scholarly journals Physics of nonlinear oscillations with nonlocal variables

2021 ◽  
Vol 2056 (1) ◽  
pp. 012010
Author(s):  
O A Volkova ◽  
M H Khamis Hassan ◽  
T F Kamalov
Keyword(s):  
Nonlinear Equations ◽  
Nonlinear Oscillations ◽  
Linear Equations ◽  
Approximate Solutions ◽  
Physical Processes ◽  
Suspension Point ◽  
Higher Derivatives

Abstract In cases where physical processes cannot be described by linear equations, and nonlinear equations are difficult to solve mathematically, we have to use approximate solutions to such problems. One such example is the description of the Kapitsa pendulum, which is a pendulum with a vibrating suspension point. In contrast to the previously known methods of describing such a problem, in this paper we propose to use additional variables in the form of higher derivatives, which allows us to obtain corrections that give a more detailed contribution to the description of this problem.

scholarly journals Approximate solution of system of equations arising in interior-point methods for bound-constrained optimization

2021 ◽  
Author(s):  
David Ek ◽  
Anders Forsgren
Keyword(s):  
Approximate Solution ◽  
Interior Point ◽  
Interior Point Methods ◽  
Linear Equations ◽  
Approximate Solutions ◽  
System Of Nonlinear Equations ◽  
Intermediate Step ◽  
System Of Linear Equations ◽  
Constrained Nonlinear Optimization ◽  
Asymptotic Error

AbstractThe focus in this paper is interior-point methods for bound-constrained nonlinear optimization, where the system of nonlinear equations that arise are solved with Newton’s method. There is a trade-off between solving Newton systems directly, which give high quality solutions, and solving many approximate Newton systems which are computationally less expensive but give lower quality solutions. We propose partial and full approximate solutions to the Newton systems. The specific approximate solution depends on estimates of the active and inactive constraints at the solution. These sets are at each iteration estimated by basic heuristics. The partial approximate solutions are computationally inexpensive, whereas a system of linear equations needs to be solved for the full approximate solution. The size of the system is determined by the estimate of the inactive constraints at the solution. In addition, we motivate and suggest two Newton-like approaches which are based on an intermediate step that consists of the partial approximate solutions. The theoretical setting is introduced and asymptotic error bounds are given. We also give numerical results to investigate the performance of the approximate solutions within and beyond the theoretical framework.

scholarly journals Compactness theorems for problems with unknown boundary

2021 ◽  
Vol 21 (1) ◽  
pp. 105-112
Author(s):  
A.G. Podgaev ◽  
◽  
T.D. Kulesh ◽  
Keyword(s):  
Phase Transitions ◽  
Approximate Solutions ◽  
Compactness Theorem ◽  
Unknown Boundary ◽  
Entire Domain ◽  
Higher Derivatives ◽  
Compactness Theorems

The compactness theorem is proved for sequences of functions that have estimates of the higher derivatives in each subdomain of the domain of definition, divided into parts by a sequence of some curves of class W_2^1. At the same time, in the entire domain of determining summable higher derivatives, these sequences do not have. These results allow us to make limit transitions using approximate solutions in problems with an unknown boundary that describe the processes of phase transitions.

scholarly journals PENGARUH PERUBAHAN NILAI PARAMETER TERHADAP NILAI ERROR PADA METODE RUNGE-KUTTA ORDE 3

2017 ◽  
Vol 3 (2) ◽  
Author(s):  
Tornados P Silaban ◽  
Faiz . Ahyaningsih
Keyword(s):  
Linear Equations ◽  
Research Process ◽  
Kutta Method ◽  
System Of Linear Equations ◽  
Runge Kutta Method ◽  
A Value ◽  
Fixed Parameter ◽  
Higher Derivatives ◽  
Runge Kutta ◽  
Parameter Values

ABSTRACTRunge-Kutta method is a numerical method used to find the solution of an equation. This method seeks to obtain a higher degree of precision, and at the same time seeking to avoid the need of higher derivatives by evaluating the function f (x, y) at the selected point in each interval step. In this paper discussed the effect of changes in the value of the parameter (h) to the value of the error in the Runge-Kutta method Order-3. The equation to be discussed is a linear ordinary differential equation of the two levels that have been changed into a system of linear equations. In the research process was not found fixed parameter values to get the minimum error value, because each parameter has a value of error varied for each equation.Keywords: Runge-Kutta, parameters, error.

scholarly journals A computational method for solving differential equations with quadratic nonlinearity by using Bernoulli polynomials

2019 ◽  
Vol 23 (Suppl. 1) ◽  
pp. 275-283
Author(s):  
Kubra Bicer ◽  
Mehmet Sezer
Keyword(s):  
Differential Equations ◽  
Linear Equations ◽  
Bernoulli Polynomials ◽  
Approximate Solutions ◽  
Mean Value ◽  
Computational Method ◽  
Mean Value Theorem ◽  
Residual Functions ◽  
Non Linear ◽  
Linear Differential

In this paper, a matrix method is developed to solve quadratic non-linear differential equations. It is assumed that the approximate solutions of main problem which we handle primarily, is in terms of Bernoulli polynomials. Both the approximate solution and the main problem are written in matrix form to obtain the solution. The absolute errors are applied to numeric examples to demonstrate efficiency and accuracy of this technique. The obtained tables and figures in the numeric examples show that this method is very sufficient and reliable for solution of non-linear equations. Also, a formula is utilized based on residual functions and mean value theorem to seek error bounds.

Kinematic Analysis and Synthesis of Three-Input, Eight-Bar Mechanisms Driven by Relatively Small Cranks

1992 ◽  
Author(s):  
Kambiz Farhang ◽  
Partha Sarathi Basu
Keyword(s):  
Nonlinear Equations ◽  
Kinematic Analysis ◽  
Linear Equations ◽  
Output Link ◽  
Analysis And Design ◽  
Constraint Equations ◽  
Analysis And Synthesis ◽  
The Mean ◽  
Approximate Equations

Abstract Approximate kinematic equations are developed for the analysis and design of three-input, eight-bar mechanisms driven by relatively small cranks. Application of a method in which an output link is presumed to be comprised of a mean and a perturbational motions, along with the vector loop approach facilitates the derivation of the approximate kinematic equations. The resulting constraint equations are, (i) in the form of a set of four nonlinear equations relating the mean link orientations, and (ii) a set of four linear equations in the unknown perturbations (output link motions). The latter set of equations is solved, symbolically, to obtain the output link motions. The approximate equations are shown to be effective in the synthesis of three-input, small-crank mechanisms.

Quantum Newton’s Method for Solving the System of Nonlinear Equations

SPIN ◽  
2021 ◽  
pp. 2140004
Author(s):  
Cheng Xue ◽  
Yuchun Wu ◽  
Guoping Guo
Keyword(s):  
Quantum Computing ◽  
Nonlinear Equations ◽  
Newton’S Method ◽  
Newton's Method ◽  
Linear Equations ◽  
Data Conversion ◽  
System Of Nonlinear Equations ◽  
Dimensional System ◽  
System Of Linear Equations ◽  
Whole Process

While quantum computing provides an exponential advantage in solving the system of linear equations, there is little work to solve the system of nonlinear equations with quantum computing. We propose quantum Newton’s method (QNM) for solving [Formula: see text]-dimensional system of nonlinear equations based on Newton’s method. In QNM, we solve the system of linear equations in each iteration of Newton’s method with quantum linear system solver. We use a specific quantum data structure and [Formula: see text] tomography with sample error [Formula: see text] to implement the classical-quantum data conversion process between the two iterations of QNM, thereby constructing the whole process of QNM. The complexity of QNM in each iteration is [Formula: see text]. Through numerical simulation, we find that when [Formula: see text], QNM is still effective, so the complexity of QNM is sublinear with [Formula: see text], which provides quantum advantage compared with the optimal classical algorithm.

Synthesis of the four-bar linkage as path generation by choosing the shape of the connecting rod

2020 ◽  
Vol 234 (13) ◽  
pp. 2643-2652 ◽  
Author(s):  
SM Varedi-Koulaei ◽  
H Rezagholizadeh
Keyword(s):  
Analytical Solution ◽  
Nonlinear Equations ◽  
Linear Equations ◽  
Solution Process ◽  
Current Method ◽  
Path Generation ◽  
Connecting Rod ◽  
Complicated Process ◽  
Four Bar Linkage ◽  
Analytical Synthesis

This paper presents a method for path generation synthesis of a four-bar linkage that includes both graphical and analytical synthesis and both cases of with and without prescribed timing. The advantage of the proposed method over available techniques is that it is easier and does not need the complicated process (especially in graphical case). In an analytical solution, this method needs the solution of the linear equations, unlike the previous methods, in that they have required the solution of the nonlinear equations. Moreover, in the current method, one can choose the shape of the coupler, while, in other methods, the shape of the coupler is the result of the solution process. The proposed algorithm can be used for path generation synthesizing of a four-bar linkage for three precision points.

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