An efficient multistep iteration scheme for systems of nonlinear algebraic equations associated with integral equations

2020 ◽  
Vol 43 (14) ◽  
pp. 8105-8115
Author(s):  
Yaser Seif ◽  
Taher Lotfi
Keyword(s):  
Integral Equations ◽  
Iteration Scheme ◽  
Algebraic Equations ◽  
Nonlinear Algebraic Equations

scholarly journals An Iteration Scheme Suitable for Solving Limit Cycles of Nonsmooth Dynamical Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Q. X. Liu ◽  
Y. M. Chen ◽  
J. K. Liu
Keyword(s):  
Dynamical Systems ◽  
Limit Cycles ◽  
Nonlinear Damping ◽  
Iteration Scheme ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Nonsmooth Systems ◽  
Nonlinear Algebraic Equations ◽  
Nonsmooth Dynamical Systems ◽  
Previous Iteration

The Mickens iteration method (MIM) is modified to solve self-excited systems containing nonsmooth nonlinearities and/or nonlinear damping terms. If the MIM is implemented routinely, the unknown frequency and amplitude of limit cycle (LC) would couple to each other in complicated nonlinear algebraic equations at each iteration. It is cumbersome to solve these algebraic equations, especially for nonsmooth systems. In the modified procedures, the unknown frequency is substituted by the determined value obtained at the previous iteration. By this means, the frequency is decoupled from the nonlinear terms. Numerical examples show that the LCs obtained by the modified MIM agree well with numerical results. The presented method is very suitable for solving self-excited systems, especially those with nonlinear damping and nonsmooth nonlinearities.

scholarly journals A New Wavelet Method for Solving a Class of Nonlinear Volterra-Fredholm Integral Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Xiaomin Wang
Keyword(s):  
Galerkin Method ◽  
Integral Equations ◽  
Approximation Scheme ◽  
Nonlinear Term ◽  
Fredholm Integral Equations ◽  
Algebraic Equations ◽  
New Approach ◽  
Wavelet Approximation ◽  
Connection Coefficients ◽  
Nonlinear Algebraic Equations

A new approach, Coiflet-type wavelet Galerkin method, is proposed for numerically solving the Volterra-Fredholm integral equations. Based on the Coiflet-type wavelet approximation scheme, arbitrary nonlinear term of the unknown function in an equation can be explicitly expressed. By incorporating such a modified wavelet approximation scheme into the conventional Galerkin method, the nonsingular property of the connection coefficients significantly reduces the computational complexity and achieves high precision in a very simple way. Thus, one can obtain a stable, highly accurate, and efficient numerical method without calculating the connection coefficients in traditional Galerkin method for solving the nonlinear algebraic equations. At last, numerical simulations are performed to show the efficiency of the method proposed.

scholarly journals Bernoulli Wavelets Operational Matrices Method for the Solution of Nonlinear Stochastic Itô-Volterra Integral Equations

2020 ◽  
pp. 395-410
Author(s):  
S. C. Shiralashetti ◽  
Lata Lamani
Keyword(s):  
Integral Equations ◽  
Operational Matrix ◽  
Volterra Integral Equations ◽  
Operational Matrices ◽  
Effective Strategy ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Stochastic Operational Matrix ◽  
Nonlinear Algebraic Equations ◽  
Operational Matrix Of Integration

This article gives an effective strategy to solve nonlinear stochastic Itô-Volterra integral equations (NSIVIE). These equations can be reduced to a system of nonlinear algebraic equations with unknown coefficients, using Bernoulli wavelets, their operational matrix of integration (OMI), stochastic operational matrix of integration (SOMI) and these equations can be solved numerically. Error analysis of the proposed method is given. Moreover, the results obtained are compared to exact solutions with numerical examples to show that the method described is accurate and precise.

scholarly journals Frequency–Amplitude Relationship of a Nonlinear Symmetric Panel Absorber Mounted on a Flexible Wall

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1188
Author(s):  
Yiu-Yin Lee
Keyword(s):  
Elliptic Integral ◽  
Algebraic Equations ◽  
Cavity Depth ◽  
Nonlinear Algebraic Equations ◽  
Flexible Panel ◽  
Flexible Wall ◽  
Elliptic Integral Method ◽  
Flexible Panels ◽  
Relationship Of ◽  
Panel Absorber

This study addresses the frequency–amplitude relationship of a nonlinear symmetric panel absorber mounted on a flexible wall. In many structural–acoustic works, only one flexible panel is considered in their models with symmetric configuration. There are very limited research investigations that focus on two flexible panels coupled with a cavity, particularly for nonlinear structural–acoustic problems. In practice, panel absorbers with symmetric configurations are common and usually mounted on a flexible wall. Thus, it should not be assumed that the wall is rigid. This study is the first work employing the weighted residual elliptic integral method for solving this problem, which involves the nonlinear multi-mode governing equations of two flexible panels coupled with a cavity. The reason for adopting the proposed solution method is that fewer nonlinear algebraic equations are generated. The results obtained from the proposed method and finite element method agree reasonably well with each other. The effects of some parameters such as vibration amplitude, cavity depth and thickness ratio, etc. are also investigated.

scholarly journals A Pseudospectral Algorithm for Solving Multipantograph Delay Systems on a Semi-Infinite Interval Using Legendre Rational Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
E. H. Doha ◽  
D. Baleanu ◽  
A. H. Bhrawy ◽  
R. M. Hafez
Keyword(s):  
Numerical Solutions ◽  
Rational Functions ◽  
Absolute Error ◽  
Delay Systems ◽  
Infinite Interval ◽  
Algebraic Equations ◽  
Collocation Points ◽  
Nonlinear Algebraic Equations ◽  
Linear And Nonlinear ◽  
Pseudospectral Scheme

A new Legendre rational pseudospectral scheme is proposed and developed for solving numerically systems of linear and nonlinear multipantograph equations on a semi-infinite interval. A Legendre rational collocation method based on Legendre rational-Gauss quadrature points is utilized to reduce the solution of such systems to systems of linear and nonlinear algebraic equations. In addition, accurate approximations are achieved by selecting few Legendre rational-Gauss collocation points. The numerical results obtained by this method have been compared with various exact solutions in order to demonstrate the accuracy and efficiency of the proposed method. Indeed, for relatively limited nodes used, the absolute error in our numerical solutions is sufficiently small.

scholarly journals Modal Perturbation Method for the Dynamic Characteristics of Timoshenko Beams

2005 ◽  
Vol 12 (6) ◽  
pp. 425-434 ◽  
Author(s):  
Menglin Lou ◽  
Qiuhua Duan ◽  
Genda Chen
Keyword(s):  
Perturbation Method ◽  
Dynamic Characteristics ◽  
Timoshenko Beam ◽  
Natural Frequencies ◽  
Solution Process ◽  
Equation Of Motion ◽  
Timoshenko Beams ◽  
Algebraic Equations ◽  
Nonlinear Algebraic Equations ◽  
Shear Distortion

Timoshenko beams have been widely used in structural and mechanical systems. Under dynamic loading, the analytical solution of a Timoshenko beam is often difficult to obtain due to the complexity involved in the equation of motion. In this paper, a modal perturbation method is introduced to approximately determine the dynamic characteristics of a Timoshenko beam. In this approach, the differential equation of motion describing the dynamic behavior of the Timoshenko beam can be transformed into a set of nonlinear algebraic equations. Therefore, the solution process can be simplified significantly for the Timoshenko beam with arbitrary boundaries. Several examples are given to illustrate the application of the proposed method. Numerical results have shown that the modal perturbation method is effective in determining the modal characteristics of Timoshenko beams with high accuracy. The effects of shear distortion and moment of inertia on the natural frequencies of Timoshenko beams are discussed in detail.

Periodic Response of a Link Coupling With Clearance

1989 ◽  
Vol 111 (2) ◽  
pp. 253-259 ◽  
Author(s):  
Y. S. Choi ◽  
S. T. Noah
Keyword(s):  
Steady State ◽  
Boundary Conditions ◽  
Solution Procedure ◽  
Contact Forces ◽  
Steady State Response ◽  
Algebraic Equations ◽  
State Response ◽  
Contact Points ◽  
Integration Schemes ◽  
Nonlinear Algebraic Equations

The nonlinear, steady-state response of a displacement-forced link coupling with clearance with finite stiffness is determined. The solution procedure is derived from satisfying the boundary conditions at the contact points and then solving the resulting nonlinear algebraic equations by setting the duration of contact as a parameter. This direct approach to determining periodic solutions for systems with clearances with finite stiffness is substantially more efficient than numerical integration schemes. Results in terms of contact forces and durations of contact are pertinent to fatigue and wear studies. Parametric relations are presented for effects of the variation of damping, stiffness, exciting displacement, and gap length on the dynamic behavior of the link pair.

Periodic Solutions in Rotor Dynamic Systems With Nonlinear Supports: A General Approach

1989 ◽  
Vol 111 (2) ◽  
pp. 187-193 ◽  
Author(s):  
C. Nataraj ◽  
H. D. Nelson
Keyword(s):  
Dynamic Systems ◽  
Original Problem ◽  
Squeeze Film ◽  
Algebraic Equations ◽  
Trigonometric Collocation ◽  
Nonlinear Algebraic Equations ◽  
The Stability ◽  
Future Work ◽  
Rotor Dynamic ◽  
Large Rotor

A new quantitative method of estimating steady state periodic behavior in nonlinear systems, based on the trigonometric collocation method, is outlined. A procedure is developed to analyze large rotor dynamic systems with nonlinear supports by the use of the above method in conjunction with Component Mode Synthesis. The algorithm discussed is seen to reduce the original problem to solving nonlinear algebraic equations in terms of only the coordinates associated with the nonlinear supports and is a big improvement over commonly used integration methods. The feasibility and advantages of the procedure so developed are illustrated with the help of an example of a typical rotor dynamic system with an uncentered squeeze film damper. Future work on the investigation of the stability of the periodic response so obtained is outlined.

scholarly journals Solution of nonlinear Fredholm-Hammerstein integral equations by using semiorthogonal spline wavelets

2005 ◽  
Vol 2005 (1) ◽  
pp. 113-121 ◽  
Author(s):  
M. Lakestani ◽  
M. Razzaghi ◽  
M. Dehghan
Keyword(s):  
Integral Equations ◽  
Algebraic Equations ◽  
B Spline ◽  
Compactly Supported ◽  
Hammerstein Integral Equations ◽  
Spline Wavelets

Compactly supported linear semiorthogonal B-spline wavelets together with their dual wavelets are developed to approximate the solutions of nonlinear Fredholm-Hammerstein integral equations. Properties of these wavelets are first presented; these properties are then utilized to reduce the computation of integral equations to some algebraic equations. The method is computationally attractive, and applications are demonstrated through an illustrative example.

The Influence of Small Particles on the Structure of a Turbulent Shear Flow

2004 ◽  
Vol 126 (4) ◽  
pp. 613-619 ◽  
Author(s):  
David I. Graham
Keyword(s):  
Shear Flow ◽  
General Trend ◽  
Turbulent Shear ◽  
Mass Loading ◽  
Turbulent Shear Flow ◽  
Algebraic Equations ◽  
Reynolds Stresses ◽  
Temporal Correlations ◽  
Fluid Velocities ◽  
Nonlinear Algebraic Equations

In this paper, an analytical solution is found for the Reynolds equations describing a simple turbulent shear flow carrying small, wake-less particles. An algebraic stress model is used as the basis of the model, the particles leading to source terms in the equations for the turbulent stresses in the flow. The sources are proportional to the mass loading of the particles and depend on the temporal correlations of the fluid velocities seen by particles, Rijτ. The resulting set of equations is a system of nonlinear algebraic equations for the Reynolds stresses and the dissipation. The system is solved exactly and the influence of the particles can be quantified. The predictions are compared with DNS results and are shown to predict trends quite well. Different scenarios are investigated, including the effects of isotropic, anisotropic and non-equilibrium time scales and negative loops in Rijτ. The general trend is to increase anisotropy and attenuate turbulence with higher mass loadings. The occurrence of turbulence enhancement is investigated and shown to be theoretically possible, but physically unlikely.

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