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Journal ArticleDOI

Approximate Solution of Homogeneous and Nonhomogeneous 5αth-Order Space-Time Fractional KdV Equations

01 Feb 2021-International Journal of Computational Methods (World Scientific Publishing Company)-Vol. 18, Iss: 01, pp 2050018
TL;DR: In this paper, the semi-inverse method is applied to derive the Lagrangian of the 5αth Korteweg de Vries equation (KdV).
Abstract: In this paper, the semi-inverse method is applied to derive the Lagrangian of the 5αth Korteweg de Vries equation (KdV). Then the time and space differential operators of the Lagrangian are replace...
Citations
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TL;DR: In this article , the authors established a variational principle for a fractal nano/microelectromechanical (N/MEMS) system by using the semi-inverse method.
Abstract: Purpose The variational principle views a complex problem in an energy way, it gives good physical understanding of an iteration method, and the variational-based numerical methods always have a conservation scheme with a fast convergent rate. The purpose of this paper is to establish a variational principle for a fractal nano/microelectromechanical (N/MEMS) system. Design/methodology/approach This paper begins with an approximate variational principle in literature for the studied problem, and a genuine variational principle is obtained by the semi-inverse method. Findings The semi-inverse method is a good mathematical tool to the search for a genuine fractal variational formulation for the N/MEMS system. Research limitations/implications The established variational principle can be used for both analytical and numerical analyses of the N/MEMS systems, and it can be extended to some more complex cases. Practical implications The variational principle can be used for variational-based finite element methods and energy-based analytical methods. Originality/value The new and genuine variational principle is obtained. This paper discovers the missing piece of the puzzle for the establishment of a variational principle from governing equations for a complex problem by the semi-inverse method. The new variational theory opens a new direction in fractal MEMS systems.

30 citations

Journal ArticleDOI
TL;DR: In this article , the authors designed granular metamaterials considering the granular structures of discrete particles which are different from elastic metammaterials consisting of continuous media, and formulated the fractional granular equation directly in a precompressed spherical chain adopting Hertz law and long wave approximation theory.
Abstract: The present article designs the granular metamaterials considering the granular structures of discrete particles which are different from elastic metamaterials consisting of continuous media. In granular metamaterials, the wave propagates through contact with neighboring particles. To identify the propagating properties of wave quantities in the rough granular medium the fractional granular equation is formulated directly in a pre-compressed spherical chain adopting Hertz law and long wave approximation theory. Using phase and group velocities, Caputo fractional derivatives are used to illustrate normal and anomalous dispersion wave dependence. To demonstrate in depth the dynamical behavior of the wave profile, various types of complex solutions like multi-shock, multi-solitons, lump, and breather solutions of the one-dimensional time fractional granular equations are explored employing Hirota’s bilinear approach. Finally, the more complicated hybrid solutions such as kink with the lump, soliton with the lump, etc. are exhibited from numerical understanding. The numerical graphs and figures demonstrate the crucial role of the order of derivative (roughness parameter) in the formation of different types of soliton solutions.
References
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Journal ArticleDOI
Ji-Huan He1
TL;DR: In this paper, the homotopy perturbation technique does not depend upon a small parameter in the equation and can be obtained uniformly valid not only for small parameters, but also for very large parameters.

3,058 citations

Journal ArticleDOI
TL;DR: In this article, the Green's function of fractional diffusion is shown to be a probability density and the corresponding Green's functions are obtained in closed form for arbitrary space dimensions in terms of Fox functions and their properties are exhibited.
Abstract: Diffusion and wave equations together with appropriate initial condition(s) are rewritten as integrodifferential equations with time derivatives replaced by convolution with tα−1/Γ(α), α=1,2, respectively. Fractional diffusion and wave equations are obtained by letting α vary in (0,1) and (1,2), respectively. The corresponding Green’s functions are obtained in closed form for arbitrary space dimensions in terms of Fox functions and their properties are exhibited. In particular, it is shown that the Green’s function of fractional diffusion is a probability density.

1,046 citations

Journal ArticleDOI
TL;DR: In this article, a simple homotopy is constructed by the modified Lindstedt-Poincare method, by the solution and the coefficient of linear term are expanded into series of the embedding parameter.
Abstract: A simple homotopy is constructed, by the modified Lindstedt-Poincare method(He,J.H. International Journal of Non-Linear Mechanics , 37, 2002, 309-314 ), the solution and the coefficient of linear term are expanded into series of the embedding parameter. Only one iteration leads to accurate solution.

907 citations

Journal ArticleDOI
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.

643 citations