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An analytic solution of projectile motion with the quadratic resistance law using the homotopy analysis method
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In this paper, the authors considered the problem of two-dimensional projectile motion in which the resistance acting on an object moving in air is proportional to the square of the velocity of the object (quadratic resistance law).Abstract:
We consider the problem of two-dimensional projectile motion in which the resistance acting on an object moving in air is proportional to the square of the velocity of the object (quadratic resistance law). It is well known that the quadratic resistance law is valid in the range of the Reynolds number: 1 × 103 ~ 2 × 105 (for instance, a sphere) for practical situations, such as throwing a ball. It has been considered that the equations of motion of this case are unsolvable for a general projectile angle, although some solutions have been obtained for a small projectile angle using perturbation techniques. To obtain a general analytic solution, we apply Liao's homotopy analysis method to this problem. The homotopy analysis method, which is different from a perturbation technique, can be applied to a problem which does not include small parameters. We apply the homotopy analysis method for not only governing differential equations, but also an algebraic equation of a velocity vector to extend the radius of convergence. Ultimately, we obtain the analytic solution to this problem and investigate the validation of the solution.read more
Citations
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Book
Homotopy Analysis Method in Nonlinear Differential Equations
TL;DR: In this paper, a convergence series for Divergent Taylor Series is proposed to solve nonlinear initial value problems and nonlinear Eigenvalue problems with free or moving boundary in heat transfer.
Journal ArticleDOI
Notes on the homotopy analysis method: Some definitions and theorems
TL;DR: In this article, the basic ideas and current developments of the homotopy analysis method, an analytic approach to get convergent series solutions of strongly nonlinear problems, which recently attracts interests of more and more researchers, are described.
Journal ArticleDOI
An optimal homotopy-analysis approach for strongly nonlinear differential equations
TL;DR: In this paper, an optimal homotopy analysis approach is described by means of the nonlinear Blasius equation as an example, which can be used to get fast convergent series solutions of different types of equations with strong nonlinearity.
Journal ArticleDOI
Optimal homotopy asymptotic method with application to thin film flow
TL;DR: In this paper, the Optimal Homotopy Asymptotic Method (OHAM) has been applied to thin film flow of a fourth grade fluid down a vertical cylinder and the results reveal that the proposed method is very accurate, effective and easy to use.
Journal ArticleDOI
On the selection of auxiliary functions, operators, and convergence control parameters in the application of the Homotopy Analysis Method to nonlinear differential equations: A general approach
TL;DR: In this paper, the authors discuss the selection of the initial approximation, auxiliary linear operator, auxiliary function, and convergence control parameter in the application of the homotopy analysis method, in a fairly general setting.
References
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Viscous Fluid Flow
TL;DR: In this article, the stability of Laminar Boundary Layer Flow Appendices has been investigated in Cylindrical and Spherical Coordinates of Incompressible Newtonian Fluids.
Journal ArticleDOI
The application of homotopy analysis method to nonlinear equations arising in heat transfer
TL;DR: In this paper, the homotopy analysis method (HAM) is compared with the numerical and HPM in the heat transfer file and the auxiliary parameter ℏ, which provides a simple way to adjust and control the convergence region of solution series.
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A kind of approximate solution technique which does not depend upon small parameters — II. An application in fluid mechanics
TL;DR: In this paper, the homotopy analysis method was further improved by introducing a non-zero parameter into the traditional way of constructing a homhotopy, which can converge even in the whole region η ϵ [0, + ∞].
Journal ArticleDOI
The application of homotopy analysis method to solve a generalized Hirota–Satsuma coupled KdV equation
TL;DR: In this article, an analytic technique, namely the homotopy analysis method (HAM), is applied to solve a generalized Hirota-Satsuma coupled KdV equation, which provides a simple way to adjust and control the convergence region of solution series.
Journal ArticleDOI
On analytic solution for thin film flow of a fourth grade fluid down a vertical cylinder
TL;DR: In this paper, a totally analytic solution for thin film flow of a fourth grade fluid down a vertical cylinder is obtained using homotopy analysis method (HAM), and the series solution is developed and the recurrence relations are given explicitly.