Local fractal Fourier transform and applications

Khalili Golmankhaneh, Alireza; Ali, Karmina Kamal; Yilmazer, Resat; Kaabar, Mohammed Khalid Awad

doi:10.22034/cmde.2021.42554.1832

فهرست نشریات دارای اعتبار وزارت علوم، تحقیقات و فناوری

تعداد نشریات	45
تعداد شماره‌ها	1,469
تعداد مقالات	17,958
تعداد مشاهده مقاله	58,289,681
تعداد دریافت فایل اصل مقاله	19,748,685

	Local fractal Fourier transform and applications
Computational Methods for Differential Equations
مقاله 3، دوره 10، شماره 3، مهر 2022، صفحه 595-607 اصل مقاله (594.11 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22034/cmde.2021.42554.1832
نویسندگان
Alireza Khalili Golmankhaneh^* ¹؛ Karmina Kamal Ali²؛ Resat Yilmazer³؛ Mohammed Khalid Awad Kaabar⁴^{، 5}
¹Department of Physics Islamic Azad University, Urmia Branch Urmia, Iran.
²Faculty of Science, Department of Mathematics, University of Zakho, Iraq.
³Faculty of Science, Department of Mathematics, Firat University, Elazig, Turkey.
⁴Department of Mathematics and Statistics, Washington State University, Pullman, WA, USA.
⁵Institute of Mathematical Sciences, University of Malaya, 50603 Kuala Lumpur, Malaysia.
چکیده
In this manuscript, we review fractal calculus and the analogues of both local Fourier transform with its related properties and Fourier convolution theorem are proposed with proofs in fractal calculus. The fractal Dirac delta with its derivative and the fractal Fourier transform of the Dirac delta is also defined. In addition, some important applications of the local fractal Fourier transform are presented in this paper such as the fractal electric current in a simple circuit, the fractal second order ordinary differential equation, and the fractal Bernoulli-Euler beam equation. All discussed applications are closely related to the fact that, in fractal calculus, a useful local fractal derivative is a generalized local derivative in the standard calculus sense. In addition, a comparative analysis is also carried out to explain the benefits of this fractal calculus parameter on the basis of the additional alpha parameter, which is the dimension of the fractal set, such that when α = 1, we obtain the same results in the standard calculus.
کلیدواژه‌ها
Fractal calculus؛ Fractal local Fourier transform؛ Fractal differential equation؛ Fractal Fourier convolution theorem؛ Fractal Dirac delta function

مراجع
[1] A. S. Balankin, A continuum framework for mechanics of fractal materials I: From fractional space to continuum with fractal metric, Eur. Phys. J. B, 88(90) (2015). Doi:10.1140/epjb/e2015-60189-y. [2] M. T. Barlow and E. A. Perkins, Brownian motion on the Sierpinski gasket, Probab. Theory Relat. Fields, 79 (1988), 543–623. [3] M. F. Barnsley, Fractals everywhere, Academic press, 2014. [4] M. Czachor, Waves along fractal coastlines: From fractal arithmetic to wave equations, Acta Phys. Pol. B, 50 (2019), 813–831. [5] L. Debnath and D. Bhatta, Integral transforms and their applications, CRC Press, 2014. [6] R. DiMartino and W. Urbina, On Cantor-like sets and Cantor-Lebesgue singular functions, arXiv preprint arXiv:1403.6554 (2014). [7] K. Falconer, Fractal geometry: mathematical foundations and applications, John Wiley & Sons, 2004. [8] K. Falconer, Techniques in Fractal Geometry, John Wiley and Sons, Hoboken, New York, 1997. [9] A. K. Golmankhane, K. K. Ali, R. Yilmazer, and M. K. A. Kaabar, Economic models involving time fractal, J. Math. Model. Financ, 1 (2020), 181–200. [10] A. K. Golmankhaneh and C. Cattani, Fractal logistic equation, Fractal Fract 3(3) (2019), 41–53. Doi:10.3390/fractalfract3030041. [11] A. K. Golmankhaneh and C. Tun¸c, Sumudu transform in fractal calculus, Appl.Math. Comput., 350 (2019), 386–401. [12] A. K. Golmankhaneh and D. Baleanu, Non-local integrals and derivatives on fractal sets with applications, Open Physics, 14(1) (2016), 542–548. [13] A. K. Golmankhaneh and A. Fernandez, Fractal Calculus of Functions on Cantor Tartan Spaces, Fractal Fract, 2(4) (2018), 30-40, Doi:10.3390/fractalfract2040030. [14] A. K. Golmankhaneh and A. Fernandez, Random Variables and Stable Distributions on Fractal Cantor Sets, Fractal Fract., 3(2) (2019), 31–44, Doi:10.3390/fractalfract3020031. [15] A. K. Golmankhaneh and A. S. Balankin, Sub-and super-diffusion on Cantor sets: Beyond the paradox, Phys. Lett. A, 382(14) (2018), 960–967. [16] A. K. Golmankhaneh and C. Tun¸c, On the Lipschitz condition in the fractal calculus, Chaos, Solitons & Fract., 95 (2017), 140–147. [17] A. K. Golmankhaneh and K. Welch, Equilibrium and non-equilibrium statistical mechanics with generalized fractal derivatives: A review, Mod. Phys. Lett. A, 36(14) (2021), 2140002. [18] A. K. Golmankhaneh, A review on application of the local fractal calculus, Num. Com. Meth. Sci. Eng., 1(2) (2019), 57–66. [19] A. K. Golmankhaneh, About Kepler’s Third Law on fractal-time spaces, Ain Shams Eng. J., 9(4) (2017), 2499– 2502. [20] A. K. Golmankhaneh, On the Fractal Langevin Equation, Fractal Fract., 3(1), (2019) , 11-20. Doi:10.3390/fractalfract3020031. [21] A. K. Golmankhaneh, A. Fernandez, A. K. Golmankhaneh, and D. Baleanu, Diffusion on middle-ξ Cantor sets, Entropy, 20(7) (2018), 504–517, Doi: 10.3390/e20070504. [22] C. P. Haynes and A. P. Roberts, Generalization of the fractal Einstein law relating con-duction and diffusion on networks, Phys. Rev. Lett., 103 (2009), 020601. [23] J. Kigami, Analysis on Fractals, Cambridge University Press, 2001. [24] M. L. Lapidus and M. Van Frankenhuijsen, Fractal geometry, complex dimensions and zeta functions: geometry and spectra of fractal strings, Springer Science & Business Media, 2012. [25] B. B. Mandelbrot, The fractal geometry of nature, New York, WH freeman, 1983. [26] T. Myint-U and L. Debnath, Linear partial differential equations for scientists and engineers, Springer Science & Business Media, 2007. [27] L. Nottale and J. Schneider, Fractals and nonstandard analysis, J. Math. Phys., 25 (1998), 1296–1300. [28] F. W. Olver, D. W. Lozier, R. F. Boisvert, and C. W Clark, editors, NIST handbook of mathematical functions hardback and CD-ROM, Cambridge University Press, 2010. [29] A. Parvate and A. D. Gangal, Calculus on fractal subsets of real-line I: Formulation, Fractals, 17(01) (2009), 53–81. [30] A. Parvate and A. D. Gangal, Calculus on fractal subsets of real-line II: Conjugacy with ordinary calculus, Fractals, 19(03) (2011), 271–290. [31] A. Parvate, S. Satin, and A. D. Gangal, Calculus on fractal curves in Rn, Fractals, 19(01) (2011), 15–27. [32] Y. B. Pesin, Dimension theory in dynamical systems: contemporary views and applications, University of Chicago Press, 2008. [33] L. Pietronero and E. Tosatti, Fractals in physics, Elsevier, 2012. [34] R.D. Richtmyer, Principles of Advanced Mathematical Physics, Vol. I, Springer-Verlag, New York, 1978. [35] M. F. Shlesinger, Fractal time in condensed mattar, Ann. Rev. Phys. Chern. 39 (1988), 269–290. [36] R. S. Strichartz, Differential Equations on Fractals: A Tutorial, Princeton University Press, Princeton, 2006. [37] V.E. Tarasov, Fractional dynamics: applications of fractional calculus to dynamics of particles, fields and media, Springer Science Business Media, 2011. [38] S. Vrobel, Fractal time: Why a watched kettle never boils, World Scientific, 2011. [39] K. Welch, A Fractal Topology of Time: Deepening into Timelessness, Fox Finding Press, 2nd Edition, 2020.
آمار تعداد مشاهده مقاله: 1,292 تعداد دریافت فایل اصل مقاله: 1,418

سامانه مدیریت نشریات علمی. قدرت گرفته از سیناوب

پیوندهای مفید

آمار

Local fractal Fourier transform and applications