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 Analytic solutions of the deterministic SEIA worm model by homotopy perturbation method

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

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.

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 Linearized Homotopy Perturbation Method for Two Nonlinear Problems of Duffing Equations

2021 ◽  
Vol 13 (6) ◽  
pp. 10
Author(s):  
Chein-Shan Liu
Keyword(s):  
Periodic Solution ◽  
Perturbation Method ◽  
Trivial Solution ◽  
Homotopy Perturbation Method ◽  
Nonlinear Problems ◽  
Analytic Solutions ◽  
Higher Order ◽  
Duffing Equation ◽  
Duffing Equations ◽  
Homotopy Perturbation

In the paper, we solve two nonlinear problems related to the Duffing equations in space and in time. The first problem is the bifurcation of Duffing equation in space, wherein a critical value of the parameter initiates the bifurcation from a trivial solution to a non-trivial solution. The second problem is an unconventional periodic problem of Duffing equation in time to determine period and periodic solution. To save computational cost and even enhance the accuracy in seeking higher order analytic solutions of these two problems, a modified homotopy perturbation method is invoked after a linearization technique being exerted on the Duffing equation, whose nonlinear cubic term is decomposed at two sides via a weight factor, such that the Duffing equation is linearized as the Mathieu type differential equation. The constant preceding the displacement is expanded in powers of homotopy parameter and the coefficients are determined to avoid secular solutions appeared in the derived sequence of linear differential equations. Consequently, after setting the homotopy parameter equal to unity and solving the amplitude equation, the higher order bifurcated solutions can be derived explicitly. For the second problem, we can determine the period and periodic solution in closed-form, which are very accurate. For the sake of comparison the results obtained from the fourth-order Runge-Kutta numerical method are used to evaluate the presented analytic solutions.

On the Primary Resonance of an Electrostatically Actuated MEMS Using the Homotopy Perturbation Method

2009 ◽  
Author(s):  
Mahdi Mojahedi ◽  
Mahdi Moghimi Zand ◽  
Mohammad Taghi Ahmadian
Keyword(s):  
Differential Equation ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Multiple Scales ◽  
Electrostatic Force ◽  
Nonlinear Partial Differential Equation ◽  
Primary Resonance ◽  
Analytic Solutions ◽  
Design Parameters ◽  
Homotopy Perturbation

In this paper, primary resonance of a double-clamped microbeam has been investigated. The Microbeam is predeformed by a DC electrostatic force and then driven to vibrate by an AC harmonic electrostatic force. Effects of midplane stretching, axial loads and damping are considered in modeling. Galerkin’s approximation is utilized to convert the nonlinear partial differential equation of motion to a nonlinear ordinary differential equation. Afterward, a combination of homotopy perturbation method and the method of multiple scales are utilized to find analytic solutions to the steady-state motion of the microbeam, far from pull-in. The effects of different design parameters on dynamic behavior are discussed. The results obtained by the presented method are validated by comparing with literature.

On Spectral-Homotopy Perturbation Method Solution of Nonlinear Differential Equations in Bounded Domains

2013 ◽  
Vol 1 (1) ◽  
pp. 25-37
Author(s):  
Ahmed A. Khidir
Keyword(s):  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Nonlinear Boundary ◽  
Nonlinear Differential Equations ◽  
Iteration Algorithm ◽  
Test Problems ◽  
Nonlinear Boundary Value Problems ◽  
Homotopy Perturbation ◽  
Spectral Technique ◽  
Homotopy Analysis

In this study, a combination of the hybrid Chebyshev spectral technique and the homotopy perturbation method is used to construct an iteration algorithm for solving nonlinear boundary value problems. Test problems are solved in order to demonstrate the efficiency, accuracy and reliability of the new technique and comparisons are made between the obtained results and exact solutions. The results demonstrate that the new spectral homotopy perturbation method is more efficient and converges faster than the standard homotopy analysis method. The methodology presented in the work is useful for solving the BVPs consisting of more than one differential equation in bounded domains. 

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