Fatma Ayça ÇETİNKAYA, Alireza Khalili GOLMANKHANEH

General characteristics of a fractal Sturm–Liouville problem

In this paper, we consider a regular fractal Sturm–Liouville boundary value problem. We prove the selfadjointness of the differential operator which is generated by the F α -derivative introduced in [32]. We obtained the F α - analogue of Liouville’s theorem, and we show some properties of eigenvalues and eigenfunctions. We present examples to demonstrate the efficiency and applicability of the obtained results. The findings of this paper can be regarded as a contribution to an emerging field

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[1] Balankin AS. A continuum framework for mechanics of fractal materials I: from fractional space to continuum with fractal metric. The European Physical Journal B 2015; 88 (90). doi: 10.1140/epjb/e2015-60189-y
[2] Barlow MT. Diffusions on fractals. In: Bernard P (editor). Lectures on Probability Theory and Statistics (SaintFlour, 1995), Volume 1690 of Lecture Notes in Mathematics. Berlin, Germany: Springer, 1998, pp. 1-121.
[3] Bohner M, Peterson A. Dynamic Equations on Time Scales, An Introduction with Applications. Boston, MA, USA: Birkhäuser Boston, Inc., 2001.
[4] Bunde A, Havlin S. Fractals in Science. Berlin, Germany: Springer-Verlag, 1994.
[5] Czachor M. Waves along fractal coastlines: from fractal arithmetic to wave equations. Acta Physica Polonica B 2019; 50 (4): 813-831. doi: 10.5506/APhysPolB.50.813
[6] Dalrymple K, Strichartz RS, Vinson JP. Fractal differential equations on the Sierpinski gasket. Journal of Fourier Analysis and Applications 1999; 5 (2-3): 203-284. doi: 10.1007/BF01261610
[7] El-Nabulsi RA. Dirac equation with position-dependent mass and Coulomb-like field in Hausdorff dimension. FewBody Systems 2020; 61 (1): 1-9. doi: 10.1007/s00601-020-1544-6
[8] Freiberg U, Zähle M. Harmonic calculus on fractals—a measure geometric approach I. Potential Analysis 2002; 16 (3): 265-277.
[9] Freiling G, Yurko VA. Inverse Sturm–Liouville Problems and Their Applications. New York, NY, USA: NOVA Science Publishers, 2001.
[10] Godfain E. Fractal space-time as underlying structure of the standart model. Quantum Matter 2014; 3 (3): 256-263. doi: 10.1166/qm.2014.1121
[11] Golmankhaneh AK. About Kepler’s third law on fractal time-spaces. Ain Shams Engineering Journal 2017; 9 (4): 2499-2502. doi: 10.1016/j.asej.2017.06.005
[12] Golmankhaneh AK, Tunç C. Sumudu transform in fractal calculus. Applied Mathematics and Computation 2017; 350: 386-401. doi: 10.1016/j.amc.2019.01.025
[13] Golmankhaneh AK, Fernandez A. Fractal calculus of functions on Cantor tartan spaces. Fractal and Fractional 2018; 3 (31). doi: 10.3390/fractalfract2040030
[14] Golmankhaneh AK. Statistical mechanics involving fractal temperature. Fractal and Fractional 2019; 3 (20). doi: 10.3390/fractalfract3020020
[15] Golmankhaneh AK, Cattani C. Fractal logistic equation. Fractal and Fractional 2019; 3 (1). doi: 10.3390/fractalfract3030041
[16] Golmankhaneh AK, Fernandez A. Random variables and state distributions on fractal cantor sets. Fractal and Fractional 2019; 3 (31). doi: 10.3390/fractalfract3020031
[17] Golmankhaneh AK, Tunç C. On the Lipschitz condition in fractal calculus. Chaos Solitons & Fractals 2017; 95: 140-147. doi: 10.1016/j.chaos.2016.12.001
[18] Golmankhaneh AK, Baleanu D. On a new measure on fractals. Journal of Inequalities and Applications 2013; 2013 (522). doi: 10.1186/1029-242X-2013-522
[19] Golmankhaneh AK, Welch K. Equilibrium and non-equilibrium statistical mechanics with generalized fractal derivatives: a review. Modern Physics Letters A 2021; 36 (14): 2140002. doi: 10.1142/S0217732321400022
[20] Iomin A, Sandev T. Fractional diffusion to a cantor set in 2d. Fractal and Fractional 2020; 4 (4): 52. doi: 10.3390/fractalfract4040052
[21] Kempfle S, Schäfer I, and Beyer H. Fractional calculus via functional calculus: theory and applications, Nonlinear Dynamics 2002; 29: 99-127. doi: 10.1023/A:1016595107471
[22] Kigami J. Analysis on Fractals, Volume 143 of Cambridge Tracts in Mathematics. Cambridge, UK: Cambridge University Press, 2001. doi: 10.1017/S0013091503214619
[23] Kolwankar KM, Gangal AD. Fractional differentiability of nowhere differentiable functions and dimensions. Chaos 1996; 6 (4): 505-513. doi: 10.1063/1.166197
[24] Kolwankar KM, Gangal AD. Hölder exponents of irregular signals and local fractional derivatives. Pramana 1997; 48: 49-68. doi: 10.1007/BF02845622
[25] Lapidus M, Frankenhuijsen VM. Fractal geometry, complex dimensions and zeta functions, geometry and spectra of fractal strings. New York, NY, USA: Springer Monographs in Mathematics, Springer, 2013.
[26] Levitan BM, Sargsjan IS. Sturm–Liouville and Dirac Operators, Volume 59 of Mathematics and its Applications. Dordrecht: Kluwer Academic Publishers Group, 1991.
[27] Mandelbrot BB. Fractals: Form, Chance, and Dimension. San Francisco: W. H. Freeman and Co., 1977.
[28] Marchenko VA. Sturm–Liouville Operators and Their Applications. Providence, RI, USA: AMS, 2011.
[29] Nigmatullin RR, Baleanu D. Relationships between 1D and space fractals and fractional integrals and their applications in physics. In: Tarasov VE (editor). Handbook of Fractional Calculus with Applications, Volume 4: Applications in Physics, Part A. Berlin, Germany: De Gruyter, 2019, pp. 183-219.
[30] Nigmatullin RR, Zhang W, Gubaidullin I. Accurate relationships between fractals and fractional integrals: new approaches and evaluations. Fractional Calculus and Applied Analysis 2017; 20 (5): 1263-1280. doi: 10.1515/fca2017-0066
[31] Nottale L. Fractals in the quantum theory of space time. In: Meyers R (editor). Mathematics of Complexity and Dynamical Systems. New York, NY, USA: Springer, 2012, pp. 571-590. doi: 10.1007/978-1-4614-1806-1-37
[32] Parvate A, Gangal AD. Calculus on fractal subsets of real line I: formulation. Fractals 2009; 17 (1): 53-81. doi: 10.1142/S0218348X09004181
[33] Parvate A, Gangal AD. Calculus on fractal subsets of real line II: conjugacy with ordinary calculus. Fractals 2011;
19 (3): 271-290. doi: 10.1142/S0218348X11005440 [34] Rocco A, West BJ. Fractional calculus and the evolution of fractal phenomena. Physica A 1999; 265 (3-4): 535-546. doi: 10.1016/S0378-4371(98)00550-0
[35] Serpa C, Buescu J. Explicitly defined fractal interpolation functions with variable parameters. Chaos Solitons & Fractals 2015; 75: 76-83. doi: 10.1016/j.chaos.2015.01.023
[36] Stillinger FH. Axiomatic basis for spaces with noninteger dimension. Journal of Mathematical Physics 1977; 18 (6): 1224-1234. doi: 10.1063/1.523395
[37] Strichartz RS. Taylor approximations on Sierpinski gasket type fractals. Journal of Functional Analysis 2000; 174 (1): 76-127. doi: 10.1006/jfan.2000.3580
[38] Tarasov VE. Fractional dynamics of relativistic particle. International Journal of Theoretical Physics 2010; 49 (2): 293-303. doi: 10.1007/s10773-009-0202-z
[39] Uchaikin V, Sibatov R. Fractional Kinetics in Space: Anomalous Transport Models. Hackensack, NJ: World Scientific Publishing Co. Pte. Ltd., 2018.
[40] Wu J, Wang C. Fractal Stokes’ theorem based on integrals on fractal manifolds. Fractals 2020; 28 (1) : 20500103. doi: 10.1142/S0218348X20500103
[41] Zettl A. Sturm–Liouville Theory, Volume 121 of Mathematical Surveys and Monographs. Providence, RI, USA: American Mathematical Society, 2005.
[42] Zubair M, Mughal MJ, Naqvi QA. Electromagnetic Fields and Waves in Fractional Dimensional Space. Heidelberg, Germany: Springer Briefs in Applied Sciences and Technology, Springer, 2012.