A New Type of Conjugate Gradient Technique for Solving Fuzzy Nonlinear Algebraic Equations

The operational matrices-based computational algorithms are the promising tools to tackle the problems of non-integer derivatives and gained a substantial devotion among the scientific community. Here, an accurate and efficient computational scheme based on another new type of polynomial with the help of collocation method (CM) is presented for different nonlinear multi-order fractional differentials (NMOFDEs) and Bagley–Torvik (BT) equations. The methods are proposed utilizing some new operational matrices of derivatives using Chelyshkov polynomials (CPs) through Caputo’s sense. Two different ways are adopted to construct the approximated (AOM) and exact (EOM) operational matrices of derivatives for integer and non-integer orders and used to propose an algorithm. The understudy problems have been transformed to an equivalent nonlinear algebraic equations system and solved by means of collocation method. The proposed computational method is authenticated through convergence and error-bound analysis. The exactness and effectiveness of said method are shown on some fractional order physical problems. The attained outcomes are endorsing that the recommended method is really accurate, reliable and efficient and could be used as suitable tool to attain the solutions for a variety of the non-integer order differential equations arising in applied sciences.

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Assessment of Apple Quality based on scaled conjugate gradient technique, using Artificial Neural Network model

International Journal of Digital Contents and Applications for Smart Devices ◽

10.21742/ijdcasd.2014.1.1.01 ◽

2014 ◽

Vol 1 (1) ◽

pp. 1-10

Author(s):

Ashutosh Kumar Bhatt ◽

◽

KunwarS ingh Vaisla ◽

Keyword(s):

Neural Network ◽

Artificial Neural Network ◽

Network Model ◽

Neural Network Model ◽

Conjugate Gradient ◽

Artificial Neural Network Model ◽

Scaled Conjugate Gradient ◽

Gradient Technique ◽

Artificial Neural ◽

Apple Quality

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

Symmetry ◽

10.3390/sym13071188 ◽

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.

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

Abstract and Applied Analysis ◽

10.1155/2014/816473 ◽

2014 ◽

Vol 2014 ◽

pp. 1-9 ◽

Cited By ~ 3

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.

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Modal Perturbation Method for the Dynamic Characteristics of Timoshenko Beams

Shock and Vibration ◽

10.1155/2005/824616 ◽

2005 ◽

Vol 12 (6) ◽

pp. 425-434 ◽

Cited By ~ 6

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.

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Periodic Response of a Link Coupling With Clearance

Journal of Dynamic Systems Measurement and Control ◽

10.1115/1.3153044 ◽

1989 ◽

Vol 111 (2) ◽

pp. 253-259 ◽

Cited By ~ 10

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.

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

Journal of Vibration and Acoustics ◽

10.1115/1.3269840 ◽

1989 ◽

Vol 111 (2) ◽

pp. 187-193 ◽

Cited By ~ 51

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.

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

Journal of Fluids Engineering ◽

10.1115/1.1779662 ◽

2004 ◽

Vol 126 (4) ◽

pp. 613-619 ◽

Cited By ~ 1

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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