On the fractal variational principle for the telegraph equation. (English) Zbl 1482.35005


35A15 Variational methods applied to PDEs
35A08 Fundamental solutions to PDEs
35R02 PDEs on graphs and networks (ramified or polygonal spaces)
35R11 Fractional partial differential equations
28A80 Fractals
Full Text: DOI


[1] Abdusalam, H. A. and Abdusalam, H. A., Exact analytic solution of the simplified telegraph model of propagation and dissipation of excitation fronts, Int. J. Theor. Phys.43(4) (2004) 1161-1167. · Zbl 1175.92010
[2] Ye, J., Hu, B. S., Jin, Y.et al., Interface engineering integrates fractal-tree structured nitrogendoped graphene/carbon nanotubes for supercapacitors, Electrochim. Acta349 (2020) 136372.
[3] Li, X. X., Tian, D., He, C. H.et al., A fractal modification of the surface coverage model for an electrochemical arsenic sensor, Electrochim. Acta296 (2019) 491-493.
[4] Kumar, A., Bhardwaj, A. and Dubey, S., A local meshless method to approximate the time-fractional telegraph equation, Eng. Comput. (2020), https://doi.org/10.1007/s00366-020-01006-x.
[5] Sweilam, N. H., Nagy, A. M. and El-Sayed, A. A., Solving time-fractional order telegraph equation via Sinc-Legendre collocation method, Mediterr. J. Math.13(6) (2016) 5119-5133. · Zbl 1349.35410
[6] Wang, K. L., Yao, S. W., Liu, Y. P.et al., A fractal variational principle for the Telegraph equation with fractal derivatives, Fractals28(4) (2020) 2050058.
[7] Borodich, F. M., Fractals and fractal scaling in fracture mechanics, Int. J. Fract.95 (1999) 239-259.
[8] Yavari, A., Sarkani, S. and Moyer, E. T., The mechanics of self-similar and self-affine fractal cracks, Int. J. Fract.114(1) (2002) 1-27.
[9] Tanaka, M., The fractal dimension of grain-boundary fracture in high-temperature creep of heat-resistant alloys, J. Mater. Sci.28(21) (1993) 5753-5758.
[10] Miao, T. J., Cheng, S. J., Chen, A. M.et al., Seepage properties of rock fractures with power law length distribution, Fractals27(4) (2019) 1950057.
[11] Pande, C. S., Richards, L. R. and Smith, S., Fractal characteristics of fractured surfaces, J. Mater. Sci. Lett.6(3) (1987) 295-297.
[12] Bona, A. D., Hill, T. J. and Mecholsky, J. J., The effect of contour angle on fractal dimension measurements for brittle materials, J. Mater. Sci.36(11) (2001) 2645-2650.
[13] Lisovskii, F. V., Lukashenko, L. I. and Mansvetova, E. G., Thermodynamically stable fractal-like domain structures in magnetic films, JETP Lett.79(7) (2004) 352-354.
[14] Mcqueen, P. G., Physics and fractal structures, J. Stat. Phys.86(5-6) (1997) 1397-1398.
[15] Yang, S. S., Fu, H. H. and Yu, B. M., Fractal analysis of flow resistance in tree-like branching networks with roughened microchannels, Fractals25(1) (2017) 1750008.
[16] Wu, Y. K. and Liu, Y., Fractal-like multiple jets in electrospinning process, Therm. Sci.24(4) (2020) 2499-2505.
[17] Grassberger, P., On the Hausdorff dimension of fractal attractors, J. Stat. Phys.26(1) (1981) 173-179.
[18] Sidletskii, V. A. and Kolupaev, B. B., Application of the fractal approach to determination of the poisson coefficient of polymeric systems, J. Eng. Phys. Thermophys.76 (2003) 937-941.
[19] Altaiski, M. V. and Sidharth, B. G., Quantization of fractal systems: One-particle excitation states, Int. J. Theor. Phys.34(12) (1995) 2343-2351. · Zbl 0843.58055
[20] Stanley, H. E., Application of fractal concepts to polymer statistics and to anomalous transport in randomly porous media, J. Stat. Phys.36(5-6) (1984) 843-860.
[21] Silva, L. R. D., Fractal dimension at the phase transition of inhomogeneous cellular automata, J. Stat. Phys.53(3-4) (1988) 985-990.
[22] Leonenko, N. and Vaz, J. Jr., Spectral analysis of fractional hyperbolic diffusion equations with random data, J. Stat. Phys.179(1) (2020) 155-175. · Zbl 1436.35007
[23] Pande, C. S., Richards, L. R. and Smith, S., Fractal characteristics of fractured surfaces, J. Mater. Sci. Lett.6(3) (1987) 295-297.
[24] Bona, A. D., Hill, T. J. and Mecholsky, J. J., The effect of contour angle on fractal dimension measurements for brittle materials, J. Mater. Sci.36(11) (2001) 2645-2650.
[25] Saouma, V. E. and Fava, G., On fractals and size effects, Int. J. Fract.137(1-4) (2006) 231-249. · Zbl 1197.74176
[26] Polyakov, V. V. and Kucheryavskii, S. V., The fractal analysis of a porous material structure, Tech. Phys. Lett.27(7) (2001) 592-593.
[27] Miao, T. J., Chen, A. M., Zhang, L. W.et al., A novel fractal model for permeability of damaged tree-like branching networks, Int. J. Heat Mass Transfer127(A) (2018) 278-285.
[28] Miao, T. J., Long, Z. C., Chen, A. M.et al., Analysis of permeabilities for slug flow in fractal porous media, Int. Commun. Heat Mass Transfer88 (2017) 194-202.
[29] Miao, T. J., Cheng, S. J., Chen, A. M.et al., Analysis of axial thermal conductivity of dual-porosity fractal porous media with random fractures, Int. J. Heat Mass Transfer102 (2016) 884-890.
[30] Xu, L. Y., Li, Y., Li, X. X.et al., Detection of cigarette smoke using a fiber membrane filmed with carbon nanoparticles and a fractal current law, Therm. Sci.24(4) (2020) 2469-2474.
[31] Yang, Z. P., Zhang, L., Dou, F.et al., A fractal model for pressure drop through a cigarette filter, Therm. Sci.24(4) (2020) 2653-2659.
[32] Yang, Z. P., Dou, F., Yu, T.et al., On the cross-section of shaped fibers in the dry spinning process: Physical explanation by the geometric potential theory, Results Phys.14 (2019) 102347.
[33] El-Nabulsi, R. A., Emergence of quasiperiodic quantum wave functions in Hausdorff dimensional crystals and improved intrinsic carrier concentrations, J. Phys. Chem. Solids127 (2019) 224-230.
[34] El-Nabulsi, R. A., Geostrophic flow and wind-driven ocean currents depending on the spatial dimensionality of the medium, Pure Appl. Geophys.176(6) (2019) 2739-2750.
[35] El-Nabulsi, R. A., Dirac equation with position-dependent mass and coulomb-like field in hausdorff dimension, Few-Body Syst.61(1) (2020) 10.
[36] El-Nabulsi, R. A., Spectrum of Schrödinger Hamiltonian operator with singular inverted complex and Kratzer’s molecular potentials in fractional dimensions, Eur. Phys. J. Plus133(7) (2018) 277.
[37] El-Nabulsi, R. A., Modifications at large distances from fractional and fractal arguments, Fractals18(2) (2010) 185-190. · Zbl 1196.28014
[38] El-Nabulsi, R. A., Path integral formulation of fractionally perturbed Lagrangian oscillators on fractal, J. Stat. Phys.172(6) (2018) 1617-1640. · Zbl 1401.83005
[39] Wang, Y., An, J. and Wang, X., A variational formulation for anisotropic wave travelling in a porous medium, Fractals27(4) (2019) 1950047.
[40] Wang, K. L. and He, C. H., A remark on Wang’s fractal variational principle, Fractals27(8) (2019) 1950134. · Zbl 1434.35267
[41] Shen, Y. and He, J. H., Variational principle for a generalized KdV equation in a fractal space, Fractals28(4) (2020) 2050069. · Zbl 1441.35009
[42] He, J. H., A fractal variational theory for one-dimensional compressible flow in a microgravity space, Fractals28(2) (2020) 2050024.
[43] He, J. H. and Ain, Q. T., New promises and future challenges of fractal calculus: From two-scale Thermodynamics to fractal variational principle, Therm. Sci.24(2A) (2020) 659-681.
[44] He, J. H., Variational principle for the generalized KdV-Burgers equation with fractal derivatives for shallow water waves, J. Appl. Comput. Mech.6(4) (2020) 735-740, https://doi.org/10.22055/JACM.2019.14813.
[45] He, J. H.. Fractal calculus and its geometrical explanation, Results Phys.10 (2018) 272-276.
[46] He, J. H. and Ji, F. Y., Two-scale mathematics and fractional calculus for thermodynamics, Therm. Sci.23(4) (2019) 2131-2133.
[47] Ain, Q. T. and He, J. H., On two-scale dimension and its applications, Therm. Sci.23(3B) (2019) 1707-1712.
[48] Ji, F.-Y., He, C.-H., Zhang, J.-J.et al., A fractal Boussinesq equation for nonlinear transverse vibration of a nanofiber-reinforced concrete pillar, Appl. Math. Model.82 (2020) 437-448. · Zbl 1481.74266
[49] He, J. H., A short review on analytical methods for to a fully fourth-order nonlinear integral boundary value problem with fractal derivatives, Int. J. Numer. Methods Heat Fluid Flow (2020), https://doi.org/10.1108/HFF-01-2020-0060.
[50] He, C. H., Shen, Y., Ji, F. Y.et al., Taylor series solution for fractal Bratu-type equation arising in electrospinning process, Fractals28(1) (2020) 2050011.
[51] He, J. H., A simple approach to one-dimensional convection-diffusion equation and its fractional modification for E reaction arising in rotating disk electrodes, J. Electroanal. Chem.854 (2019) 113565.
[52] Liu, H. Y., Yao, S. W., Yang, H. W. and Liu, J., A fractal rate model for adsorption kinetics at solid/solution interface, Therm. Sci.23(4) (2019) 2477-2480.
[53] Wang, Y., Yao, S. W. and Yang, H. W., A fractal derivative model for snow’s thermal insulation property, Therm. Sci.23(4) (2019) 2351-2354.
[54] He, J. H., Generalized variational principles for buckling analysis of circular cylinders, Acta Mech.231 (2020) 899-906. · Zbl 1434.74058
[55] He, J. H., Variational principle and periodic solution of the Kundu-Mukherjee-Naskar equation, Results Phys.17 (2020) 103031.
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