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Analysis of Electromagnetic Fields andWaves in Fractional Dimensional Space
ByMuhammad Junaid Mughal
Associate Professor Faculty of Electronic Engineering
GIK Institute of Engineering Sciences and TechnologyPakistan
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Motivation
Introduction to Fractional Space
Problem Statement
Research Work Differential Electromagnetic Equations in Fractional Space
Electromagnetic Wave Propagation in Fractional SpaceConclusions
References
O utline
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Wave Propagation in 3-D Dielectric Media
Solution of wave propagation problems inEuclidean space requires
Classical MaxwellEquations Continuity Equation Wave Equation Boundary Conditions
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Wave Propagation in Complex Fractal Structures
Wave propagation in suchcomplex structures cannot bestudied easily using classicalMaxwell Equations inEuclidean space.
But, in fact, the concept of fractional dimensional spacecan applied in order to studythe wave propagation in suchcomplex geometry usingModified Maxwell Equationsin fractional space . Menger Sponge
(3D View)
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If we take an objectembedded in Euclideandimension D and reduce itslinear size by 1/r in eachspatial direction, its measure(length, area, or volume)would increase by
N= r D times the original.
D = log(N)/log(r)
D could be a fraction, if it isfractal geometry.
What is Dimension of space?
Introduction Fractional Space (contd.)
N = 8=2 3
N = 27=3 3
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A square may be brokeninto N2 self-similar pieces,each with magnificationfactor N can produce original
square. So, we can write
What is Dimension of space?
Introduction Fractional Space (contd.)
N = 2 N = 3
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Similarly, A cube may be broken into N3 self-similar pieces, each withmagnification factor N can
produce original cube. So, wecan write the dimension of acube is:
What is Dimension of space?
Introduction Fractional Space (contd.)
N = 2 N = 3
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Now consider theSierpinski triangleconsists of 3 self-similar
pieces, each with
magnification factor 2can generate originaltriangle. So the fractaldimension is
Fractal Dimension and Fractional Dimensional Space
Introduction Fractional Space (contd.)
This estimates that the Sierpinski triangle is somewhere in between lines and planes. Similarly, many fractal structures are known in literature that possess afractal dimension. Roughly speaking, we state that the space embedding suchfractal curves or surfaces is known as fractional dimensional space .
N = 2
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Fractal Dimension and Fractional Dimensional Space
Introduction Fractional Space (contd.)
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Another example of fractal shape isMenger Sponge.Its fractal dimension
is 2.727 (approx).
Fractal Dimension and Fractional Dimensional Space
Introduction Fractional Space (contd.)
Replacement number = 20Scale = 1/3Fractal dimension = log 20 / log 3 = 2.727
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Modified MaxwellsEquations in D-dimensional
fractional Space Realizable/ O rdinary
Boundary Conditions
Wave Propagation in Complex Fractal Structures
Classical MaxwellsEquations in 3-dimensional
Euclidean Space Unrealizable/Difficult
Boundary Conditions
D=2.727D=3 D=3Apply FractionalSpace Concept
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Euclidean Space Differential Volume
Elements are
Volume is given by
Dimensional Regularization
Introduction Fractional Space (contd.)
321 d xd xd xdV !
;! dV V
Fractional Space Differential Volume
Elements are [13]
where,
Volume is given by
321321
xd xd xd dV DEEE
!
;!
D
D DdV V
L imits will be discontinuous in caseof fractals like Menger Sponge,Sierpinski triangle etc.
[13] Muslih, Sami I.,Om P. Agrawal,A scaling method and its applications to problems in fractional dimensional space,"Journal of Mathematical Physics, Volume 50,Issue 12, pp. 123501-123501-11, 2009.
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Problem Statement
A theoretical investigation of electromagnetic fields and waves infractional dimensional space or simply fractional space can beachieved by
Establishing novel differential electromagnetic equations in fractionalspaceStudying electromagnetic wave propagation in fractional spaceDefining potentials for static and time-varying fields in fractional spaceStudying electromagnetic radiations from sources in fractional spac e
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1. Differential Electromagnetic Equations inFractional Space
This problem is further sub-divided as:a. Fractional Space Generalization of L aplacian Operator (Review)
b. Fractional Space Generalization of Del operator and Related DifferentialOperators
i. Del Operator in Fractional Space
ii. Gradient Operator in Fractional Spaceiii. Divergence Operator in Fractional Spaceiv. Curl Operator in Fractional Space
c. Fractional Space Generalization of Differential Maxwell's Equationsd. Fractional Space Generalization of the Helmholtz's Equatione. Fractional Space Generalization of Potentials for Static & Time-Varying
Fields, Poisson's and L aplace's Equations
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Stillinger [1] provided a formalism for integration of radially symmetricfunction f(r) in an D -dimensional fractional space is given by
where,
with
Using this formulation a single variable
L
aplacian operator is derived inD
-dimensional fractional space as:
1a. Fractional Space Generalization of Laplacian O perator
Stillingers Formalism
[1] F. H. Stillinger, Axiomatic basis for spaces with noninteger dimension,J. Math, Phys.,18 (6), 1224-1234, 1977.
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Plamer and Stavrinou [3] generalized the Stillingers results to n orthogonalcoordinates and L aplacian operator in D -dimensional fractional space in three-spatial coordinates is given as:
where, parameters (0 < 1 1, 0 < 2 1 and 0 < 3 1) are used to describethe measure distribution of space where each one is acting independently on asingle coordinate and the total dimension of the system is D = 1 + 2 + 3 .
[3] C. Palmer and P.N. Stavrinou, Equations of motion in a noninteger- dimension space, J. Phys. A, 37 , 6987-7003, 2004.
1a. Fractional Space Generalization of Laplacian O perator (contd.)
Palmer and Stavrinous Formalism
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From [3] we have
We wish to find: We assume
For ,we get,
So in single-variable
Extending above procedure to three-variable case, for
[3] C. Palmer and P.N. Stavrinou, Equations of motion in a noninteger- dimension space, J. Phys. A, 37 , 6987-7003, 2004.
1b. Fractional Space Generalization of DelO perator and Related O perators
Del O perator in Fractional Space
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1b. Fractional Space Generalization of DelO perator and Related O perators (contd.)
Gradient O perator in Fractional Space
Divergence O perator in Fractional Space
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1b. Fractional Space Generalization of DelO perator and Related O perators (contd.)
Curl O perator in Fractional Space
or
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1c. Fractional Space Generalization of Differential Maxwells Equations
Differential form of Maxwell's equations in far field region in the fractionalspace as follows
and the Continuity Equation in fractional space is
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1d. Fractional Space Generalization of Helmholtzs Equation
Helmholtzs wave Equation in fractional space for electric field:
Helmholtzs wave Equation in fractional space for magnetic field:
in source-free region
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1e. Potentials for Static and Time-VaryingFields in Fractional Space
Poissons Equation in the fractional space:
Vector Equivalent of Poissons Equation in fractional space:
L aplaces Equation in the fractional space:
= Scalar electric potential
= Scalar magnetic potential
= Vector magnetic potential
Fractional Space Generalization of Potentials for Time-Varying
Fields :
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A novel fractional space generalization of the differential electromagneticequations, that is helpful in studying the behavior of electric and magneticfields in fractal media, is provided.
A new form of vector differential operator Del and its related differentialoperators is formulated in fractional space. U sing these modified vector
differential operators, the classical Maxwell's electromagnetic equationshave been worked out. The L aplace's, Poisson's and Helmholtz's equations in fractional space are
derived by using modified vector differential operators. For all investigated cases, when integer dimensional space is considered, the
classical results can be recovered. The provided fractional space generalization of differential electromagnetic
equations is valid in far-field region only.
_____________________________________________________
1. Summary: Differential Electromagnetic Equations in Fractional Space
[RP 1] M. Zubair, M. J. Mughal, Q.A. Naqvi and A.A. Rizvi Differential electromagnetic equations in fractional
space. Progress in Electromagnetic Research, Vol. 114, page 255-269, 2011.
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2. Electromagnetic Wave Propagation in FractionalSpace
This problem is further sub-divided as:a. General Plane Wave Solutions in Fractional Space: L ossless Medium Case
b. General Plane Wave Solutions in Fractional Space: L ossy Medium Casec. Cylindrical Wave Propagation in Fractionald. Spherical Wave Propagation in Fractional Space
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2a. General Plane Wave Solutions in FractionalSpace : Lossless Medium Case
Helmholtzs equation in source free and lossless medium:
, , Once the solution to any one of above equations in fractional space is
known, the solution to the other can be written by an interchange of E withH or H with E due to duality.
In rectangular coordinates, a general solution for E can be written as
In expanded form Helmholtzs equation is
An Exact Solution of Helmholtzs Equation in Fractional Space
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2a. General Plane Wave Solutions in FractionalSpace : Lossless Medium Case
which reduces to three ODEs as:
where,
The solution for any one of them in fractional space can be replicated for others by inspection.
An Exact Solution of Helmholtzs Equation in Fractional Space (contd.)
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2a. General Plane Wave Solutions in FractionalSpace : Lossless Medium Case
U sing separation of variables the final solution:
where,
An Exact Solution of Helmholtzs Equation in Fractional Space (contd.)
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2a. General Plane Wave Solutions in FractionalSpace : Lossless Medium Case
As a special case, for three-dimensional space, this problem reduces toclassical wave propagation concept. This validates our solution.
Discussion on Fractional Space Solutions
As an example, an infinite sheet of surface currentcan be considered as a source of plane waves in D -dimensional fractional space. Then Correspondingwave equations and their solution is :
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2a. General Plane Wave Solutions in FractionalSpace : Lossless Medium Case
Discussion on Fractional Space Solutions (contd.)
Figure 2: Wave propagation in fractional space ( D =2.5).
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2a. General Plane Wave Solutions in FractionalSpace : Lossless Medium Case
The solution for the usual wave for z > 0 with D = 3 is shown in Figure 1,which is comparable to well known plane wave solutions in 3-dimensionalspace [42].
Similarly, for D = 2.5 we have fractal medium wave for z > 0 as shown in
Figure 2, where amplitude variations are described in terms of Besselfunctions.
This shows a localization of plane waves in fractal media.
[42] C. A. Balanis, `AdvancedEngineeringElectromagnetics", New York: Wiley, 1989.
Discussion on Fractional Space Solutions (contd.)
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2b. General Plane Wave Solutions in FractionalSpace : Lossy Medium Case
which reduces to three ODEs as:
where,
The solution for any one of them in fractional space can be replicated for others by inspection.
An Exact Solution of Helmholtzs Equation in Fractional Space (contd.)
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2b. General Plane Wave Solutions in FractionalSpace : Lossy Medium Case
U sing separation of variables the final solution:
where,
An Exact Solution of Helmholtzs Equation in Fractional Space (contd.)
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2b. General Plane Wave Solutions in FractionalSpace : Lossy Medium Case
Discussion on Fractional Space Solutions As an example, an infinite sheet of surface current
can be considered as a source of plane waves in D -dimensional fractional space. Then Correspondingwave equations and their solution is :
where,
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2b. General Plane Wave Solutions in FractionalSpace : Lossy Medium Case
Discussion on Fractional Space Solutions (contd.)
Figure 3: U sual wave propagation ( D =3) in lossy medium.
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2b. General Plane Wave Solutions in FractionalSpace : Lossy Medium Case
Discussion on Fractional Space Solutions (contd.)
Figure 4: Wave propagation in fractional space ( D =2.5) or lossy fractal media.
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2c. Cylindrical Wave Propagation in FractionalSpace
The scalar Helmholtzs equation describes the phenomenon of cylindricalwave propagation in fractional space and give as:
where, is a scalar function that can represent a field or vector potential component.
In cylindrical coordinate system the L aplacian operator is given by
Now we using separation of variables :
An Exact Solution of Cylindrical Wave Equation in Fractional Space
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2c. Cylindrical Wave Propagation in FractionalSpace
the resulting ordinary differential equations are obtained as follows:
where, The final solution is found as:
where, and
An Exact Solution of Cylindrical Wave Equation in Fractional Space (contd.)
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2c. Cylindrical Wave Propagation in FractionalSpace
As a special case, for three dimensional space D = 3, this problem reducesto classical wave propagation results [42].
Now, we assume that a cylindrical wave exists in a fractional space due tosome infinite line source. The radial amplitude variations of scalar field infractional space which are given by
U sing asymptotic expansions we see that
[42] C. A. Balanis, `AdvancedEngineering Electromagnetics", New York: Wiley, 1989.
Discussion on Fractional Space Solutions
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2c. Cylindrical Wave Propagation in FractionalSpace
Discussion on Fractional Space Solutions (contd.)
Figure 5: Cylindrical wave propagation in Euclidean space ( D =3).
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2c. Cylindrical Wave Propagation in FractionalSpace
Discussion on Fractional Space Solutions (contd.)
Figure 6: Cylindrical wave propagation in fractional space ( D =2.5).
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2d. Spherical Wave Propagation in FractionalSpace
The scalar Helmholtzs equation describes the phenomenon of sphericalwave propagation in fractional space and give as:
where, is a scalar function that can represent a field or vector potential component.
In spherical coordinate system the L aplacian operator is given by [3]
where, Now we using separation of variables :
[3] C. Palmer and P.N. Stavrinou, Equations of motion in a noninteger- dimension space, J. Phys. A, 37 , 6987-7003,2004.
An Exact Solution of Spherical Wave Equation in Fractional Space
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2d. Spherical Wave Propagation in FractionalSpace
the resulting ordinary differential equations are obtained as follows:
The final solution is found as:
An Exact Solution of Spherical Wave Equation in Fractional Space (contd.)
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2d. Spherical Wave Propagation in FractionalSpace
As a special case, for three dimensional space D = 3, this problem reducesto classical wave propagation results [42].
Now, we assume that a spherical wave exists in a fractional space due tosome point source. The radial amplitude variations of scalar field infractional space which are given by
U sing asymptotic expansions we see that
[42] C. A. Balanis, `AdvancedEngineering Electromagnetics", New York: Wiley, 1989.
Discussion on Fractional Space Solutions
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2d. Spherical Wave Propagation in FractionalSpace
Discussion on Fractional Space Solutions (contd.)
Figure 8: Spherical wave propagation in Euclidean space ( D =3).
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2d. Spherical Wave Propagation in FractionalSpace
Discussion on Fractional Space Solutions (contd.)
Figure 9: Spherical wave propagation in fractional space ( D =2.5).
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2d. Spherical Wave Propagation in FractionalSpace
Discussion on Fractional Space Solutions (contd.)
Figure 10: Spherical wave propagation in fractional space ( D =2.1).
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Analytical solutions of the plane-, cylindrical- and spherical wave equationsare obtained in D -dimensional fractional space.
The obtained fractional space solution provides a generalization of electromagnetic wave propagation phenomenon from integer space tofractional space.
For all investigated results when integer dimension is considered, theclassical results are recovered.
The presented fractional space solutions of the wave equation can be used todescribe the phenomenon of wave propagation in any fractal media
________________________________________________________________
2. Summary: Electromagnetic Wave Propagation in Fractional Space
[RP2 ] M. Zubair , M. J. Mughal, and Q.A. Naqvi, The wave equation and general plane wave solutions in fractionalspace. Progress in Electromagnetic Research L etters, Vol. 19, 137-146, 2010.
[RP3 ] M. Zubair , M. J. Mughal, Q.A. Naqvi , On electromagnetic wave propagation in fractional space, Non-linear Analysis B: Real World Applications, 2011 (in Press), doi:10.1016/j.norwa.2011.04.010
[RP4 ] M. Zubair , M. J. Mughal, and Q.A. Naqvi, An exact solution of cylindrical wave equation for electromagneticfield in fractional dimensional space. Progress in Electromagnetic Research, Vol. 114, page 443-455, 2011.
[RP5 ] M. Zubair , M. J. Mughal, and Q.A. Naqvi, An exact solution of spherical wave in D-dimensional fractionalspace. Journal of Electromagnetic Waves and Applications Vol. 25, 14811491, 2011.
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Summary: O verall
Integration over D-dimensional Fractionalspace
DimensionalRegularization
Fractional SpaceRepresentation
L aplacian operator Gradient operator Divergence operator Curl operator
Differential operatorsin Fractional Space
Modified MaxwellsEquations Helmholtzs Equation
L aplaces Equation Poissons Equation
Modified differentialelectromagnetic
Equations
Solution of modifiedelectromagneticequations in Cartesian,Cylindrical andSpherical coordinates
EM wave propagationand radiation in
Fractional Space
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This work describes a theoretical investigation of electromagnetic fields andwaves in fractional dimensional space which is useful to study the behavior of electromagnetic fields and waves in fractal media.
A novel fractional space generalization of the differential electromagneticequations was provided.
Most of the further work was related to solution of the establisheddifferential electromagnetic equations in fractional space.
The phenomenon of electromagnetic wave propagation in fractional spacewas studied in detail by providing full analytical plane-, cylindrical- andspherical-wave solutions of the vector wave equation in D -dimensionalfractional pace.
An analytical solution procedure for radiation problems in fractional spacehas also been proposed.
Conclusions
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Zubair, Muhammad, Mughal ,Muhammad Junaid , Naqvi, Q. A.(Authors), Electromagnetic Fieldsand Waves in Fractional DimensionalSpace, SpringerBriefs in AppliedSciences and Technology , Springer
ISBN 978-3-642-25357-7, 2012, XII,76 p.
Published Work
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M. Zubair, M. J. Mughal, and Q.A. Naqvi, An exact solution of spherical wave in D-dimensional fractional space. Journal of Electromagnetic Waves and Applications Vol.25, 14811491, 2011. (Impact Factor:1.376)
M. Zubair, M. J. Mughal, Q.A. Naqvi , On electromagnetic wave propagation infractional space, Non-linear Analysis B: Real World Applications, 2011, Volume 12,Issue 5, 2844-2850, 2011. (Impact Factor:2.138)
M. Zubair, M. J. Mughal, Q.A. Naqvi and A.A. Rizvi Differential electromagneticequations in fractional space. Progress in Electromagnetic Research, Vol. 114, page255-269, 2011. (Impact Factor:3.745)
M. Zubair, M. J. Mughal, and Q.A. Naqvi, An exact solution of cylindrical waveequation for electromagnetic field in fractional dimensional space. Progress inElectromagnetic Research, Vol. 114, page 443-455, 2011. (Impact Factor:3.745)
M. Zubair, M. J. Mughal, and Q.A. Naqvi, The wave equation and general plane wavesolutions in fractional space. Progress in Electromagnetic Research L etters, Vol. 19,137-146, 2010.
M. J. Mughal, M. Zubair, Fractional space solutions of antenna radiation problems: Anapplication to Hertzian Dipole, Proc. 19 th IEEE Conference on Signal Processing andCommunications Applications, Antalya Turkey, 2011.
List of Publications
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[1] F. H. Stillinger, Axiomatic basis for spaces with noninteger dimension,J. Math, Phys.,18 (6), 1224-1234, 1977.[2] He, X., Anisotropy and isotropy: a model of fraction-dimensional space, PSolid State Commun., 75, 111-114, 1990.[3] Palmer, C., P.N. Stavrinou, \Equations of motion in a noninteger-dimension space, "J. Phys. A,37 , 6987-7003, 2004.[4] Willson, K.G. , Quantum eld-theory, models in less than 4 dimensions," Phys. Rev. D 7 (10), 2911-2926, 1973[5] Mandelbrot, B.,The Fractal Geometry of Nature," W.H. Freeman, New York, 1983.[6] Bollini, C.G., J .J. Giambiagi, \Dimensional renormalization: The number of dimensions as a regularizing parameter, Nuovo Cimento B, 12, 20-26, 1972.[7] Ashmore, J.F., On renormalization and complex space-time dimensions, Commun. Math. Phys., 29, 177-187, 1973.[8] Agrawal, O.P., Formulation of Euler- L agrange equations for fractional varia- tional problems," J. Math. Anal. Appl., 271 (1),368-379, 2002.[9] Baleanu, D., S. Muslih, " L agrangian formulation of classical fields within Riemann- L iouville fractional derivatives," Phys. Scripta, 72 (23), 119-121, 2005.[10] Tarasov, V.E., Electromagnetic elds on fractals," Modern Phys. L ett. A, 21 (20), 1587-1600, 2006.[11] Tarasov, V.E.,Continuous medium model for fractal media,"Physics L etters A, Volume 336, Issues 2-3, 2005.[12] Muslih S., D. Baleanu, Fractional Multipoles in fractional space," Nonlinear Analysis: Real World Applications,8, 198-203, 2007.[13] Muslih, Sami I.,Om P. Agrawal,A scaling method and its applications to problems in fractional dimensional space,"Journal of Mathematical Physics,
Volume 50, Issue 12, pp. 123501-123501-11, 2009.[14] Baleanu, D., AK Golmankhaneh and Ali K. Golmankhaneh, On electromagnetic field in fractional space," Nonlinear Analysis: Real World Applications,
Volume 11, Issue 1, 288-292 , 2010.[15] Wang, Zhen-song and L u, Bao-wei,The scattering of electromagnetic waves in fractal media," Waves in Random and Complex Media, 4: 1, 97-103,
1994.[16] Bender, C.M.,K.A. Milton, Scalar Casimir effect for a D-dimensional sphere,"Phys. Rev. D,50, 6547-6555, 1994.[17] Muslih S., D. Baleanu, Mandelbrot scaling and parametrization invariant theories," Romanian Reports in Physics, Volume 62, Issue 4, 689-696, 2010.[18] Muslih S., M. Saddallah, D. Baleanu and E. Rabe, L agrangian formulation of maxwell's field in fractional D dimensional space-time," Romanian Reports
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49, Number 2, 270-275, 2010.[20] Muslih S. , Solutions of a Particle with Fractional [delta]-Potential in a Fractional Dimensional Space," International Journal of Theoretical Physics,
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