Numerical solution of fractionally damped beam by homotopy perturbation method

Behera, Diptiranjan; Chakraverty, Snehashish

doi:10.2478/s11534-013-0201-9

Cited by 11 publications

(8 citation statements)

References 32 publications

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“…For both the cases, for r = 1 , fuzzy initial conditions convert into crisp initial conditions. It is interesting to note that for both the responses, lower and upper bounds of the fuzzy solutions are same for r = 1 And those are found to be same as Behera and Chakraverty (2013c) and Liang and Tang (2007).…”

Section: Numerical Results and Discussionmentioning

confidence: 80%

“…We start with an initial approximation: and from variational iterational equation (15) with the application of Caputo derivative as defined in Chakraverty and Behera (2013), Behera and Chakraverty (2013c) for fractional differentiation, we have: and so on, where f(i)=∂if∂xi.…”

Section: Uncertain Response Analysismentioning

confidence: 99%

“…Simplifying the above expression for ṽ 1 ( x,t ; r ,β) using the definition for fractional derivative in Caputo sense as defined in Chakraverty and Behera (2013), Behera and Chakraverty (2013c) for the second term and by considering general differentiation for first and third term involved in expression of φ we have: …”

Section: Uncertain Response Analysismentioning

confidence: 99%

“…The same type of problem is also studied by Yuan and Agrawal (2002) using a numerical technique when the damping factor is defined as fractional. Recently, Behera and Chakraverty (2013c, 2015) studied numerical solution of fractionally damped beam using homotopy perturbation method. Also, dynamic responses of fractionally damped spring mass system have been investigated by Chakraverty and Behera (2013).…”

Section: Introductionmentioning

confidence: 99%

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Uncertain dynamic responses of imprecisely defined arbitrary order fractionally damped beam subject to various loads

Behera

Huang

Tapaswini

2018

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Purpose Recently, fractional differential equations have been used to model various physical and engineering problems. One may need a reliable and efficient numerical technique for the solution of these types of differential equations, as sometimes it is not easy to get the analytical solution. However, in general, in the existing investigations, involved parameters and variables are defined exactly, whereas in actual practice it may contain uncertainty because of error in observations, maintenance induced error, etc. Therefore, the purpose of this paper is to find the dynamic response of fractionally damped beam approximately under fuzzy and interval uncertainty. Design/methodology/approach Here, a semi analytical approach, variational iteration method (VIM), has been considered for the solution. A newly developed form of fuzzy numbers known as double parametric form has been applied to model the uncertainty involved in the system parameters and variables. Findings VIM has been successfully implemented along with double parametric form of fuzzy number to find the uncertain dynamic responses of the fractionally damped beam. The advantage of this approach is that the solution can be written in power series or compact form. Also, this method converges rapidly to have the accurate solution. The uncertain responses subject to impulse and step loads have also been computed and the behaviours of the responses are analysed. Applying the double parametric form, it reduces the computational cost without separating the fuzzy equation into coupled differential equations as done in traditional approaches. Originality/value Uncertain dynamic responses of fuzzy fractionally damped beam using the newly developed double parametric form of fuzzy numbers subject to unit step and impulse loads have been obtained. Gaussian fuzzy numbers are used to model the uncertainties. In the methodology using the alpha cut form, corresponding beam equation is first converted to an interval-based fuzzy equation. Next, it has been transformed to crisp form by applying double parametric form of fuzzy numbers. Finally, VIM has been applied to solve the same for the general fuzzy responses. Various numerical examples have been taken in to consideration.

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Section: Numerical Results and Discussionmentioning

confidence: 80%

Section: Uncertain Response Analysismentioning

confidence: 99%

Section: Uncertain Response Analysismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Uncertain dynamic responses of imprecisely defined arbitrary order fractionally damped beam subject to various loads

Behera

Huang

Tapaswini

2018

Self Cite

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“…Recently, homotopy perturbation method is found to be a powerful tool for linear and nonlinear differential equations. The HPM was first developed by He in [22,23], and many authors applied this method to solve various linear and nonlinear functional equations of scientific and engineering problems [24,25]. The solution is considered as the sum of infinite series, which converges rapidly to accurate solutions.…”

Section: Introductionmentioning

confidence: 99%

Numerical Solution of Uncertain Beam Equations Using Double Parametric Form of Fuzzy Numbers

Tapaswini

Chakraverty

2013

Applied Computational Intelligence and Soft Computing

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Present paper proposes a new technique to solve uncertain beam equation using double parametric form of fuzzy numbers. Uncertainties appearing in the initial conditions are taken in terms of triangular fuzzy number. Using the single parametric form, the fuzzy beam equation is converted first to an interval-based fuzzy differential equation. Next, this differential equation is transformed to crisp form by applying double parametric form of fuzzy number. Finally, the same is solved by homotopy perturbation method (HPM) to obtain the uncertain responses subject to unit step and impulse loads. Obtained results are depicted in terms of plots to show the efficiency and powerfulness of the methodology.

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A higher order numerical scheme for solving fractional Bagley‐Torvik equation

Ding

Wong

2021

Math Methods in App Sciences

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In this paper, we develop a higher order numerical method for the fractional Bagley‐Torvik equation. The main tools used include a new fourth‐order approximation for the fractional derivative based on the weighted shifted Grünwald‐Letnikov difference operator and a discrete cubic spline approach. We show that the theoretical convergence order improves those of previous work. Five examples are further presented to illustrate the efficiency of our method and to compare with other methods in the literature.

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Numerical solution of fractionally damped beam by homotopy perturbation method

Cited by 11 publications

References 32 publications

Uncertain dynamic responses of imprecisely defined arbitrary order fractionally damped beam subject to various loads

Uncertain dynamic responses of imprecisely defined arbitrary order fractionally damped beam subject to various loads

Numerical Solution of Uncertain Beam Equations Using Double Parametric Form of Fuzzy Numbers

A higher order numerical scheme for solving fractional Bagley‐Torvik equation

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