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 Approximate representations of shaped pulses using the homotopy analysis method

2021 ◽  
Vol 2 (1) ◽  
pp. 175-186
Author(s):  
Timothy Crawley ◽  
Arthur G. Palmer III
Keyword(s):  
Magnetic Resonance ◽  
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 hyperbolic-secant pulses. The homotopy analysis method is used to obtain new approximate solutions to the Riccati equation for shaped radiofrequency pulses in nuclear magnetic resonance (NMR) spectroscopy. The results of even relatively low orders of approximation are highly accurate and can be calculated very efficiently. The results are extended in a second application of the homotopy analysis method to represent relaxation as a perturbation of the magnetization trajectory calculated in the absence of relaxation. The homotopy analysis method is powerful and flexible and is likely to have other applications in magnetic resonance.

scholarly journals Constructing analytic solutions on the Tricomi equation

Open Physics ◽  
2018 ◽  
Vol 16 (1) ◽  
pp. 143-148 ◽  
Author(s):  
Emran Khoshrouye Ghiasi ◽  
Reza Saleh
Keyword(s):  
Homotopy Analysis Method ◽  
Series Solution ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Analytic Solutions ◽  
Analysis Method ◽  
Auxiliary Parameter ◽  
Tricomi Equation ◽  
Homotopy Analysis ◽  
Optimal Values

AbstractIn this paper, homotopy analysis method (HAM) and variational iteration method (VIM) are utilized to derive the approximate solutions of the Tricomi equation. Afterwards, the HAM is optimized to accelerate the convergence of the series solution by minimizing its square residual error at any order of the approximation. It is found that effect of the optimal values of auxiliary parameter on the convergence of the series solution is not negligible. Furthermore, the present results are found to agree well with those obtained through a closed-form equation available in the literature. To conclude, it is seen that the two are effective to achieve the solution of the partial differential equations.

scholarly journals Usage of the homotopy analysis method for solving fractional Volterra-Fredholm integro-differential equation of the second kind

2018 ◽  
Vol 49 (4) ◽  
pp. 301-315 ◽  
Author(s):  
Ahmen Hamoud ◽  
Kirtiwant Ghadle
Keyword(s):  
Differential Equation ◽  
Homotopy Analysis Method ◽  
Existence And Uniqueness ◽  
Approximate Solutions ◽  
Analytical Approximation ◽  
Analysis Method ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Computational Work ◽  
Homotopy Analysis

The reliability of the homotopy analysis method (HAM) and reduction in the size of the computational work give this method a wider applicability. In this paper, HAM has been successfully applied to find the approximate solutions of Caputo fractional Volterra-Fredholm integro-differential equations. Also, the behavior of the solution can be formally determined by analytical approximation. Moreover, the study proves the existence and uniqueness results and the convergence of the solution. This paper concludes with an example to demonstrate the validity and applicability of the proposed technique.

The Analysis Approach of Boundary Layer Equations of Power-Law Fluids of Second Grade

2008 ◽  
Vol 63 (9) ◽  
pp. 564-570 ◽  
Author(s):  
Saeid Abbasbandy ◽  
Muhammet Yürüsoy ◽  
Mehmet Pakdemirli
Keyword(s):  
Power Law ◽  
Homotopy Analysis Method ◽  
Nonlinear Problems ◽  
Analytic Solutions ◽  
Second Grade ◽  
Analysis Method ◽  
Boundary Layer Equations ◽  
Power Law Fluids ◽  
Homotopy Analysis ◽  
Constant Coefficients

A powerful analytic technique for nonlinear problems, the homotopy analysis method (HAM), is employed to give analytic solutions of power-law fluids of second grade. For the so-called secondorder power-law fluids, the explicit analytic solutions are given by recursive formulas with constant coefficients. Also, for the real power-law index in a quite large range an analytic approach is proposed. It is demonstrated that the approximate solution agrees well with the finite difference solution. This provides further evidence that the homotopy analysis method is a powerful tool for finding excellent approximations to nonlinear equations of the power-law fluids of second grade.

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