Journal of Applied and Computational Mechanics
Fractal Variational Theory for Chaplygin-He Gas in a Microgravity Condition
Document Type : Research Paper
Authors
School of Mathematic and Information Science, Henan Polytechnic University, Jiaozuo, Henan 454000, P. R. China
Abstract
On the microgravity condition, gravity affects the motion of objects and the flow of fluids, and the continuum assumption is not valid, therefore, a fractal Chaplygin-He gas model is developed by a new fractal derivative in microgravity space. A fractal variational principle is successfully established via the fractal semi-inverse method.
Keywords
- Fractal derivative
- Chaplygin-He gas model
- Microgravity space
- Fractal variational theory
- Fractal semi-inverse method
Main Subjects
[1] Liu,Y., Li, G.H., Kallio, S., Hydrodynamic modeling of dense gas-particle turbulence flows under microgravity space environments, Microgravity Science and Technology,23(1),2011, 1-11.
[2] He, J.H., A fractal variational theory for one-dimensional compressible flow in a microgravity space, Fractals, 28(2), 2020, 2050024.
[3] He, J.H., A Tutorial Review on Fractal Spacetime and Fractional Calculus, International Journal of Theoretical Physics, 53(11), 2014, 3698-3718.
[4] He, J.H. et al., A new fractional derivative and its application to explanation of polar bear hairs, Journal of King Saud University Science, 28(2), 2016, 190-192.
[5] He, J.H., A new fractal derivation, Thermal Science,15(1), 2011, S145-S147.
[6] Wang, Q. L., Fractal calculus and its application to explanation of biomechanism of polar bear hairs, Fractals, 27(5), 2019, 1992001.
[7] Wang, Y., Deng, Q.G., Fractal derivative model for tsunami travelling, Fractals, 27(1), 2019, 1950017.
[8] Wang, Y., A fractal derivative model for snow’s thermal insulation property, Thermal Science,23(4), 2019, 2351-2354.
[9] He, J.H., A short review on analytical methods for to a fully fourth-order nonlinear integral boundary value problem with fractal derivatives, International Journal of Numerical Methods for Heat and Fluid Flow, 2020, doi: 10.1108/HFF-01-2020-0060.
[10] He, J.H., Generalized Variational Principles for Buckling Analysis of Circular Cylinders, Acta Mechanica, 231, 2020, 899–906.
[11] He, J.H., Variational Principle for the Generalized KdV-Burgers Equation with Fractal Derivatives for Shallow Water Waves, Journal of Applied and Computational Mechanics, 6(4), 2020, 735-740.
[12] He, J.H., A simple approach to one-dimensional convection-diffusion equation and its fractional modification for E reaction arising in rotating disk electrodes, Journal of Electroanalytical Chemistry, 854, 2019, 113565.
[13] He, J.H., Lagrange Crisis and Generalized Variational Principle for 3D unsteady flow, International Journal of Numerical Methods for Heat and Fluid Flow, 2019, doi: 10.1108/HFF-07-2019-0577.
[14] He, J.H., Sun, C., A variational principle for a thin film equation, Journal of Mathematical Chemistry, 57(9), 2019, 2075–2081.
[15] Wang, K.L., He, C.H., A remark on Wang's fractal variational principle, Fractals, 27(8), 2019, 1950134.
[16] Wang, K.L., Yao, S.W., A fractal variational principle for the telegraph equation with fractal derivatives, Fractals, 28(4), 2020, 2050058.
[17] Ain, Q.T., He, J.H., On two-scale dimension and its applications, Thermal Science, 23(3B), 2019, 1707-1712.
[18] Wang, Y., et al, A short review on analytical methods for fractional equations with He’s fractional derivative, Thermal Science, 21(4), 2017, 1567-1574.
[19] He, J.H., Ain, Q.T., New promises and future challenges of fractal calculus: from two-scale Thermodynamics to fractal variational principle, Thermal Science, 24(2), 2020, 659-681.
[20] He, J.H., Ji, F.Y., Two-scale mathematics and fractional calculus for thermodynamics, Thermal Science, 23(4), 2019, 2131-2133.
[21] Li, X.J., Liu, Z., He, J.H., A fractal two-phase flow model for the fiber motion in a polymer filling process, Fractals, 2020, doi:10.1142/S0218348X20500930.
[22] Wang, K.L., Wang, K.J., He, C.H., Physical insight of local fractional calculus and its application to fractional Kdv-Burgers-Kuramoto equation, Fractals, 27(7), 2019, 1950122.
[23] He, J.H., Homotopy perturbation technique, Computer Methods in Applied Mechanics and Engineering, 178, 1999, 257-262.
[24] Yu, D.N., He, J.H, Garcia, A.G., Homotopy perturbation method with an auxiliary parameter for nonlinear oscillators, Journal of Low Frequency Noise, Vibration and Active Control, 38, 2019, 3-4.
[25] Sedighi, H.M., Daneshmand, F., Nonlinear transversely vibrating beams by the homotopy perturbation method with an auxiliary term, Journal of Applied and Computational Mechanics, 1(1), 2015, 1-9.
[26] Wang, K.L., Yao, S.W., Numerical method for fractional Zakharov-Kuznetsov equation with He’s fractional derivative, Thermal Science, 23(4), 2019, 2163-2170.
[27] Sedighi, H.M., Shirazi, K.H., Using homotopy analysis method to determine profile for disk cam by means of optimization of dissipated energy, International Review of Mechanical Engineering, 5(5), 2011, 941-946.
[28] Ravichandran, C., Valliammal, N., Nieto, J.J., New results on exact controllability of a class of fractional neutral integro-differential systems with state-dependent delay in Banach spaces, Journal of the Franklin Institute, 356(3), 2019, 1535-1565.
[29] He, C.H., et al., Taylor series solution for fractal Bratu-type equation arising in electrospinning process, Fractals, 2019 doi: 10.1142/S0218348X20500115.
[30] He, J.H., Ji, F.Y., Taylor series solution for Lane-Emden equation, Journal of Mathematical Chemistry, 57(8), 2019, 1932–1934.
[31] He, J.H., Taylor series solution for a third order boundary value problem arising in architectural engineering, Ain Shams Engineering Journal, 2020, doi: 10.1016/j.asej.2020.01.016.
[32] He, C.H., Shen, Y., Ji, F.Y., He, J.H., Taylor series solution for fractal Bratu-type equation arising in electrospinning process, Fractals, 28(1), 2020, 2050011.
[33] Wang, K.L., Wang, K.J., A modification of the reduced differential transform method for fractional calculus, Thermal Science, 22(4), 2018, 1871-1875.
[34] Nadeem,M., Li, F.Q., He-Laplace method for nonlinear vibration systems and nonlinear wave equation, Journal of Low Frequency Noise, Vibration and Active Control,38, 2019, 1060-1074.
[35] Nadeem, M., Li, F.Q. , Ahmad, H., Modified Laplace variational iteration method for solving fourth-order parabolic partial differential equation with variable coefficients, Computer and Mathematics with applications,78(6), 2019, 2052-2062.
[36] Kumar, S., A new fractional modeling arising in engineering sciences and its analytical approximate solution, Alexandria Engineering Journal,52, 2013, 813-819.
[37] He, J.H., Latifizadeh, H., A general numerical algorithm for nonlinear differential equations by the variational iteration method, International Journal of Numerical Methods for Heat and Fluid Flow, 2020, doi:10.1108/HFF-01-2020-0029.
[38] Ahmad, H., Khan, T.A., Variational iteration algorithm-I with an auxiliary parameter for wave-like vibration equations, Journal of Low Frequency Noise, Vibration and Active Control, 38(3-4), 2019, 1113-1124.
[39] Ahmad, H., Seadawy, A.R., Khan, T.A., Study on numerical solution of dispersive water wave phenomena by using a reliable modification of variational iteration algorithm, Mathematics and Computers in Simulation, 17, 2020, 13-23
[40] Nadeem, M., Li, F.Q., Ahmad, H. Modified Laplace variational iteration method for solving fourth-order parabolic partial differential equation with variable coefficients, Computers and Mathematics with Applications, 78, 2019, 2052-2062.
[41] Sedighi, H.M., Shirazi, K.H., Bifurcation analysis in hunting dynamical behavior in a railway bogie: Using novel exact equivalent functions for discontinuous nonlinearities, Scientia Iranica, 19(6), 2012, 1493-1501.
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