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The generalized variational iteration method to solve the fractal partial differential equations | ||
| Journal of New Researches in Mathematics | ||
| دوره 8، شماره 40، مهر و آبان 2023، صفحه 13-28 اصل مقاله (407.07 K) | ||
| نوع مقاله: research paper | ||
| شناسه دیجیتال (DOI): 10.30495/jnrm.2022.69663.2328 | ||
| نویسندگان | ||
Homa Afraz   1؛ Alireza Khalili Golmankhaneh 2
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| 1department mathematic, Payame Noor University: Tehran, IR | ||
| 2Department of Physics, Urmia Branch, Islamic Azad University, Urmia, Iran | ||
| چکیده | ||
| Fractional calculus is a branch of classical mathematics, which deals with the generalization of fractional order derivative and integral operator. Recently, a great deal of research has been carried out on the fractional calculus to study the phenomena associated with fractal structures and processes. Fractals have a fractional dimension and occur naturally in non-linear and imbalanced phenomena in various forms and contexts. In recent years, various types of derivatives and fractional and fractal calculus have been proposed by many scientists and have been extensively utilized. Measurements are localized in physical processes, and local fractional calculus is a useful tool for solving some type of physical and engineering problems. Gangal studied the local fractional calculus and got the relation between it and the fractals. Using the local fractional calculus and fractal properties, he defined the fractal-alpha calculus on a subset of the real line, which is a simple calculs, useful, structural and algorithmic. In this study, we first describe the fractal-F alpha calculus. Next, we propose The generalized variational iteration method based on the fractal calculus. To show the efficiency of fractal calculus and the new method, we solve several fractal partial differential equations with this method and show that this method is better, easier and more suitable than the two other methods mention the above. | ||
| کلیدواژهها | ||
| Fractal Calculus؛ Generalized Variational Iteration Method؛ fractal؛ fractal partial differential equations | ||
| مراجع | ||
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[1] Kolwankar, K.M., Gangal, A.D, Hölder exponents of irregular signals and local fractional derivatives, Pramana J. Phys., 1997, 49-68. [2] Tatom F. B, The Relationship between Fractional Calculus and Fractals, Fractals, 3(1), 1995, 217-229. [3] Golmankhaneh.A.K, On the calculus of the parameterized fractal curves, Turk. J. Phys. 41 (2017) 418-425. [4] Golmankhaneh.A.K, Fernandez.A, Fractal Calculus of Functions on Cantor Tartan Spaces. Fractal Fract 2(30) (2008) 1-13. [5]Golmankhaneh.A. K, Fernandez.A, Golmankhaneh. A. K, Baleanu.D, Diffusion on middle-X Cantor sets, Entropy, 20(504) (2018) 1-13. [6] Parvate.A, Gangal.A.D, Calculus on fractal subsets of real-line I: Formulation, Fractals 17(01) (2009) 53-148. [7] Parvate.A, Gangal.A.D, Calculus on fractal subsets of real line II: Conjugacy with ordinary calculus, Fractals 19(03) (2011) 271-290. [8] Parvate.A, Satin.S, Gangal.A.D, Calculus on fractal curves in R^n, Fractals, 19(01) (2011), 15-27. [9] Satin.S, Gangal.A.D, Langevin Equation on Fractal Curves, Fractals 24(03) (2016), 1650028. [10] Ashrafi.s , Golmankhaneh. A.K., Energy Straggling Function by Fa -Calculus, ASME J. Comput. Nonlin. Dyn. 12(5) (2017), 051010. [11]Golmankhaneh.A.K,Golmankhaneh.A.K, Baleanu. D, About Maxwell’s equations on fractal subsets of R3, Cent. Eur. J. Phys., 11(6) (2013) 863-867. [12] Golmankhaneh. A.K, On the Fractal Langevin Equation, Fractal Fract.,3(1):11 (2019) 1-9. [13] Golmankhaneh.A.K, Statistical Mechanics Involving Fractal Temperature, Fractal Fract. 3(2):20 (2019) 1-12. [14] Golmankhaneh.A.K, Baleanu.D, Fractal calculus involving gauge function. Commun. Nonlinear Sci, 37, (2016) 125-130. [15]Golmankhaneh.A.K,Golmankhaneh.A.K, Baleanu.D, Lagrangian and Hamiltonian mechanics on fractals subset of real-line, Int.J. Theor. Phys., 52(11), (2013) 4210-4217. [16]Golmankhaneh.A.K.Golmankhaneh.A.K,Baleanu.D,About Schr¨odinger equation on fractals curves imbedding in R3, Int. J. Theor. Phys., 54(4) (2015), 1275-1282. [17] Golmankhaneh.A.K, Baleanu.D, Diffraction from fractal grating Cantor sets, J. Mod. Opt., 63(14) (2016), 1364-1369. [18] Golmankhaneh.A.K, Cattani.C, Fractal Logistic Equation, Fractal Fract.,DOI:10.3390/fractalfract3030041. [19]Golmankhaneh.A.K, Fazlollahi.V, Baleanu.D, Newtonian mechanics on fractals subset of real-line, Rom. Rep. Phys., 65(1) (2013) 84-93. [20]Golmankhaneh.A.K,Fernandez.A, Random Variables and Stable Distributions on Fractal Cantor Sets, Fractal Fract., 3(2):31(2019) 1-13. [21] Golmankhaneh.A.K, Tunc.C, On the Lipschitz condition in the fractal calculus, Chaos, Soliton Fract. , 95 (2017), 140-147. [22] Golmankhaneh.A.K, Tunc.C, Sumudu Transform in Fractal Calculus, Appl. Math. Comput., 350 (2019) 386-401. [23] Golmankhaneh. A. K, KamalAli. K, Fractal Kronig-Penney model involving fractal comb potential, DOI: 10. 22124/Journal of Mathematical Modeling, Vol. 9, No. 3, 2021, pp. 331-345. [24] Golmankhaneh. A. K, Fractal Calculus and its Applications, World Scientific, 2022. doi:10.1142/12988. [25] Ashrafi. S, Golmankhaneh. A. K , Dimension of quantum mechanical path, chain rule, and extension of Landau’s energy straggling method using F^α-Calculus, Turk. J. Phys., 42(2) (2018), 104-115 | ||
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