about maxwell s equations on fractal subsets of 3

Cent Eur J Phys DOI: 10.2478/s11534-013-0192-6 Central European Journal of Physics About Maxwell’s equations on fractal subsets of R3 Research Article Alireza K Golmankhaneh1∗ , Ali K Golmankhaneh Dumitru Baleanu 2,3,4† Departments of Physics, Urmia Branch, Islamic Azad University, P.O.BOX 969, Oromiyeh, Iran Çankaya University, Faculty of Art and Sciences, Department of Mathematics and Computer Sciences Balgat 0630, Ankara, Turkey Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O Box: 80204, Jeddah, 21589, Saudi Arabia Institute of Space Sciences, P.O.BOX, MG-23, R 76900, Magurele-Bucharest, Romania Received 13 november 2012; accepted 14 February 2013 Abstract: In this paper we have generalized F ξ -calculus for fractals embedding in R3 F ξ -calculus is a fractional local derivative on fractals It is an algorithm which may be used for computer programs and is more applicable than using measure theory In this Calculus staircase functions for fractals has important role F ξ -fractional differential form is introduced such that it can help us to derive the physical equation Furthermore, using the F ξ -fractional differential form of Maxwell’s equations on fractals has been suggested PACS (2008): 02.30.Cj,02.30.Sa,02.40.Ma, 41.20.-q, 41.90.+e, Keywords: fractal integral• fractal derivativeã fractal differential equationsã fractional Maxwell equation â Versita sp z o.o Introduction Fractal patterns appear in many natural phenomena Fractal geometry plays important roles in some branches of science, engineering and art Fractals have complicated structures thus ordinary calculus does not apply For example, The Lebesgue-Cantor staircase function is zero almost everywhere, therefore this function is not a solution of any ordinary differential equations Fractional derivatives are non-local therefore are suitable for frac∗ † E-mail: alireza@physics.unipune.ac.in (Corresponding Author) E-mail: dumitru@cankaya.edu.tr tal functions Several authors have recognized the need to use fractional derivatives and integrals to explore the characteristic features of fractal walks, anomalous diffusion, transport, etc (see references [1–35] and references therein) During recent decades, analysis on fractals has developed and been applied in many important cases, for example, heat conduction, fractal-time diffusion equation, waves, etc (see[30, 34, 35] and several references therein) Using measure theory people have introduced calculus on fractals [32, 33] It consists of defining a derivative as an inverse of an integral with respect to a measure and defining other operators using this derivative Taking into account all the above we conclude that a simple method About Maxwell’s equations on fractal subsets of R3 ξ of fractional order operators on fractal sets is only a moderate survey Riemann integration like procedures have their own place and are better and more useful algorithmic methods [38].Recently, the Maxwell fractional-order differential calculus addressed the skin effect and developed a new method for implementing fractional-order inductive elements [39] In this manuscript we have generalized F ξ calculus on fractal subsets of R3 The organization of the manuscript is as follows: We begin the Section by defining the generalized F α -calculus In Section we introduce F ξ -differential form Section presents to our conclusions mass γδ (F , a, b, c, d, e, f) of F ∩ [a, b] × [c, d] × [e, f] is given by ξ γδ (F , a, b, c, d, e, f) = Λξ [F , I] (3) where |P| = max1≤i≤n (xi −xi−1 )(yi −yi−1 )(zi −zi−1 ) Taking infimum over all subdivisions P of [a, b] × [c, d] × [e, f] satisfying |P| ≤ δ Definition The mass function γ ξ (F , a, b, c, d, e, f) is given by [38] ξ Fractional F ξ -calculus inf P[a,b]×[c,d]×[e,f] :|P|≤δ γ ξ (F , a, b, c, d, e, f) = lim γδ (F , a, b, c, d, e, f) δ→0 (4) In this section we will introduce new definitions on fractals subsets of R3 We begin by defining the integral staircase function [38] Definition 2.1 The mass function and the integral staircase  ξ  if γ (F , a0 , b0 , c0 , x, y, z) ξ SF (x, y, z) = x ≥ a0 , y ≥ b0 , z ≥ c0   ξ −γ (F , a0 , b0 , c0 , x, y, z) otherwise (5) Now, we introduce the following new definitions: Let a0 , b0 , c0 be fixed and real numbers The integral ξ staircase function SF (x, y, z) of order ξ for a set F is [38] Definition Definition F be a fractal subset of The flag function Θ(F , I) for a fractal set F is define as [36–38] Θ(F , I) = if F ∩ I = ∅, otherwise where I = [a, b] × [c, d] × [e, f] is a subset in We say that a point (x, y, z) is a point of change of a function f if f is not constant over any open set (a, d) × (c, d) × (e, f) containing (x, y, z) The set of all points of change of f is called the set of change of f and is denoted by Schf [38] (1) Definition The η-dimension of F ∩ [a, b] × [c, d] × [e, f] denoted by dimη (F ∩ [a, b] × [c, d] × [e, f]) and define dimη (F ∩ [a, b] × [c, d] × [e, f]) Definition = inf{ξ : γ ξ (F , a, b, c, d, e, f) = 0} For a set F and I= [a, b] × [c, d] × [e, f], with subdivision P[a,b]×[c,d]×[e,f] , a < b, c < d, e < f we define n ξ Λ [F , I] = i=1 Definition (xi − xi−1 )α (yi − yi−1 )β (zi − zi−1 )ε Γ(α + 1) Γ(β + 1) Γ(ε + 1) Θ(F , [xi−1 , xi ] × [yi−1 , yi ] × [zi−1 , zi ]) = sup{ξ : γ ξ (F , a, b, c, d, e, f) = ∞} (2) where ξ = α + β + ε Let < α ≤ 1, < β ≤ 1, and < ε ≤ (6) ξ Let F ⊂ R3 be such that SF (x, y, z) is finite for all ξ (x, y, z) ∈ R3 for ξ = dimγ F Then the Sch(SF ) is said ξ to be ξ-perfect (Closed and every point of Sch(SF ) is its limit point) 2.2 F ξ -limit and F ξ - continuity Definition Definition Given δ > and a ≤ b, c ≤ d, e ≤ f the coarse-grained Let F ⊂ R3 , f : R3 → R3 and (x, y, z) ∈ F A number l is said to be the limit of f through the points of F or simply Alireza K Golmankhaneh, Ali K Golmankhaneh Dumitru Baleanu , , F ξ -limit of f as (x , y , z ) → (x, y, z) if given any ε > there exists δ > such that [38] (x , y , z ) ∈ F Definition 13 If f be a bounded function on F we say that f is F ξ integrable on on I = [a, b] × [c, d] × [e, f] if [38] and ξ |(x , y , z ) − (x, y, z)| < δ ⇒ |f(x , y , z ) − l| < ε (7) I ξ ξ f(x, y, z)dF xdF ydF z = sup P[a,b]×[c,d]×[e,f] (14) If such a number exists then it is denoted by ξ ξ l=F − lim (x ,y ,z )→(x,y,z) = f(x , y , z ) lim (x ,y ,z )→(x,y,z) f(x , y , z ) (9) Let f be a bounded function on F and I be a closed ball [38] Then f(x, y, z) if F ∩ I = (x,y,z)∈F ∩I = inf P[a,b]×[c,d]×[e,f] In that case the F ξ -integral of f on [a, b]×[c, d]×[e, f] deξ ξ ξ noted by f(x, y, z)dF xdF ydF z is given by the comI mon value [38] F α -Differentiation 2.4 otherwise (10)  f(x , y, z) − f(x, y, z)   lim  ξ ξ F− (x ,y,z)→(x,y,z) SF (x , y, z) − SF (x, y, z) x ξ DF f(x, y, z) = if (x, y, z) ∈ F     otherwise (16) and similarly y m[f, F , I] = inf (x,y,z)∈F ∩I = U ξ [f, F , P] If F is an ξ-perfect set then the F ξ -partial derivative of f respect to x is [38] Definition 11 sup ξ Definition 14 F ξ -integration M[f, F , I] = ξ f(x, y, z)dF xdF ydF z = (15) A function f : R3 → R3 is said to be F ξ -continues at (x, y, z) ∈ F if 2.3 I (8) Definition 10 f(x, y, z) = F ξ − Lξ [f, F , P] f(x, y, z) if the limit exists Likewise the ξ DF f(x, y, z) can be defined z if F ∩ I = otherwise (11) ξ DF f(x, y, z) and F ξ -differential form Definition 12 ξ Let SF (x, y, z) be finite for (x, y, z) ∈ [a, b] × [c, d] × [e, f] Let P be a subdivision of [a, b] × [c, d] × [e, f] with points x0 , y0 , z0 , xn , yn , zn The upper F ξ -sum and the lower F ξ sum for function f over the subdivision P are given respectively by [38] n U ξ [f, F , P] = M[f, F , [(xi−1 , yi−1 , zi−1 ), (xi , yi , zi )]] i=1 ξ ξ (SF (xi , yi , zi ) − SF (xi−1 , yi−1 , zi−1 )) (12) In this section we have generalized the F ξ -fractional calculus on fractals subset of R3 F ξ -Fractional 1-forms 3.1 A differential fractional 1-form on an set F subset of R3 is a expression H(x, y, z)dFα x +G(x, y, z)dFβ y+N(x, y, z)dFε z where H, G, N are functions on the open set If f(x, y, z) is Cξ1 function, then its F ξ -fractional total differential (or exterior derivative) is ξ dF f(x, y, z) = x DFα f(x, y, z)dFα x + y DFβ f(x, y, z)dFβ y + z DFε f(x, y, z)dFε z and ξ =α + β + ε n Lξ [f, F , P] = (17) m[f, F , [(xi−1 , yi−1 , zi−1 ), (xi , yi , zi )]] i=1 ξ ξ (SF (xi , yi , zi ) − SF (xi−1 , yi−1 , zi−1 )) (13) In the same manner Eq (17) can generalized to a higher dimension About Maxwell’s equations on fractal subsets of R3 F ξ - Fractional Exactness 3.2 Suppose that HdFα x + GdFβ y + NdFε z is a F ξ -fractional differential on F with Cξ1 coefficients We will say that it is exact if one can find a Cξ2 function f(x, y, z) with to Eq (24) and supposing α = β = ε = κ = µ Since ξ ξ dF ωF = 0, so Eq (24) leads {x1 DFµ E2 − ξ dF f = HdFα x +GdFβ y+NdFε z We will call a F ξ -fractional differential closed if x ξ y DF f = H ξ z DF f = G DFβ f = N DFβ N z = DFε G, x DFα G y = DFβ H, z DFε H x = A F ξ -Fractional 2-forms is like a M(x, y, z)dFα x dFβ y + W (x, y, z)dFβ y dFε z + L(x, y, z)dFε z dFα x where M, W and L are functions And wedge product of two F ξ Fractional 1-forms with following properties dFβ y = −dFβ y dFα x dFα x dFα x = (20) ξ dF ξ So far we have applied to functions to obtain F Fractional 1-forms, and then to F ξ -Fractional 1-forms to get 2-forms, so that Eq (20) can been generalized as a standard calculus One can define F ξ -Fractional gradient, divergence, and curl as follows, respectively: ξ gradF f = ix DFα f + jy DFβ f + kz DFε f ξ =α +β+ε (21) ξ divF V x = DFα Vx x + DFβ Vy z + DFε Vz (22) ξ curlF V = i( y DFβ Vz + k( x − DFα Vy z DFε Vy ) − y + j( z DFε Vx − x dFµ t + {x1 DFµ E3 − x3 DFµ E1 }dFµ x1 dFµ x3 dFµ t + {x1 DFµ E2 − x2 DFµ E3 }dFµ x3 dFµ x2 dFµ t dFµ x2 dFµ t − t DFµ B2 dFµ x1 dFµ x3 dFµ t −t DFµ B1 dFµ x3 dFµ x2 dFµ t, (26) and, (x1 DFµ B1 + F ξ -Fractional 2-forms dFα x dFµ x2 DFα N (19) 3.3 DFµ E1 }dFµ x1 = −t DFµ B3 dFµ x1 (18) Therefor HdFα x + GdFβ y + NdFε z is exact if we have y x2 x2 DFµ B2 + DFµ B3 )dFµ x1 dFµ x2 dFµ x3 = (27) x3 In the vector notation it will be curlµF E = −t DFµ B (28) divµF B (29) = Now we define a fractional form as πFµ = J1 dFµ x2 + J3 dFµ x1 dFµ x3 + J2 dFµ x3 dFµ x2 dFµ x1 dFµ t − ρdFµ x1 dFµ x2 (30) where Ji , i = 1, 2, are components of current and ρ density of charge Likewise applying dFµ to the left side of Eq (30) and dFµ πFµ = we have x1 DFµ J1 +x2 DFµ J2 + dFµ x3 x3 DFµ J3 + t DFµ ρ dFµ x1 (31) DFα Vz ) Furthermore, conservation of charge on fractals is DFβ Vx ) (23) Maxwell’s equation on fractals = (E1 dFα x1 + E2 dFβ x2 + dFε x3 + B2 dFε x3 E3 dFε x3 ) dFα x1 + dFκ t + B3 dFβ x1 (32) Consider the following fractional form We obtain the fractional Maxwell’s equation on fractals as follows ξ ωF dFµ x2 dFµ t = divµF J + t DFµ ρ = dFµ x3 B1 dFβ x2 dFα x2 ζFµ = A1 dFµ x1 + A2 dFµ x2 + A3 dFµ x3 + φdFµ t, (33) where Ai , i = 1, 2, is vector potential Using dFµ ζFµ = one can obtain curlµF A = B gradµF φ − t DFµ A = E (34) (24) where Ei , Bj , i, j = 1, 2, are components of electroξ magnetic field Applying dF that is ξ dF = x1 DFµ dFµ x1 + x2 DFµ dFµ x2 + x3 DFµ dFµ x3 + t DFµ dFµ t, (25) Therefore, we arrive at the wave equation as following c curlµF B = t DFµ E, where c is speed of light (35) Alireza K Golmankhaneh, Ali K Golmankhaneh Dumitru Baleanu , , Conclusion Fractal calculus is still an open problem In this manuscript we have generalized F ξ -calculus for fractals embedding in R3 F ξ -calculus is a local derivative of fractals and has an algorithm which may be used in computer programs F ξ -fractional differential form is introduced to derive the physical equation By using the F ξ -fractional differential form, the form of the Maxwell’s equation on fractals has been suggested References [1] B.B Mandelbrot, The Fractal Geometry of Nature (Freeman and company, 1977) [2] A Bunde and S.Havlin (Eds), Fractal in Science (Springer, 1995) [3] K 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on fractal subsets of R3 ξ of fractional order operators on fractal sets is only a moderate survey Riemann integration like procedures have their own... Furthermore, conservation of charge on fractals is DFβ Vx ) ( 23) Maxwell? ? ?s equation on fractals = (E1 dFα x1 + E2 dFβ x2 + dFε x3 + B2 dFε x3 E3 dFε x3 ) dFα x1 + dFκ t + B3 dFβ x1 (32 ) Consider the... ξ ξ (SF (xi , yi , zi ) − SF (xi−1 , yi−1 , zi−1 )) ( 13) In the same manner Eq (17) can generalized to a higher dimension About Maxwell? ? ?s equations on fractal subsets of R3 F ξ - Fractional