Vibrations of non-uniform rings studied by means of the differential quadrature method

Gutiérrez, R.H.; Laura, P.A.A.

doi:10.1006/jsvi.1995.0396

Cited by 30 publications

(32 citation statements)

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“…Gutierrez and Laura [6]), some points adjacent to the boundaries are introduced and enforced to act as boundary points (δ−point technique) or in the Generalized Differential Quadrature Rule (GDQR) (e.g. Wu and Liu [7,8,9]), the derivatives at boundary points are also regarded as independent variables.…”

Section: Numerical Solution Of Odesmentioning

confidence: 99%

“…The vibration problem governed by the sixth order differential equation was studied by Gutierrez and Laura [6] using the differential quadrature method (DQM) and the optimized Rayleigh-Ritz method and by Wu and Liu [7] using the generalized differential quadrature rule (GDQR).…”

Section: Sixth Order Ode -Vibrations Of Ring-like Structuresmentioning

confidence: 99%

“…The unsymmetric IRBF collocation method is further verified here in the solution of eighth order ODE y [8] + y [7] + y [6] + y [5] + y [4] + y + y + y + y = 9 exp(x),…”

Section: Eighth Order Ode -Initial Value Problem and Boundary Value Pmentioning

confidence: 99%

“…Firstly, the initial value problem governed by (55) with initial values y(0) = y (0) = y (0) = y (0) = y [4] (0) = y [5] (0) = y [6] (0) = y [7] (0) = 1,…”

Section: Eighth Order Ode -Initial Value Problem and Boundary Value Pmentioning

confidence: 99%

“…Sallam and El-Hawary [4]; Esmail et al [5]; Gutierrez and Laura [6]; Wu and Liu [7,8,9]). For example, the Cauchy problems governed by second order and fourth order equations were solved successfully using deficient spline function approximations by Sallam and El-Hawary [4] and Esmail et al [5] respectively.…”

Section: Introductionmentioning

confidence: 99%

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Solving high order ordinary differential equations with radial basis function networks

Mai‐Duy

2005

Int. J. Numer. Meth. Engng

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SUMMARYThis paper is concerned with the application of radial basis function networks (RBFNs) for numerical solution of high order ordinary differential equations (ODEs).Two unsymmetric RBF collocation schemes, namely the usual direct approach based on a differentiation process and the proposed indirect approach based on an integration process, are developed to solve high order ODEs directly and the latter is found to be considerably superior to the former. Good accuracy and high rate of convergence are obtained with the proposed indirect method.

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Section: Numerical Solution Of Odesmentioning

confidence: 99%

Section: Sixth Order Ode -Vibrations Of Ring-like Structuresmentioning

confidence: 99%

“…The unsymmetric IRBF collocation method is further verified here in the solution of eighth order ODE y [8] + y [7] + y [6] + y [5] + y [4] + y + y + y + y = 9 exp(x),…”

Section: Eighth Order Ode -Initial Value Problem and Boundary Value Pmentioning

confidence: 99%

“…Firstly, the initial value problem governed by (55) with initial values y(0) = y (0) = y (0) = y (0) = y [4] (0) = y [5] (0) = y [6] (0) = y [7] (0) = 1,…”

Section: Eighth Order Ode -Initial Value Problem and Boundary Value Pmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Solving high order ordinary differential equations with radial basis function networks

Mai‐Duy

2005

Int. J. Numer. Meth. Engng

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Application of generalized differential quadrature rule to sixth-order differential equations

Liu

2000

Commun. Numer. Meth. Engng.

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A generalized di erential quadrature rule (GDQR) has been proposed as a general numerical method to solve high-order di erential equations. Applications are given to sixth-order di erential equations that govern the free vibration analysis of ring structures. The DQM uses the function values at all grid points as independent variables in the establishment of algebraic equations. This leads to di culties in implementing the boundary conditions at a point. The GDQR regards the function values at all grid points and their derivatives at grid points wherever necessary as independent variables. The given multiple conditions at any point can therefore be imposed in the GDQR in a direct manner. A procedure is also proposed for calculating GDQR's explicit weighting coe cients for an e ective establishment of algebraic equations.

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An effective spectral collocation method for the direct solution of high‐order ODEs

Mai‐Duy

2005

Commun. Numer. Meth. Engng.

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SUMMARYThis paper reports a new Chebyshev spectral collocation method for directly solving high-order ordinary differential equations (ODEs). The construction of the Chebyshev approximations is based on integration rather than conventional differentiation. This use of integration allows the multiple boundary conditions to be incorporated more efficiently. Numerical results show that the proposed formulation significantly improves the conditioning of the system and yields more accurate results and faster convergence rates than conventional formulations.

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Vibrations of non-uniform rings studied by means of the differential quadrature method

Cited by 30 publications

References 0 publications

Solving high order ordinary differential equations with radial basis function networks

Solving high order ordinary differential equations with radial basis function networks

Application of generalized differential quadrature rule to sixth-order differential equations

An effective spectral collocation method for the direct solution of high‐order ODEs

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