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.

scholarly journals Homotopy Analysis Method for the First Order Fuzzy Volterra-Fredholm Integro-differential Equations

2018 ◽  
Vol 11 (3) ◽  
pp. 857 ◽  
Author(s):  
Ahmed A. Hamoud ◽  
Kirtiwant P. Ghadle
Keyword(s):  
Differential Equation ◽  
Homotopy Analysis Method ◽  
Existence And Uniqueness ◽  
Analysis Method ◽  
First Order ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Homotopy Analysis ◽  
Fuzzy Solution ◽  
Parametric Case

A fuzzy Volterra-Fredholm integro-differential equation (FVFIDE) in a parametric case is converted to its related crisp case.  We use homotopy analysis method to find the approximate solution of this system and hence obtain an approximation for the fuzzy solution of the  FVFIDE. This paper discusses existence and uniqueness results and convergence of the proposed method.

scholarly journals Application of Extended Homotopy Analysis Method to the Two-Degree-of-Freedom Coupled van der Pol-Duffing Oscillator

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Y. H. Qian ◽  
S. M. Chen ◽  
L. Shen
Keyword(s):  
Homotopy Analysis Method ◽  
Duffing Oscillator ◽  
Approximate Solutions ◽  
Analytical Approximation ◽  
Degree Of Freedom ◽  
Analysis Method ◽  
Van Der Pol ◽  
Different Types ◽  
Homotopy Analysis ◽  
Two Degree Of Freedom

The extended homotopy analysis method (EHAM) is presented to establish the analytical approximate solutions for two-degree-of-freedom (2-DOF) coupled van der Pol-Duffing oscillator. Meanwhile, the comparisons between the results of the EHAM and standard Runge-Kutta numerical method are also presented. The results demonstrate that the analytical approximate solutions of the EHAM agree well with the numerical integration solutions. For EHAM as an analytical approximation method, we are not sure whether it can apply to all of the nonlinear systems; we can only verify its effectiveness through specific cases. As a result of the existence of nonlinear terms, we must study different types of systems, no matter from the complication of calculation and physical significance.

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 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 Approximate Solutions of Nonlinear Partial Differential Equations by Modifiedq-Homotopy Analysis Method

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Shaheed N. Huseen ◽  
Said R. Grace
Keyword(s):  
Differential Equations ◽  
Homotopy Analysis Method ◽  
Series Solution ◽  
Nonlinear Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Approximate Solutions ◽  
Power Series Solution ◽  
Analysis Method ◽  
Exactly Solvable ◽  
Homotopy Analysis

A modifiedq-homotopy analysis method (mq-HAM) was proposed for solvingnth-order nonlinear differential equations. This method improves the convergence of the series solution in thenHAM which was proposed in (see Hassan and El-Tawil 2011, 2012). The proposed method provides an approximate solution by rewriting thenth-order nonlinear differential equation in the form ofnfirst-order differential equations. The solution of thesendifferential equations is obtained as a power series solution. This scheme is tested on two nonlinear exactly solvable differential equations. The results demonstrate the reliability and efficiency of the algorithm developed.

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