FIT2011-Dr Mughal Fractional Space

8/3/2019 FIT2011-Dr Mughal Fractional Space

1/60

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


2/60

Ghulam Ishaq Khan Institute of Engineering Sciences & Technology, Pakistan 2

Motivation

Introduction to Fractional Space

Problem Statement

Research Work Differential Electromagnetic Equations in Fractional Space

Electromagnetic Wave Propagation in Fractional SpaceConclusions

References

O utline


3/60


Wave Propagation in 3-D Dielectric Media

Solution of wave propagation problems inEuclidean space requires

Classical MaxwellEquations Continuity Equation Wave Equation Boundary Conditions


4/60


5/60


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)


6/60


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


7/60


A square may be brokeninto N2 self-similar pieces,each with magnificationfactor N can produce original

square. So, we can write



N = 2 N = 3


8/60


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:



N = 2 N = 3


9/60


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


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


10/60





11/60


Another example of fractal shape isMenger Sponge.Its fractal dimension

is 2.727 (approx).



Replacement number = 20Scale = 1/3Fractal dimension = log 20 / log 3 = 2.727


12/60


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


13/60


Euclidean Space Differential Volume

Elements are

Volume is given by

Dimensional Regularization


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.


14/60


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


15/60


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


16/60


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.


17/60


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


18/60


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


19/60


1b. Fractional Space Generalization of DelO perator and Related O perators (contd.)

Gradient O perator in Fractional Space

Divergence O perator in Fractional Space


20/60


1b. Fractional Space Generalization of DelO perator and Related O perators (contd.)

Curl O perator in Fractional Space

or


21/60


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


22/60


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


23/60


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 :


24/60


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.


25/60


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


26/60


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


27/60



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.)


28/60



U sing separation of variables the final solution:

where,



29/60



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 :


30/60


31/60



Discussion on Fractional Space Solutions (contd.)

Figure 2: Wave propagation in fractional space ( D =2.5).


32/60



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.



33/60


34/60


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.



35/60



U sing separation of variables the final solution:

where,



36/60



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,


37/60




Figure 3: U sual wave propagation ( D =3) in lossy medium.


38/60




Figure 4: Wave propagation in fractional space ( D =2.5) or lossy fractal media.


39/60


40/60


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


41/60



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.)


42/60



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.



43/60




Figure 5: Cylindrical wave propagation in Euclidean space ( D =3).


44/60




Figure 6: Cylindrical wave propagation in fractional space ( D =2.5).


45/60


46/60


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


47/60



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.)


48/60



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.



49/60




Figure 8: Spherical wave propagation in Euclidean space ( D =3).


50/60




Figure 9: Spherical wave propagation in fractional space ( D =2.5).


51/60




Figure 10: Spherical wave propagation in fractional space ( D =2.1).


52/60


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.


53/60


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


54/60


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


55/60


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


56/60


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


57/60


58/60


[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

in Physics, Volume 55, Issue 7-8, 659-663, 2010.[19] Muslih S. and Om P. Agrawal, Riesz Fractional Derivatives and Fractional Dimensional Space ," International Journal of Theoretical Physics, Volume

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,

Volume 49, Number 9, 2095-2104, 2010.[21] Muslih S. and Om P. Agrawal, A scaling method and its applications to problems in fractional dimensional space," Journal of Mathematical Physics,

50,123501, 2009.[22] Rajeh Eid, Sami I. Muslih, Dumitru Baleanu and E. Rabei , On fractional Schrodinger equation in [alpha]-dimensional fractional space," Nonlinear

Analysis: Real World Applications, Volume 10, Issue 3, 1299-1304, 2009.[23] Sadallah, M. and Sami I. Muslih , Solution of the Equations of Motion for Einsteins Field in Fractional D Dimensional Space-Time ," International

Journal of Theoretical Physics, Volume 48, Number 12, 3312-3318, 2009.[24] Muslih S., D. Baleanu, E. Rabe, E.M. , Solutions of Massless Conformal Scalar Field in an n-Dimensional Einstein Space," Acta Physica Polonica Series

B, Vol 39, Numb 4, Pages 887-892, 2008.[25] Oldham, K.B., J. Spanier,The Fractional Calculus,"Academic Press, New York,1974.

References


59/60


[26] Hussain, A. and Q. A. Naqvi, Fractional rectangular impedance waveguide, Progress In Electromagnetics Research, Vol. 96, 101-116, 2009.[27] Naqvi Q. A., Planar slab of chiral nihility metamaterial backed by fractional dual/PEMC interface," Progress In Electromagnetics Research, Vol. 85,

381-391, 2008.[28] Naqvi Q. A.,Fractional dual interface in chiral nihility medium,"Progress In Electromagnetics Research L etters, Vol. 8, 135-142, 2009.[29] Naqvi Q. A.,Fractional Dual Solutions in Grounded Chiral Nihility Slab and their Effect on Outside Fields ,"Journal of Electromagnetic Waves and

Applications, Volume 23, Numbers 5-6, pp. 773-784(12), 2009.[30] Naqvi, A., S. Ahmed, Q.A. Naqvi,Perfect Electromagnetic Conductor and Fractional Dual Interface Placed in a Chiral Nihility Medium ,"Journal of

Electromagnetic Waves and Applications, Volume 24, Numbers 14-15, pp. 1991-1999(9), 2010.[31] Naqvi, A., A. Hussain, Q.A. Naqvi,Waves in Fractional Dual Planar Waveguides Containing Chiral Nihility Metamaterial ,"Journal of Electromagnetic

Waves and Applications, Volume 24, Numbers 11-12, pp. 1575-1586(12), 2010.[32] Veliev, E. I., M. V. Ivakhnychenko, and T. M. Ahmedov,``Fractional Boundary Conditions in Plane Waves Diffraction on a Strip,"Progress In

Electromagnetics Research, Vol. 79, 443-462, 2008.[33] Naqvi, S. A., M. Faryad, Q. A. Naqvi, and M. Abbas,``Fractional Duality in Homogeneous Bi-Isotropic Medium ,"Progress In Electromagnetics

Research, Vol. 78, 159-172, 2008.[34] Sangawa, U .,``The Origin of Electromagnetic Resonances in Three-Dimensional Photonic Fractals ,"Progress In Electromagnetics Research, Vol.94, 153-

173, 2009.[35] Teng, H. T., H.T. Ewe, and S. L . Tan,``Multifractal Dimension and its Geometrical Terrain Properties for Classification of Multi-Band Multi-Polarized

SAR Image,"Progress In Electromagnetics Research, Vol. 104, 221-237, 2010.[36] Mahatthanajatuphat, C., S. Saleekaw, P. Akkaraekthalin, and M. Krairiksh,`À Rhombic Patch Monopole Antenna with Modified Minkowski Fractal

Geometry for U MTS, W L AN, and Mobile WiMAX Application ,"Progress In Electromagnetics Research, Vol. 89, 57-74, 2009.[37] Mahatthanajatuphat, C., P. Akkaraekthalin, S. Saleekaw, and M. Krairiksh,`À Bidirectional Multiband Antenna with Modied Fractal Slot FED by CPW

,"Progress In Electromagnetics Research, Vol. 95, 59-72, 2009.[38] Karim, M. N. A, M. K. A. Rahim, H. A. Majid, O. B. Ayop, M. Abu, and F. Zubir,`` L og Periodic Fractal Koch Antenna for U HF Band Applications

,"Progress In Electromagnetics Research, Vol. 100, 201-218, 2010.[39] Siakavara, K.,``Novel Fractal Antenna Arrays for Satellite Networks: Circular Ring Sierpinski Carpet Arrays Optimized by Genetic Algorithms

,"Progress In Electromagnetics Research, Vol. 103, 115-138, 2010.[40] He, Y., L . L i, C.H. L iang, and Q. H. L iu ,`ÈBG Structures with Fractal Topologies for U ltra-Wideband Ground Bounce Noise Suppression ,"Journal of

Electromagnetic Waves and Applications, Volume 24, Numbers 10, pp. 1365-1374(10), 2010.[41] Abramowitz, M., Stegun, I.A., ``Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables.", U .S. Department of

Commerce., 1972[42] Balanis, C.A. , `Àdvanced Engineering Electromagnetics", New York: Wiley, 1989.[43] Jackson. J. D,``Classical Electrodynamics,"John Wiley, New York, 3rd ed., 1999.[44] C. A. Balanis,`Àntenna Theory: Analysis and Design", New York: Wiley, 1982.[45] Polyanin A. D., ``Handbook of Exact Solutions for Ordinary Differential Equations", second edition,CRC Press,2003.

References


60/60

Thank you.

FIT2011-Dr Mughal Fractional Space

Documents

Transcript of FIT2011-Dr Mughal Fractional Space