In addition, a paradox for stochastic circuit equation is also given.

Nonlinear Response of a Sandwich Plate Subjected to Blast Load

2007 ◽  
Author(s):  
Ali Bas¸ ◽  
Zafer Kazancı ◽  
Zahit Mecitog˘lu
Keyword(s):  
Sandwich Plate ◽  
Virtual Work ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Approximate Solutions ◽  
Blast Load ◽  
Thin Plates ◽  
Inertia Effects ◽  
The Time Domain ◽  
The Galerkin Method

Present work includes in-plane stiffness and inertia effects on the motion of a sandwich plate under blast load. The geometric nonlinearity effects are taken into account with the von Ka´rma´n large deflection theory of thin plates. All edges clamped boundary conditions are considered in the analyses. The equations of motion for the plate are derived by the use of the virtual work principle. Approximate solutions are assumed for the space domain and substituted into the equations of motion. Then the Galerkin Method is used to obtain the nonlinear differential equations in the time domain. The finite difference method is applied to solve the system of coupled nonlinear equations. The results of theoretical analyses are obtained.

Dynamics of Infinite Beams Described by Fractional Time Derivatives

2003 ◽  
Author(s):  
Peter Ruge ◽  
Carolin Trinks
Keyword(s):  
Frequency Domain ◽  
Time Domain ◽  
Dynamic Stiffness ◽  
Time Derivative ◽  
Low Frequency ◽  
Internal Variables ◽  
Closed Form Solutions ◽  
Rational Polynomial ◽  
The Time Domain ◽  
Fractional Time Derivative

Closed-form solutions of infinite Bernoulli-Euler beams on a viscoelastic foundation are available for harmonic excitations with frequency Ω. For more general time-dependent loadings and beam-systems with local perturbations, for example caused by non-linear effects an overall treatment of the system in the time-domain is highly appropriated. Here the analytical dynamic stiffness of the infinite beam in the frequency-domain is approximated by a rational polynomial in the low frequency-domain and by an irrational part representing the asymptotic behaviour for Ω tending towards infinity. Thus, the corresponding description in the time-domain contains a fractional time derivative part and additonal internal variables due to splitting the rational polynomial into a linear system with respect to Ω.

scholarly journals The Novel Integral Homotopy Expansive Method

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1204
Author(s):  
Uriel Filobello-Nino ◽  
Hector Vazquez-Leal ◽  
Jesus Huerta-Chua ◽  
Jaime Ramirez-Angulo ◽  
Darwin Mayorga-Cruz ◽  
...  
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Initial Conditions ◽  
Nonlinear Differential Equations ◽  
Approximate Solutions ◽  
The Novel ◽  
Practical Applications ◽  
Linear And Nonlinear ◽  
Scientific Phenomena ◽  
Analytical Expressions

This work proposes the Integral Homotopy Expansive Method (IHEM) in order to find both analytical approximate and exact solutions for linear and nonlinear differential equations. The proposal consists of providing a versatile method able to provide analytical expressions that adequately describe the scientific phenomena considered. In this analysis, it is observed that the proposed solutions are compact and easy to evaluate, which is ideal for practical applications. The method expresses a differential equation as an integral equation and expresses the integrand of the equation in terms of a homotopy. As a matter of fact, IHEM will take advantage of the homotopy flexibility in order to introduce adjusting parameters and convenient functions with the purpose of acquiring better results. In a sequence, another advantage of IHEM is the chance to distribute one or more of the initial conditions in the different iterations of the proposed method. This scheme is employed in order to introduce some additional adjusting parameters with the purpose of acquiring accurate analytical approximate solutions.

scholarly journals Modeling And Characterization of Carrier Mobility For Truncated Conical Quantum Dot Infrared Photodetectors

2021 ◽  
Author(s):  
Yasser M. El-Batawy ◽  
Marwa Feraig
Keyword(s):  
Differential Equation ◽  
Quantum Dot ◽  
Time Domain ◽  
Carrier Mobility ◽  
Carrier Transport ◽  
Infrared Photodetectors ◽  
Generic Model ◽  
Quantum Dot Infrared Photodetectors ◽  
The Time Domain

Abstract In the present paper, a full theoretical model for calculating the carrier mobility coming as a result of the existence of a truncated conical quantum dots of n-type quantum dot infrared photodetectors (QDIPs) is developed. This model is built on solving Boltzmann’s transport equation that is a complex integro-differential equation describing the carrier transport. The time-domain finite-difference method is used in this numerical solution. The influences of dimensions and density of the QDs for this structure on the carrier mobility are studied. Eventually, the calculated mobility for truncated conical InAs/GaAs QDIP is contrasted to other conical, spherical, and hemispherical QD structures. The model put forward is a generic model that is applicable to various structures of truncated conical QDs devices.

scholarly journals Time domain solution of electromagnetic radiation model of the grounding system excited by pulse current

2020 ◽  
Vol 35 (1) ◽  
pp. 74-81
Author(s):  
Nedis Dautbasic ◽  
Adnan Mujezinovic
Keyword(s):  
Differential Equation ◽  
Electromagnetic Radiation ◽  
Time Domain ◽  
Gauge Condition ◽  
Pulse Excitation ◽  
Solution Technique ◽  
Grounding System ◽  
Integro Differential Equation ◽  
Transient Voltage ◽  
The Time Domain

This paper deals with an advanced electromagnetic radiation approach for analyzing the time-domain performance of grounding systems under pulse excitation currents. The model of the grounding systems presented within this paper is based on the homogeneous Pocklington integro-differential equation for the calculation of the current distribution on the grounding system and Lorentz gauge condition which is used for the grounding system transient voltage calculation. For the solution of the Pocklington integro-differential equation, the indirect boundary element method and marching on-in time method are used. Fur- thermore, the solution technique for the calculation of the grounding system transient voltage is presented. The numerical model for the calculation of the grounding system transients was verified by comparing it with onsite measurement results.

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