scholarly journals Ninety years of Duffing’s equation

2013 ◽  
Vol 40 (1) ◽  
pp. 49-63 ◽  
Author(s):  
Livija Cveticanin
Keyword(s):  
Exact Solution ◽  
Approximate Solution ◽  
Elliptic Functions ◽  
Jacobi Elliptic Functions ◽  
Duffing's Equation ◽  
Homotopy Perturbation ◽  
Duffing’S Equation ◽  
Solution Methods

In the paper the origin of the so named ?Duffing?s equation? is shown. The author?s generalization of the equation, her published papers dealing with Duffing?s equation and some of the solution methods are presented. Three characteristic approximate solution procedures based on the exact solution of the strong cubic Duffing?s equation are shown. Using the Jacobi elliptic functions the elliptic-Krylov-Bogolubov (EKB), the homotopy perturbation and the elliptic-Galerkin (EG) methods are described. The methods are compared. The advantages and the disadvantages of the methods are discussed.

scholarly journals Approximate Solution of Nonlinear Integral Equations of the Second Kind by Using Homotopy Perturbation Method

2017 ◽  
Vol 65 (2) ◽  
pp. 151-155
Author(s):  
MM Hasan ◽  
MA Matin
Keyword(s):  
Integral Equation ◽  
Exact Solution ◽  
Approximate Solution ◽  
Perturbation Method ◽  
Fredholm Integral Equation ◽  
Homotopy Perturbation Method ◽  
Nonlinear Integral Equations ◽  
Numerical Examples ◽  
Homotopy Perturbation ◽  
Nonlinear Fredholm Integral Equation

In this paper, we apply Homotopy perturbation method (HPM) for obtaining approximate solution of nonlinear Fredholm integral equation of the second kind. Finally, some numerical examples are provided, and the obtained numerical approximations are compared with the corresponding exact solution. Dhaka Univ. J. Sci. 65(2): 151-155, 2017 (July)

Investigation of third-order nonlinear dynamical X-ray diffraction based on a new exact solution

2020 ◽  
Vol 27 (4) ◽  
pp. 941-952
Author(s):  
Minas K. Balyan
Keyword(s):  
Exact Solution ◽  
Elliptic Functions ◽  
Jacobi Elliptic Functions ◽  
X Ray Diffraction ◽  
Nonlinear Case ◽  
Elementary Functions ◽  
Third Order ◽  
Nonlinear Dynamical ◽  
X Ray ◽  
Free Parameters

Third-order nonlinear two-wave dynamical X-ray diffraction in a crystal is considered. For the Laue symmetrical case of diffraction a new exact solution is obtained. The solution is presented via Jacobi elliptic functions. Two input free parameters are essential: the deviation parameter from the Bragg exact angle and the intensity of the incident wave. It is shown that the behavior of the field inside the crystal is determined by the sign of a certain combination of these parameters. For negative and positive signs of this combination, the wavefield is periodic and the nonlinear Pendellösung effect takes place. For the nonlinear Pendellösung distance the appropriate expressions are obtained. When the above-mentioned combination is zero, the behavior of the field can be periodic as well as non-periodic and the solution is presented by elementary functions. In the nonperiodic case, the nonlinear case Pendellösung distance tends to infinity. The wavefield diffracts and propagates in a medium, whose susceptibility is modulated by the amplitudes of the wavefields. The behavior of the wavefield can be described also by an effective deviation from the Bragg exact angle. This deviation is also a function of the wavefields.

scholarly journals Exact solution of a difference approximation to Duffing's equation

1981 ◽  
Vol 23 (1) ◽  
pp. 64-77 ◽  
Author(s):  
Renfrey B. Potts
Keyword(s):  
Exact Solution ◽  
Closed Form ◽  
Periodic Solutions ◽  
Difference Equations ◽  
Difference Approximation ◽  
Linear Difference Equations ◽  
Duffing's Equation ◽  
Duffing’S Equation ◽  
Non Linear

AbstractDuffing's equation, in its simplest form, can be approximated by various non-linear difference equations. It is shown that a particular choice can be solved in closed form giving periodic solutions.

scholarly journals Approximate solution for fractional attractor one-dimensional Keller-Segel equations using homotopy perturbation sumudu transform method

2020 ◽  
Vol 9 (1) ◽  
pp. 370-381
Author(s):  
Dinkar Sharma ◽  
Gurpinder Singh Samra ◽  
Prince Singh
Keyword(s):  
Approximate Solution ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Simple Technique ◽  
Nonlinear Partial Differential Equations ◽  
Transform Method ◽  
Present Scheme ◽  
One Dimensional ◽  
Homotopy Perturbation ◽  
Sumudu Transform

AbstractIn this paper, homotopy perturbation sumudu transform method (HPSTM) is proposed to solve fractional attractor one-dimensional Keller-Segel equations. The HPSTM is a combined form of homotopy perturbation method (HPM) and sumudu transform using He’s polynomials. The result shows that the HPSTM is very efficient and simple technique for solving nonlinear partial differential equations. Test examples are considered to illustrate the present scheme.

scholarly journals Second Approximate Solution of Duffing Equation with Strong Nonlinearity by Homotopy Perturbation Method

1970 ◽  
Vol 30 ◽  
pp. 59-75
Author(s):  
M Alhaz Uddin ◽  
M Abdus Sattar
Keyword(s):  
Approximate Solution ◽  
Nonlinear Systems ◽  
Perturbation Method ◽  
Differential System ◽  
Homotopy Perturbation Method ◽  
Approximate Solutions ◽  
Second Order ◽  
Strong Nonlinearity ◽  
Analytical Technique ◽  
Homotopy Perturbation

 In this paper, the second order approximate solution of a general second order nonlinear ordinary differential system, modeling damped oscillatory process is considered. The new analytical technique based on the work of He’s homotopy perturbation method is developed to find the periodic solution of a second order ordinary nonlinear differential system with damping effects. Usually the second or higher order approximate solutions are able to give better results than the first order approximate solutions. The results show that the analytical approximate solutions obtained by homotopy perturbation method are uniformly valid on the whole solutions domain and they are suitable not only for strongly nonlinear systems, but also for weakly nonlinear systems. Another advantage of this new analytical technique is that it also works for strongly damped, weakly damped and undamped systems. Figures are provided to show the comparison between the analytical and the numerical solutions. Keywords: Homotopy perturbation method; damped oscillation; nonlinear equation; strong nonlinearity. GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 30 (2010) 59-75  DOI: http://dx.doi.org/10.3329/ganit.v30i0.8504

Minimum Energy Control of a Unicycle Model Robot

2021 ◽  
Author(s):  
Youngjin Kim ◽  
Tarunraj Singh
Keyword(s):  
Optimal Control ◽  
Minimum Energy ◽  
Kinematic Model ◽  
Elliptic Functions ◽  
Terminal Point ◽  
Jacobi Elliptic Functions ◽  
Differential Drive ◽  
Terminal Constraints ◽  
Differential Drive Robot ◽  
Point To Point

Abstract Point-to-point path planning for a kinematic model of a differential-drive wheeled mobile robot (WMR) with the goal of minimizing input energy is the focus of this work. An optimal control problem is formulated to determine the necessary conditions for optimality and the resulting two point boundary value problem is solved in closed form using Jacobi elliptic functions. The resulting nonlinear programming problem is solved for two variables and the results are compared to the traditional shooting method to illustrate that the Jacobi elliptic functions parameterize the exact profile of the optimal trajectory. A set of terminal constraints which lie on a circle in the first quadrant are used to generate a set of optimal solutions. It is noted that for maneuvers where the angle of the vector connecting the initial and terminal point is greater than a threshold, which is a function of the radius of the terminal constraint circle, the robot initially moves into the third quadrant before terminating in the first quadrant. The minimum energy solution is compared to two other optimal control formulations: (1) an extension of the Dubins vehicle model where the constant linear velocity of the robot is optimized for and (2) a simple turn and move solution, both of whose optimal paths lie entirely in the first quadrant. Experimental results are used to validate the optimal trajectories of the differential-drive robot.

scholarly journals Approximate Method for Solving the Linear Fuzzy Delay Differential Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
S. Narayanamoorthy ◽  
T. L. Yookesh
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Approximate Solution ◽  
Differential Equations ◽  
Approximate Method ◽  
Delay Differential Equations ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Adomian Decomposition ◽  
Delay Differential

We propose an algorithm of the approximate method to solve linear fuzzy delay differential equations using Adomian decomposition method. The detailed algorithm of the approach is provided. The approximate solution is compared with the exact solution to confirm the validity and efficiency of the method to handle linear fuzzy delay differential equation. To show this proper features of this proposed method, numerical example is illustrated.

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