Numerical solution of fractionally damped beam by homotopy perturbation method

Open Physics ◽  
2013 ◽  
Vol 11 (6) ◽  
Author(s):  
Diptiranjan Behera ◽  
Snehashish Chakraverty
Keyword(s):  
Fractional Derivative ◽  
Perturbation Method ◽  
Numerical Solution ◽  
Fractional Derivatives ◽  
Homotopy Perturbation Method ◽  
Analytic Solutions ◽  
Step Function ◽  
Homotopy Perturbation ◽  
Derivatives Of ◽  
Appl Math

AbstractThis paper investigates the numerical solution of a viscoelastic continuous beam whose damping behaviours are defined in term of fractional derivatives of arbitrary order. The Homotopy Perturbation Method (HPM) is used to obtain the dynamic response. Unit step function response is considered for the analysis. The obtained results are depicted in various plots. From the results obtained it is interesting to note that by increasing the order of the fractional derivative the beam suffers less oscillation. Similar observations have also been made by keeping the order of the fractional derivative constant and varying the damping ratios. Comparisons are made with the analytic solutions obtained by Zu-feng and Xiao-yan [Appl. Math. Mech. 28, 219 (2007)] to show the effectiveness and validation of this method.

Homotopy Perturbation Method for Time-Fractional Shock Wave Equation

2011 ◽  
Vol 3 (6) ◽  
pp. 774-783 ◽  
Author(s):  
Mithilesh Singh ◽  
Praveen Kumar Gupta
Keyword(s):  
Shock Wave ◽  
Wave Equation ◽  
Perturbation Method ◽  
Numerical Solution ◽  
Fractional Derivatives ◽  
Homotopy Perturbation Method ◽  
Initial Conditions ◽  
Numerical Results ◽  
Homotopy Perturbation

AbstractA scheme is developed to study numerical solution of the time-fractional shock wave equation and wave equation under initial conditions by the homotopy perturbation method (HPM). The fractional derivatives are taken in the Caputo sense. The solutions are given in the form of series with easily computable terms. Numerical results are illustrated through the graph.

scholarly journals The Approximate Analytic Solution of the Time-Fractional Black-Scholes Equation with a European Option Based on the Katugampola Fractional Derivative

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 214
Author(s):  
Sivaporn Ampun ◽  
Panumart Sawangtong
Keyword(s):  
Fractional Derivative ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Analytic Solution ◽  
Analytic Solutions ◽  
Approximate Analytic Solution ◽  
European Option ◽  
European Option Pricing ◽  
Homotopy Perturbation ◽  
Black Scholes

In the finance market, it is well known that the price change of the underlying fractal transmission system can be modeled with the Black-Scholes equation. This article deals with finding the approximate analytic solutions for the time-fractional Black-Scholes equation with the fractional integral boundary condition for a European option pricing problem in the Katugampola fractional derivative sense. It is well known that the Katugampola fractional derivative generalizes both the Riemann–Liouville fractional derivative and the Hadamard fractional derivative. The technique used to find the approximate analytic solutions of the time-fractional Black-Scholes equation is the generalized Laplace homotopy perturbation method, the combination of the generalized Laplace transform and homotopy perturbation method. The approximate analytic solution for the problem is in the form of the generalized Mittag-Leffler function. This shows that the generalized Laplace homotopy perturbation method is one of the most effective methods to construct approximate analytic solutions of the fractional differential equations. Finally, the approximate analytic solutions of the Riemann–Liouville and Hadamard fractional Black-Scholes equation with the European option are also shown.

scholarly journals HPM for Solving the Time-fractional Coupled Burgers Equations

2016 ◽  
Vol 12 (4) ◽  
pp. 6133-6138
Author(s):  
Khadijah Abu Alnaja
Keyword(s):  
Fractional Derivative ◽  
Perturbation Method ◽  
Fractional Derivatives ◽  
Homotopy Perturbation Method ◽  
Approximate Solutions ◽  
Burgers Equations ◽  
Homotopy Perturbation ◽  
Special Cases ◽  
Burger's Equations

In this paper, the homotopy perturbation method is implemented to derive the explicit approximate solutions for the time-fractional coupled Burger's equations. The including fractional derivative is in the Caputo sense. Special attention is given to prove the convergence of the method. The results are compared with those obtained by the exact at special cases of the fractional derivatives. The results reveal that the proposed method is very effective and simple.

scholarly journals Analytic solutions of the stochastic SEIA worm model by homotopy perturbation method

2019 ◽  
Author(s):  
M. Radha ◽  
S. Balamuralitharan ◽  
S. Geethamalini ◽  
V. Geetha ◽  
A. Rathinasamy
Keyword(s):  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Analytic Solutions ◽  
Homotopy Perturbation ◽  
Worm Model

scholarly journals Study on Space-Time Fractional Nonlinear Biological Equation in Radial Symmetry

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Yanqin Liu
Keyword(s):  
Nonlinear Equation ◽  
Fractional Derivative ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Approximate Solutions ◽  
Radial Symmetry ◽  
Space Time ◽  
Homotopy Perturbation ◽  
Initial Stage

We consider the initial stage of space-time fractional generalized biological equation in radial symmetry. Dimensionless multiorder fractional nonlinear equation was first given, and approximate solutions were derived in the form of series using the homotopy perturbation method with a new modification. And the influence of fractional derivative is also discussed.

Sign in / Sign up

Export Citation Format

Share Document