The main object of this paper is to investigate the Helmholtz and diffusion equations on the Cantor sets involving local fractional derivative operators. The Cantor-type cylindrical-coordinate method is applied to handle the corresponding local fractional differential equations. Two illustrative examples for the Helmholtz and diffusion equations on the Cantor sets are shown by making use of the Cantorian and Cantor-type cylindrical coordinates.
1. Introduction
In the Euclidean space, we observe several interesting physical phenomena by using the differential equations in the different styles of planar, cylindrical, and spherical geometries. There are many models for the anisotropic perfectly matched layers [1], the plasma source ion implantation [2], fractional paradigm and intermediate zones in electromagnetism [3, 4], fusion [5], reflectionless sponge layers [6], time-fractional heat conduction [7], singular boundary value problems [8], and so on (see also the references cited in each of these works).
The Helmholtz equation was applied to deal with problems in such fields as electromagnetic radiation, seismology, transmission, and acoustics. Kreß and Roach [9] discussed the transmission problems for the Helmholtz equation. Kleinman and Roach [10] studied the boundary integral equations for the three-dimensional Helmholtz equation. Karageorghis [11] presented the eigenvalues of the Helmholtz equation. Heikkola et al. [12] considered the parallel fictitious domain method for the three-dimensional Helmholtz equation. Fu and Mura [13] suggested the volume integrals of the inhomogeneous Helmholtz equation. Samuel and Thomas [14] proposed the fractional Helmholtz equation.
Diffusion theory has become increasingly interesting and potentially useful in solids [15, 16]. Some applications of physics, such as superconducting alloys [17], lattice theory [18], and light diffusion in turbid material [19], were considered. Fractional calculus theory (see [20–28]) was applied to model the diffusion problems in engineering, and fractional diffusion equation was discussed (see, e.g., [29–36]).
Recently, the local fractional calculus theory was applied to process the nondifferentiable phenomena in fractal domain (see [37–48] and the references cited therein). There are some local fractional models, such as the local fractional Fokker-Planck equation [37], the local fractional stress-strain relations [38], the local fractional heat conduction equation [45], wave equations on the Cantor sets [47], and the local fractional Laplace equation [48].
The main aim of this paper is present in the mathematical structure of the Helmholtz and diffusion equations within local fractional derivative and to propose their forms in the Cantor-type cylindrical coordinates by using the Cantor-type cylindrical-coordinate method [46].
Our present investigation is structured as follows. In Section 2, the Helmholtz equation on the Cantor sets with local fractional derivative is investigated. The diffusion equation on the Cantor sets based upon the local fractional vector calculus is structured in Section 3. The Helmholtz and diffusion equations in the Cantor-type cylindrical coordinates are studied in Section 4. Finally, the conclusions are presented in Section 5.
2. The Helmholtz Equation on the Cantor Sets
In order to derive the Helmholtz equation on the Cantor sets, if the local fractional derivative is defined through [43–46]
(1)f(α)(x0)=dαf(x)dxα|x=x0=limx→x0Δα(f(x)-f(x0))(x-x0)α
with
(2)Δα(f(x)-f(x0))≅Γ(1+α)Δ(f(x)-f(x0)),
then the wave equation on the Cantor sets was suggested in [44] by
(3)∇2αu(r,t)=1a2α∂2αu(r,t)∂t2α,
where the local fractional Laplace operator is given by [43, 44, 48]
(4)∇2α=∂2α∂x2α+∂2α∂y2α+∂2α∂z2α,
where 1/a2α is a constant and u(r,t) is satisfied with local fractional continuous conditions (see [47]).
Using separation of variables in nondifferentiable functions, which begins by assuming that the fractal wave function u(r,t) may be separable, namely,
(5)u(r,t)=M(r)T(t),
we have
(6)∇2αM(r)M(r)=1a2αT(t)∂2αT(t)∂t2α,
such that
(7)∇2αM(r)+ω2αM(r)=0,(8)1a2αT(t)∂2αT(t)∂t2α=-ω2α.
In the three-dimensional Cantorian coordinate system, by following (7), we have
(9)∂2αM(x,y,z)∂x2α+∂2αM(x,y,z)∂y2α+∂2αM(x,y,z)∂z2α+ω2αM(x,y,z)=0,
where the operator is a local fractional derivative operator.
For the two-dimensional Cantorian coordinate system, the local fractional homogeneous Helmholtz equation is given by
(10)∂2αM(x,y)∂x2α+∂2αM(x,y)∂y2α+ω2αM(x,y)=0.
For a fractal dimension α=1, (9) becomes
(11)∂2M(x,y,z)∂x2+∂2M(x,y,z)∂y2+∂2M(x,y,z)∂z2+ω2M(x,y,z)=0,
which is the classical Helmholtz equation [10].
In view of (9), the inhomogeneous Helmholtz equation reads as follows:
(12)∂2αM(x,y,z)∂x2α+∂2αM(x,y,z)∂y2α+∂2αM(x,y,z)∂z2α+ω2αM(x,y,z)=f(x,y,z),
where f(x,y,z) is a local fractional continuous function.
In the two-dimensional Cantorian coordinate system, following (12), the local fractional inhomogeneous Helmholtz equation can be suggested by
(13)∂2αM(x,y)∂x2α+∂2αM(x,y)∂y2α+ω2αM(x,y)=f(x,y),
where f(x,y) is a local fractional continuous function.
We notice that the fractional Helmholtz equation was applied to deal with the differentiable wave equations in [14]. However, the Helmholtz equation with local fractional derivative arises in physical problems in such areas as, for example, fractal electromagnetic radiation, seismology, and acoustics, because their wave functions are the local fractional continuous functions (nondifferentiable functions). So, the Helmholtz equation on the Cantor sets can be used to describe the fractal electromagnetic radiation, the fractal seismology, the fractal acoustics, and so on.
3. Diffusion Equation on the Cantor Sets
In this section, we derive the diffusion equation on the Cantor sets with local fractional vector calculus [44].
Let us recall Fick’s law within the local fractional derivative, which was presented as
(14)J(r,t)=-D(φ)∇αφ(r,t),
where φ(r,t) and J(r,t) are local fractional continuous functions.
It is noticed that the flux of the diffusing material in any part of the fractal system is proportional to the local fractional density gradient. If the diffusion coefficient D(φ)=D is constant, the local fractional Fick law was suggested as [44]
(15)J(r,t)=-D∇αφ(r,t),
which was expressed as [44]
(16)∯J(r,t)·dS(β)=-∯D(φ)∇αφ(r,t)·dS(β),
where the local fractional vector integral is defined as [44]
(17)∬u(rP)·dS(β)=limN→∞∑P=1Nu(rP)·nPΔSP(β),
with N elements of area with a unit normal local fractional vector nP, ΔSP(β)→0 as N→∞ for β=2α, and φ(r,t) is the density of the diffusing material in local fractional field.
The conservation of mass within local fractional vector operator was presented as [44]
(18)dαdtα∭φ(r,t)dV(γ)=-∯J(r,t)·dS(β),
where local fractional volume integral is given by [44]
(19)∭u(rP)dV(γ)=limN→∞∑P=1Nu(rP)ΔVP(γ),
with N elements of volume ΔVP(γ)→0 as N→∞ for γ=(3/2)β=3α.
Following (18), and by using the divergence theorem of local fractional field [44], we have
(20)dαφ(r,t)dtα+∇α·J(r,t)=0,
where J(r,t) is the flux of the diffusing material in local fractional field.
Submitting (14) into (20), we obtain
(21)dαφ(r,t)dtα+∇α[-D(φ)∇αφ(r,t)]=0,
which is the so-called diffusion equation on the Cantor sets. This result differs from the fractional diffusion equation [29–36].
For the diffusion coefficient D(φ)=D, (21) becomes
(22)dαφ(r,t)dtα=D∇2αφ(r,t).
In the three-dimensional Cantorian coordinate system, following (22), we have
(23)dαφ(x,y,z,t)dtα=D[∂2α∂x2αφ(x,y,z,t)+∂2α∂y2αφ(x,y,z,t)+∂2α∂z2αφ(x,y,z,t)].
In the two-dimensional Cantorian coordinate system, we get
(24)dαφ(x,y,t)dtα=D[∂2α∂x2αφ(x,y,t)+∂2α∂y2αφ(x,y,t)].
In the one-dimensional Cantorian coordinate system, we obtain [48]
(25)dαφ(x,t)dtα=D∂2α∂x2αφ(x,t).
We notice that when fractal dimension α is equal to 1, we get the classical diffusion equation [15, 16]. However, the diffusion equation on the Cantor sets with local fractional derivative is derived from local fractional field, whose quantities are local fractional continuous functions.
4. The Cantor-Type Cylindrical-Coordinate Method to the Helmholtz and Diffusion Equations on the Cantor Sets
Let us consider the Cantor-type cylindrical coordinates, which read as follows:
(26)xα=Rαcosαθα,yα=Rαsinαθα,zα=zα,
with R∈(0,+∞), z∈(-∞,+∞), θ∈(0,π], and x2α+y2α=R2α.
We now have a local fractional vector given by
(27)r=Rαcosαθαe1α+Rαsinαθαe2α+zαe3α=rReRα+rθeθα+rzezα,
such that [46]
(28)∇αϕ(R,θ,z)=eRα∂α∂Rαϕ+eθα1Rα∂α∂θαϕ+ezα∂α∂zαϕ,(29)∇2αϕ(R,θ,z)=∂2α∂R2αϕ+1R2α∂2α∂θ2αϕ+1Rα∂α∂Rαϕ+∂2α∂z2αϕ,(30)∇α·r=∂αrR∂Rα+1Rα∂αrθ∂θα+rRRα+∂αrz∂zα,(31)∇α×r=(1Rα∂αrθ∂θα-∂αrθ∂zα)eRα+(∂αrR∂zα-∂αrz∂Rα)eθα+(∂αrθ∂Rα+rRRα-1Rα∂αrR∂θα)ezα,
where
(32)eRα=cosαθαe1α+sinαθαe2α,eθα=-sinαθαe1α+cosαθαe2α,ezα=e3α.
Submitting (29) into (9) and (12), it yields
(33)∂2αM(R,θ,z)∂R2α+1R2α∂2αM(R,θ,z)∂θ2α+1Rα∂αM(R,θ,z)∂Rα+∂2αM(R,θ,z)∂z2α+ω2αM(R,θ,z)=0,∂2αM(R,θ,z)∂R2α+1R2α∂2αM(R,θ,z)∂θ2α+1Rα∂αM(R,θ,z)∂Rα+∂2αM(R,θ,z)∂z2α+ω2αM(R,θ,z)=f(R,θ,z),
which is the Helmholtz equation in the Cantor-type cylindrical coordinates.
In the like manner, from (23), we get
(34)dαφ(R,θ,z,t)dtα=D[∂2αφ(R,θ,z,t)∂R2α+1R2α∂2αφ(R,θ,z,t)∂θ2α==+1Rα∂αφ(R,θ,z,t)∂Rα+∂2αφ(R,θ,z,t)∂z2α],
which is the diffusion equation in the Cantor-type cylindrical coordinates.
5. Concluding Remarks and Observations
In the present work, we have derived the Helmholtz and diffusion equations on the Cantor sets in the Cantorian coordinates, which are based upon the local fractional derivative operators. By applying the Cantor-type cylindrical-coordinate method, we have also investigated the Helmholtz and diffusion equations on the Cantor sets in the Cantor-type cylindrical coordinates. Furthermore, we have presented two illustrative examples for the corresponding fractional Helmholtz and diffusion equations on the Cantor sets by using the Cantorian and Cantor-type cylindrical coordinates.
Acknowledgments
This work was supported by National Natural Science Foundation of China (no. 11102181) and in part by Natural Science Foundation of Hebei Province (no. A2012203117).
TeixeiraF. L.ChewW. C.Systematic derivation of anisotropic PML absorbing media in cylindrical and spherical coordinates19977113713732-s2.0-003127185410.1109/75.641424ScheuerJ. T.ShamimM.ConradJ. R.Model of plasma source ion implantation in planar, cylindrical, and spherical geometries1990673124112452-s2.0-000052860310.1063/1.345722EnghetaN.On fractional paradigm and intermediate zones in electromagnetism. I. Planar observation1999224236241EnghetaN.On fractional paradigm and intermediate zones in electromagnetism. II. Cylindrical and spherical observations1999232100103SchetselaarE. M.Fusion by the IHS transform: should we use cylindrical or spherical coordinates?19981947597652-s2.0-0032030453PetropoulosP. G.Reflectionless sponge layers as absorbing boundary conditions for the numerical solution of Maxwell equations in rectangular, cylindrical, and spherical coordinates20006031037105810.1137/S0036139998334688MR1750090ZBL1025.78016JiangX.XuM.The time fractional heat conduction equation in the general orthogonal curvilinear coordinate and the cylindrical coordinate systems2010389173368337410.1016/j.physa.2010.04.023MR2659297BuckmireR.Investigations of nonstandard, Mickens-type, finite-difference schemes for singular boundary value problems in cylindrical or spherical coordinates200319338039810.1002/num.10055MR1969200ZBL1079.76048KreßR.RoachG. F.Transmission problems for the Helmholtz equation197819614331437MR049565310.1063/1.523808ZBL0433.35017KleinmanR. E.RoachG. F.Boundary integral equations for the three-dimensional Helmholtz equation197416214236MR038008710.1137/1016029ZBL0253.35023KarageorghisA.The method of fundamental solutions for the calculation of the eigenvalues of the Helmholtz equation200114783784210.1016/S0893-9659(01)00053-2MR1849238ZBL0984.65111HeikkolaE.RossiT.ToivanenJ.A parallel fictitious domain method for the three-dimensional Helmholtz equation20032451567158810.1137/S1064827500370305MR1978150ZBL1035.65126FuL. S.MuraT.Volume integrals of ellipsoids associated with the inhomogeneous Helmholtz equation19824214114910.1016/0165-2125(82)90030-0MR650451ZBL0479.73026SamuelM. S.ThomasA.On fractional Helmholtz equations2010133295308MR2761355ZBL1223.26013ShewmonP. G.1963New York, NY, USAMcGraw-HillRichterG. R.An inverse problem for the steady state diffusion equation198141221022110.1137/0141016MR628945ZBL0501.35075UsadelK. D.Generalized diffusion equation for superconducting alloys19702585075092-s2.0-374314773410.1103/PhysRevLett.25.507Wolf-GladrowD.A lattice Boltzmann equation for diffusion1995795-6102310322-s2.0-054239036110.1007/BF02181215IshimaruA.Diffusion of light in turbid material1989281222102215PodlubnyI.1999198San Diego, Calif, USAAcademic PressMathematics in Science and EngineeringMR1658022HilferR.2000River Edge, NJ, USAWorld Scientific Publishing10.1142/9789812817747MR1890104KilbasA. A.SrivastavaH. M.TrujilloJ. J.2006204Amsterdam, The NetherlandsElsevier Science B.V.North-Holland Mathematics StudiesMR2218073SabatierJ.AgrawalO. P.MachadoJ. A. T.2007Dordrecht, The NetherlandsSpringer10.1007/978-1-4020-6042-7MR2432163MainardiF.2010London, UKImperial College Press10.1142/9781848163300MR2676137ZaslavskyG. M.Chaos, fractional kinetics, and anomalous transport2002371646158010.1016/S0370-1573(02)00331-9MR1937584ZBL0999.82053BaleanuD.MachadoJ. A. T.LuoA. C. J.2012New York, NY, USASpringer10.1007/978-1-4614-0457-6MR2905887KlafterJ.LimS. C.MetzlerR.2012SingaporeWorld Scientific PublishingMR2920446BaleanuD.DiethelmK.ScalasE.TrujilloJ. J.20123SingaporeWorld Scientific PublishingSeries on Complexity, Nonlinearity and Chaos10.1142/9789814355216MR2894576WyssW.The fractional diffusion equation198627112782278510.1063/1.527251MR861345ZBL0632.35031ChavesA. S.A fractional diffusion equation to describe Lévy flights19982391-2131610.1016/S0375-9601(97)00947-XMR1616103ZBL1026.82524SokolovI. M.ChechkinA. V.KlafterJ.Fractional diffusion equation for a power-law-truncated Lévy process20043363-42452512-s2.0-164261746210.1016/j.physa.2003.12.044ChechkinA. V.GorenfloR.SokolovI. M.Fractional diffusion in inhomogeneous media20053842L679L68410.1088/0305-4470/38/42/L03MR2186196ZBL1082.76097MainardiF.PagniniG.The Wright functions as solutions of the time-fractional diffusion equation20031411516210.1016/S0096-3003(02)00320-XMR1984227ZBL1053.35008TadjeranC.MeerschaertM. M.SchefflerH.-P.A second-order accurate numerical approximation for the fractional diffusion equation2006213120521310.1016/j.jcp.2005.08.008MR2203439ZBL1089.65089HristovJ.Approximate solutions to fractional subdiffusion equations201119312292432-s2.0-7995367959210.1140/epjst/e2011-01394-2HristovJ.A short-distance integral-balance solution to a strong subdiffusion equation: a Weak Power-Law Profile201025555563KolwankarK. M.GangalA. D.Local fractional Fokker-Planck equation199880221421710.1103/PhysRevLett.80.214MR1604435ZBL0945.82005CarpinteriA.ChiaiaB.CornettiP.Static-kinematic duality and the principle of virtual work in the mechanics of fractal media20011911-23192-s2.0-003583454210.1016/S0045-7825(01)00241-9Ben AddaF.CressonJ.About non-differentiable functions2001263272173710.1006/jmaa.2001.7656MR1866075ZBL0995.26006BabakhaniA.Daftardar-GejjiV.On calculus of local fractional derivatives20022701667910.1016/S0022-247X(02)00048-3MR1911751ZBL1005.26002JumarieG.Table of some basic fractional calculus formulae derived from a modified Riemann-Liouville derivative for non-differentiable functions200922337838510.1016/j.aml.2008.06.003MR2483503ZBL1171.26305ChenW.SunH.ZhangX.KorošakD.Anomalous diffusion modeling by fractal and fractional derivatives20105951754175810.1016/j.camwa.2009.08.020MR2595948ZBL1189.35355YangX. J.2011Hong Kong, ChinaAsian Academic PublisherYangX. J.2012New York, NY, USAWorld Science PublisherYangX. J.BaleanuD.Fractal heat conduction problem solved by local fractional variation iteration method2013172625628YangX. J.SrivastavaH. M.HeJ. H.BaleanuD.Cantor-type cylindrical-coordinate method for differential equations with local fractional derivatives201337728–3016961700HuM.-S.AgarwalR. P.YangX.-J.Local fractional Fourier series with application to wave equation in fractal vibrating string2012201215567401MR3004869ZBL1257.35193YangY. J.BaleanuD.YangX. J.A local fractional variational iteration method for Laplace equation within local fractional operators20132013620265010.1155/2013/202650