A framework for iterative analysis of non-classically damped dynamical systems

Aureli, Matteo

doi:10.1016/j.jsv.2014.07.018

Cited by 7 publications

(7 citation statements)

References 45 publications

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“…Similarly, we expand Φ(t, x, q) and Ξ(t, x, q) into power series of q, i.e. 43) subject to the boundary condition…”

Section: An Example Of Two-dimensional Bsdementioning

confidence: 99%

“…Especially, there is a convergence-control parameter c 0 in the series solutions, which provides us a convenient way to guarantee the convergence of series solutions gained by the HAM. It is these merits that distinguish the HAM from other analytic approaches, and thus enable the HAM to be successfully applied to many complicated problems with high nonlinearity [39][40][41][42][43][44][45][46][47]. Note that the HAM was successfully applied to give an analytic approximation with much longer expiry for the optimal exercise boundary of an American put option than perturbation approximations [48].…”

Section: Introductionmentioning

confidence: 99%

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On the homotopy analysis method for backward/forward-backward stochastic differential equations

Zhong

Liao

2017

Numer Algor

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In this paper, an analytic approximation method for highly nonlinear equations, namely the homotopy analysis method (HAM), is employed to solve some backward stochastic differential equations (BSDEs) and forwardbackward stochastic differential equations (FBSDEs), including one with high dimensionality (up to 12 dimensions). By means of the HAM, convergent series solutions can be quickly obtained with high accuracy for a FBSDE in a 6 dimensional case, within less than 1% CPU time used by a currently reported numerical method for the same case [34]. Especially, as dimensionality enlarges, the increase of computational complexity for the HAM is not as dramatic as this numerical method. All of these demonstrate the validity and high efficiency of the HAM for the backward/forward-backward stochastic differential equations in science, engineering and finance.

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“…Similarly, we expand Φ(t, x, q) and Ξ(t, x, q) into power series of q, i.e. 43) subject to the boundary condition…”

Section: An Example Of Two-dimensional Bsdementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On the homotopy analysis method for backward/forward-backward stochastic differential equations

Zhong

Liao

2017

Numer Algor

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“…A convergence-control parameter h is introduced into the series solutions to guarantee the convergence. HAM distinguishes itself from other analytic approaches to be applied to complicated problems with strong nonlinearity [14][15][16][17][18][19][20][21][22][23][24][25][26][27]. The Föppl-Von Kármán's plate equations were solved by HAM [28,29].…”

Section: Introductionmentioning

confidence: 99%

Explicit Solution to Large Deformation of Cantilever Beam by Improved Homotopy Analysis Method II: Vertical and Horizontal Displacements

Yin-shan

Xie

et al. 2022

Applied Sciences

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Explicit solutions to vertical and horizontal displacements are derived for large deformation of a cantilever beam under point load at the free end by an improved homotopy analysis method (IHAM). Quadratic and cubic nonlinear differential equations are adopted to construct more proficient nonlinear equations for vertical and horizontal displacements respectively combined with their currently available nonlinear displacement equations. Higher-order nonlinear iterative homotopy equations are established to solve the vertical and horizontal displacements by combining simultaneous equations of the constructed nonlinear equations and the auxiliary linear equations. The convergence range of vertical displacement is extended by the homotopy-Páde approximation. The explicit solutions to the vertical and horizontal displacements are in favorable agreements with the respective exact solutions. The convergence ranges for a relative error of 1% by the improved homotopy analysis method for vertical and horizontal displacements increases by 60% and 7%, respectively. These explicit formulas are helpful in practical engineering design for very slender structures, such as high-rise buildings and long bridges.

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“…Thus, the so-called "convergencecontrol parameter" c 0 can indeed greatly enlarge the convergence interval of solution series. As a powerful technique to solve highly nonlinear equations, the HAM has been successfully employed to solve various types of nonlinear problems over the past two decades [27][28][29][30][31][32][33][34][35][36]. Note that the Föppl-Von Kármán's plate equations was solved by means of the HAM [37].…”

Section: Introductionmentioning

confidence: 99%

Analytic Solutions of Von Kármán Plate under Arbitrary Uniform Pressure — Equations in Differential Form

Zhong

Liao

2017

Stud Appl Math

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The large deflection of a circular thin plate under uniform external pressure is a classic problem in solid mechanics, dated back to Von Kármán [1]. This problem is reconsidered in this paper using an analytic approximation method, namely the homotopy analysis method (HAM). Convergent series solutions are obtained for four types of boundary conditions with rather high nonlinearity, even in the case of w(0)/h > 20, where w(0)/h denotes the ratio of central deflection to plate thickness. Especially, we prove that the previous perturbation methods for an arbitrary perturbation quantity (including the Vincent's [2] and Chien's [3] methods) and the modified iteration method [4] are only the special cases of the HAM. However, the HAM works well even when the perturbation methods become invalid. All of these demonstrate the validity and potential of the HAM for the Von Kármán's plate equations, and show the superiority of the HAM over perturbation methods for highly nonlinear problems.

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A framework for iterative analysis of non-classically damped dynamical systems

Cited by 7 publications

References 45 publications

On the homotopy analysis method for backward/forward-backward stochastic differential equations

On the homotopy analysis method for backward/forward-backward stochastic differential equations

Explicit Solution to Large Deformation of Cantilever Beam by Improved Homotopy Analysis Method II: Vertical and Horizontal Displacements

Analytic Solutions of Von Kármán Plate under Arbitrary Uniform Pressure — Equations in Differential Form

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