Homotopy analysis method for quadratic Riccati differential equation

2008 ◽  
Vol 13 (3) ◽  
pp. 539-546 ◽  
Author(s):  
Yue Tan ◽  
Saeid Abbasbandy
Keyword(s):  
Differential Equation ◽  
Homotopy Analysis Method ◽  
Analysis Method ◽  
Riccati Differential Equation ◽  
Homotopy Analysis

scholarly journals Approximate Representations of Shaped Pulses Using the Homotopy Analysis Method

2021 ◽  
Author(s):  
Timothy Crawley ◽  
Arthur G. Palmer III
Keyword(s):  
Differential Equation ◽  
Homotopy Analysis Method ◽  
Approximate Solutions ◽  
Analytic Solutions ◽  
Euler Angles ◽  
Analysis Method ◽  
Riccati Differential Equation ◽  
Spin Magnetization ◽  
Homotopy Analysis ◽  
Radiofrequency Pulse

Abstract. The evolution of nuclear spin magnetization during a radiofrequency pulse in the absence of relaxation or coupling interactions can be described by three Euler angles. The Euler angles in turn can be obtained from the solution of a Riccati differential equation; however, analytic solutions exist only for rectangular and chirp pulses. The Homotopy Analysis Method is used to obtain new approximate solutions to the Riccati equation for shaped radiofrequency pulses in NMR spectroscopy. The results of even relatively low orders of approximation are highly accurate and can be calculated very efficiently. The Homotopy Analysis Method is powerful and flexible and is likely to have other applications in theoretical magnetic resonance.

scholarly journals On the Solutions of Fractional Cauchy Problem Featuring Conformable Derivative

2018 ◽  
Vol 22 ◽  
pp. 01045 ◽  
Author(s):  
Mehmet Yavuz ◽  
Necati Özdemir
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Exact Solutions ◽  
Detailed Analysis ◽  
Analytical Solutions ◽  
Homotopy Analysis Method ◽  
Analysis Method ◽  
Conformable Derivative ◽  
Homotopy Analysis ◽  
Fractional Cauchy Problem

In this study, we have obtained analytical solutions of fractional Cauchy problem by using q-Homotopy Analysis Method (q-HAM) featuring conformable derivative. We have considered different situations according to the homogeneity and linearity of the fractional Cauchy differential equation. A detailed analysis of the results obtained in the study has been reported. According to the results, we have found out that our obtained solutions approach very speedily to the exact solutions.

Optimum Path of a Flying Object with Exponentially Decaying Density Medium

2009 ◽  
Vol 64 (7-8) ◽  
pp. 431-438 ◽  
Author(s):  
Said Abbasbandy ◽  
Mehmet Pakdemirli ◽  
Elyas Shivanian
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Homotopy Analysis Method ◽  
Analysis Method ◽  
Dimensionless Form ◽  
Optimum Path ◽  
Homotopy Analysis

AbstractIn this paper, a differential equation describing the optimum path of a flying object is derived. The density of the fluid is assumed to be exponentially decaying with altitude. The equation is cast in to a dimensionless form and the exact solution is given. This equation is then analyzed by homotopy analysis method (HAM). The results showed in the figures reveal that this method is very effective and convenient.

Studying Dynamic Pull-In Behavior of Microbeams by Means of the Homotopy Analysis Method

2008 ◽  
Author(s):  
Mahdi Moghimi Zand ◽  
S. Ahmad Tajalli ◽  
Mohammad Taghi Ahmadian
Keyword(s):  
Differential Equation ◽  
Homotopy Analysis Method ◽  
Nonlinear Partial Differential Equation ◽  
Nonlinear Ordinary Differential Equation ◽  
Strong Nonlinearity ◽  
Analysis Method ◽  
Solution Methods ◽  
Homotopy Analysis ◽  
Fringing Field ◽  
Fringing Field Effect

In this study, the homotopy analysis method (HAM) is used to study dynamic pull-in instability in microbeams considering different sources of nonlinearity. Electrostatic actuation, fringing field effect and midplane stretching causes strong nonlinearity in microbeams. In order to investigate dynamic pull-in behavior, using Galerkin’s decomposition method, the nonlinear partial differential equation of motion is reduced to a single nonlinear ordinary differential equation. The obtained equation is solved analytically in time domain using HAM. The problem is studied by two separate manners: direct use of HAM and indirect use of HAM in conjunction with He’s Modified Lindstedt-Poincare´ Method. To demonstrate the effectiveness of the solution methods, results are compared with those in literature. The comparison between obtained results and those available in literature shows good agreement.

scholarly journals Application of Homotopy Analysis Method to the Unsteady Squeezing Flow of a Second-Grade Fluid between Circular Plates

2010 ◽  
Vol 2010 ◽  
pp. 1-18 ◽  
Author(s):  
Mohammad Mehdi Rashidi ◽  
Abdul Majid Siddiqui ◽  
Mostafa Asadi
Keyword(s):  
Differential Equation ◽  
Homotopy Analysis Method ◽  
Nonlinear Ordinary Differential Equation ◽  
Second Grade ◽  
Squeezing Flow ◽  
Circular Plates ◽  
Analysis Method ◽  
Second Grade Fluid ◽  
Homotopy Analysis ◽  
Grade Fluid

We investigated an axisymmetric unsteady two-dimensional flow of nonconducting, incompressible second grade fluid between two circular plates. The similarity transformation is applied to reduce governing partial differential equation (PDE) to a nonlinear ordinary differential equation (ODE) in dimensionless form. The resulting nonlinear boundary value problem is solved using homotopy analysis method and numerical method. The effects of appropriate dimensionless parameters on the velocity profiles are studied. The total resistance to the upper plate has been calculated.

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