Abstract
Scale or dimension is critical for all physical laws, and different scales might result in distinct and sometimes contradicting laws for the same phenomenon. In this study, the idea of the two-scale fractal is utilized to model the time-fractional tsunami wave traveling on an unsmooth surface. Mass and momentum conservation equations are established in a fractal space. To covert the considered fractional model into a differential equation, the fractional complex transform is used, then He’s variational iteration method is adopted to solve the resultant equation. The fractal model helps in studying the dynamic behavior of tsunami wave and their prevention through boundary control. The present study sheds new light on two-scale fluid mechanics.
Similar content being viewed by others
References
Ain, Q.T., He, J.H.: On two-scale dimension and its applications. Therm. Sci. 23, 1707–1712 (2019)
Ain, Q.T., He, J.H., Anjum, N., Ali, M.: The fractional complex transform: a novel approach to the time-fractional Schrodinger equation. Fractals (2020). https://doi.org/10.1142/S0218348X20501418
Ali, M., Anjum, N., Ain, Q.T., et al.: Homotopy perturbation method for the attachment oscillator arising in nanotechnology. Fibers Polym. (2021). https://doi.org/10.1007/s12221-021-0844-x
Anjum, N., He, J.H.: Laplace transform: making the variational iteration method easier. Appl. Math. Lett. 92, 134–138 (2019)
Anjum, N., Suleman, M., et al.: Numerical iteration for nonlinear oscillators by Elzaki transform. J. Low Freq. Noise Vib. Active Control (2019). https://doi.org/10.1177/1461348419873470
Anjum, N., He, J.H.: Nonlinear dynamic analysis of vibratory behavior of a graphene nano/microelectromechanical system. Math. Methods Appl. Sci. (2020a). https://doi.org/10.1002/mma.6699
Anjum, N., He, J.H.: Analysis of nonlinear vibration of nano/microelectromechanical system switch induced by electromagnetic force under zero initial conditions. Alex. Eng. J. (2020b). https://doi.org/10.1016/j.aej.2020.07.039
Anjum, N., Ain, Q.T.: Application of He’s fractional derivative and fractional complex transform for time fractional Camassa-Holm equation. Therm. Sci. 24(5A), 3023–3030 (2020)
Donadio, C., Paliaga, G., Radke, J.D.: Tsunamis and rapid coastal remodeling: linking energy and fractal dimension. Prog. Phys. Geogr. Earth Environ. 44(4), 550–571 (2019)
He, J.H.: Variational iteration method: a kind of nonlinear analytical technique: some examples. Int. J. Nonlinear Mech. 34(4), 699–708 (1999)
He, J.H., Li, Z.B.: Converting fractional differential equations into partial differential equations. Therm. Sci. 16, 331–334 (2012)
He, J.H., Elagan, S.K., Li, Z.B.: Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus. Phys. Lett. A 376, 257–259 (2012)
He, J.H., Li, Z.B., Wang, Q.: A new fractional derivative and its application to explanation of polar bear hairs. J. King Saud Univ. Sci. 28, 190–192 (2016)
He, J.H.: Fractal calculus and its geometrical explanation. Results Phys. 10, 272–276 (2018)
He, J.H.: Lagrange crisis and generalized variational principle for 3D unsteady flow. Int. J. Numer. Methods Heat Fluid Flow 0961–5539 (2019)
He, J.H., Ain, Q.T.: New promises and future challenges of fractal calculus: from two-scale thermodynamics to fractal variational principle. Therm. Sci. 24, 659–685 (2020)
He, J.H., El-Dib, Y.O.: Periodic property of the time-fractional Kundu–Mukherjee–Naskar equation. Results Phys. 19, 103345 (2020)
He, J.H.: A fractal variational theory for one-dimensional compressible flow in a microgravity space. Fractals 28(2), 2050024 (2020). https://doi.org/10.1142/S0218348X20500243
He, J.H., El-Dib, Y.O.: The enhanced homotopy perturbation method for axial vibration of strings. Facta Univ. Mech. Eng. (2021). https://doi.org/10.22190/FUME210125033H
He, C.H., Liu, C., Gepreel, K.A.: Low frequency property of a fractal vibration model for a concrete beam. Fractals (2021a). https://doi.org/10.1142/S0218348X21501176
He, C.H., Liu, C., He, J.H., Shirazi, A.H., Sedighi, H.M.: Passive Atmospheric water harvesting utilizing an ancient Chinese ink slab and its possible applications in modern architecture. Facta Univ. Mech. Eng. (2021b). https://doi.org/10.22190/FUME201203001H
He, J.H., Kou, S.J., He, C.H., et al.: Fractal oscillation and its frequency-amplitude property. Fractals (2021c). https://doi.org/10.1142/S0218348X2150105X
He, J.H., Hou, W.-F., Qie, N., Gepreel, K.A., Shirazi, A.H., Sedighi, H.M.: Hamiltonian-based frequency-amplitude formulation for nonlinear oscillators. Facta Univ. Ser. Mech. Eng. (2021d). https://doi.org/10.22190/FUME201205002H
He, J.H., Qie, N., He, C.H.: Solitary waves travelling along an unsmooth boundary. Results Phys. Article Number: 104104 (2021e)
Mandelbrot, B.B.: The Fractal Geometry of Nature. W. H. Freeman and Company, New York (1982)
Mohebbi, A., Saffarian, M.: Implicit RBF meshless method for the solution of 2-D variable order fractional cable equation. J. Appl. Comput. Mech. 6(2), 235–247 (2020)
Tian, D., Ain, Q.T., Anjum, N.: Fractal N/MEMS: from pull-in instability to pull-in stability. Fractals (2020). https://doi.org/10.1142/S0218348X21500304
Wang, Y., An, J.Y.: Amplitude–frequency relationship to a fractional Duffing oscillator arising in microphysics and tsunami motion. J. Low Freq. Noise Vib. Active Control 38, 1008–1012 (2019)
Wang, Y., An, J.Y., Wang, X.: A variational formulation for anisotropic wave traveling in a porous medium. Fractals 27, 1950047 (2019)
Wang, Y., Deng, Q.: Fractal derivative model for tsunami traveling. Fractals 27, 1950017 (2019)
Wang, K.J.: A new fractional nonlinear singular heat conduction model for the human head considering the effect of febrifuge. Eur. Phys. J. plus. 135, 871 (2020). https://doi.org/10.1140/epjp/s13360-020-00891-x
Wang, K.J.: Variational principle and approximate solution for the generalized Burgers-Huxley equation with fractal derivative. Fractals 2150044 (2021). https://doi.org/10.1142/S0218348X21500444.
Wang, K.J., Wang, G.D.: Variational principle and approximate solution for the fractal generalized Benjamin–Bona–Mahony–Burgers equation in fluid mechanics. Fractals 2150075 (2021). https://doi.org/10.1142/S0218348X21500754
Zenkour, A., Abouelregal, A.: Fractional thermoelasticity model of a 2-D problem of Mode-I crack in a fibre-reinforced thermal environment. J. Appl. Comput. Mech. 5(2), 269–280 (2019)
Zuo, Y.-T.: A gecko-like fractal receptor of a three-dimensional printing technology: a fractal oscillator. J. Math. Chem. (2021). https://doi.org/10.1007/s10910-021-01212-y
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors have no conflicts of interest to declare that are relevant to the content of this article.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Anjum, N., Ain, Q.T. & Li, XX. Two-scale mathematical model for tsunami wave. Int J Geomath 12, 10 (2021). https://doi.org/10.1007/s13137-021-00177-z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s13137-021-00177-z