# Computing the Electric and Magnetic Matrix Green’s Functions in a Rectangular Parallelepiped with a Perfect Conducting Boundary

Authors: V. G. Yakhno; Ş. Ersoy
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

A method for the approximate computation of frequency-dependent magnetic and electric matrix Green’s functions in a rectangular parallelepiped with a perfect conducting boundary is suggested in the paper. This method is based on approximation (regularization) of the Dirac delta function
and its derivatives, which appear in the differential equations for magnetic and electric Green’s functions, and the Fourier series expansion meta-approach
for solving the elliptic boundary value problems. The elements of approximate Green’s functions are found explicitly in the form of the Fourier series with a finite number of terms. The convergence analysis for finding the number of the terms is given. The computational experiments have confirmed the robustness of the method.


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## Body

## 1. Introduction

In recent years a lot of attention has been devoted to derivation of the electric and magnetic fields inside bounded domains with perfect conducting boundaries [1–4]. The study of the distribution of electromagnetic fields inside a real indoor environment (cabinets, desks, etc.) has the great practical interest (see, e.g., [5]). The theory (existence, uniqueness, and stability estimate theorems) of time-dependent Maxwell’s equations in bounded domains with perfect conducting boundary conditions and smooth current and charge densities has been described by Dautray and Lions in [6]. These theoretical results form the background for many computational methods such as finite element method, finite-difference method, finite-difference time domain method, modal expansion method, and Nitsche type method, which have been handled for computation of the electric and magnetic fields in bounded domains with a perfect conducting boundary [1–3, 6–8].We need to note that the solution (both analytical and numerical) of the initial value and initial boundary value problems in electrodynamics is often facilitated by introduction of electric and magnetic Green’s functions [9–17], but the derivations of electric and magnetic Green’s functions have been made in a free space or unbounded domain. The computation of time-dependent electric and magnetic Green’s functions for the bounded domain with perfect conducting conditions has not been achieved so far.In our paper we suggest a new method for the approximate computation of frequency-dependent magnetic and electric matrix Green’s functions in a rectangular parallelepiped with perfect conducting boundary. This method consists of the following. The equations for magnetic Green’s function are written in the special form which does not contain elements of electric Green’s function. These equations are elliptic partial differential equations and contain the components of the generalized vector functions (distributions)curlx[δ(x-x0)es→], s=1,2,3 in their right hand sides. We approximate components of these generalized vector functions by the Fourier series with a finite number of terms and then applied standard Fourier series expansion meta-approach for solving the boundary value problem for the elliptic partial differential equations in the bounded region. After calculation of all columns of magnetic matrix Green’s function the approximate computation of the columns of electric Green’s function is given by explicit calculation of the differential operator curlx from each column of magnetic matrix Green’s function with respect to 3D space variable x=(x1,x2,x3).


## 2. Frequency-Dependent Magnetic and Electric Green’s Functions

Letb1,b2,b3 be given positive numbers and let
(1)V={x∈R3:0<x1<b1,0<x2<b2,0<x3<b3}
be a rectangular parallelepiped; Γ is the boundary of V; ω∈R (frequency, ω>0), let x0=(x10,x20,x30)∈V be 1D and 3D parameters, respectively. The 3×3 matrices
(2)GH(x,ω,x0)=(H11(x,ω,x0)H12(x,ω,x0)H13(x,ω,x0)H21(x,ω,x0)H22(x,ω,x0)H23(x,ω,x0)H31(x,ω,x0)H32(x,ω,x0)H33(x,ω,x0)),GE(x,ω,x0)=(E11(x,ω,x0)E12(x,ω,x0)E13(x,ω,x0)E21(x,ω,x0)E22(x,ω,x0)E23(x,ω,x0)E31(x,ω,x0)E32(x,ω,x0)E33(x,ω,x0)),
whose columns Hs(x,ω,x0), Es(x,ω,x0), and s=1,2,3 satisfy
(3)curlxHs(x,ω,x0)=-iωϵEs(x,ω,x0)+δ(x-x0)es→,hhhhhhhhhhhhhhhhhhhhhhx∈V,(4)curlxEs(x,ω,x0)=iμωHs(x,ω,x0),(5)(Es×n→)|Γ=0,(Hs·n→)|Γ=0,s=1,2,3,
are called frequency-dependent magnetic and electric Green’s functions in V, respectively. Here i is the imaginary unit (i2=-1); ϵ>0 and μ>0 are given constants characterizing the permittivity and permeability of the electromagnetic medium, which is located inside V; es→ are the basis vectors from Euclidean space R3 (e1→=(1,0,0)T,e2→=(0,1,0)T,e3→=(0,0,1)T); x0=(x10,x20,x30)∈V is the 3D parameter; δ(x-x0)=δ(x1-x10)δ(x2-x20)δ(x3-x30), and δ(xj-xj0) are the Dirac delta distribution concentrated at xj=xj0 and j=1,2,3; n→ is an outward unit normal vector on Γ. The relations in (5) are called the perfect conducting boundary conditions.Remark 1.
We note thatdiv⁡xHs(x,ω,x0)=0 for all x∈V is a consequence of (4).Applyingcurlx to (3) and using (4), we find
(6)ω2ϵμHs-curlxcurlxHs=-curlx[δ(x-x0)es→],x∈V.
Taking the cross product of both sides of (3) and n→, and using (5) we get
(7)(curlxHs×n→)|Γ=0,(Hs·n→)|Γ=0.The following equation follows from (3)
(8)E(x,ω,x0)=iωϵ[curlxHs(x,ω,x0)-δ(x-x0)es→].Remark 2.
IfHs is a vector function satisfying (6) and (7) and Es is a vector function satisfying (8) for a fixed value of s=1,2,3, then Hs and Es satisfy (3)–(5); that is, Hs and Es are columns of frequency-dependent magnetic and electric Green’s functions. Really, from (8), we find curlxHs(x,ω,x0)=-iωϵEs(x,ω,x0)+δ(x-x0)es→. Substituting the expression for curlxHs into (6) we have curlxEs(x,ω,x0)=iμωHs(x,ω,x0). Moreover, taking the cross product of (8) and n→, letting x∈Γ and, using (7) we find that Es satisfies the first condition of (5). Therefore, we obtain that if Hs and Es satisfy (6), (7), and (8) then Hs and Es satisfy (3)–(5) also; that is, Hs and Es are s-columns of frequency-dependent electric and magnetic Green’s functions.Remark 3.
LetHs and Es satisfy (6) and (8). Then divxHs=0, x∈V.


## 3. The Eigenvalue-Eigenfunction Problem for the OperatorΔxI3 in a Rectangular Parallelepiped

Let us consider the problem of finding all values ofλ (eigenvalues) for which there exists a nonzero vector-valued function U(x)=(U1(x),U2(x),U3(x))T (vector-valued eigenfunction) satisfying
(9)ΔxI3U(x)+λU(x)=0,x∈V,(10)(curlxU(x)×n→)|Γ=0,(U·n→)|Γ=0,
where I3 is the 3×3 identity matrix.In this section we show that the eigenvalue-eigenfunction problem (9) and (10) can be decomposed into three eigenvalue-eigenfunction problems for the scalar-valued Laplace operator in the rectangular parallelepiped with the mixed boundary conditions. These subproblems can be solved by a standard technique [18].Lemma 4.
Let the boundaryΓ of the parallelepiped V be presented as Γ=Γ1∪Γ2∪Γ3, where Γ1, Γ2, and Γ3 are the sets defined by
(11)Γ1={x=(x1,x2,x3):x1=0∨x1=b1,hih0<x2<b2,0<x3<b3},Γ2={x=(x1,x2,x3):x2=0∨x2=b2,hih0<x1<b1,0<x3<b3},Γ3={x=(x1,x2,x3):x3=0∨x3=b3,hih0<x1<b1,0<x2<b2}.
The boundary condition in (10) can be written as
(12)(αj(x)∂Uj(x)∂n+βj(x)Uj(x))|Γ=0,
where the functions αj(x) and βj(x), j=1,2,3 are defined by
(13)αj(x)={0,x∈Γj,1,x∈Γ∖Γj;βj(x)={1,x∈Γj,0,x∈Γ∖Γj.Proof.
Let us prove this forj=1 and let us consider the part of the boundary where x1=b1. Thus, the outward normal unit vector is e1→=(1,0,0)T. The second condition of (10) reads
(14)U1(b1,x2,x3)=0,for0<x2<b2,0<x3<b3.
The first condition of (10) takes the form
(15)∂U2∂x1(b1,x2,x3)-∂∂x2[U1(b1,x2,x3)]=0,∂U3∂x1(b1,x2,x3)-∂∂x3[U1(b1,x2,x3)]=0.
Combining the last three equalities we can write the boundary condition (10) in the form
(16)(αj(x)∂Uj(x)∂n+βj(x)Uj(x))|x1=b1=0,j=1,2,3,
where α1=0, β1=1; α2=1, and β2=0; α3=1 and β3=0. Similar arguments hold for the lower part of Γ1 and for the parts of the boundaries Γ2 and Γ3 leading to (12).Letλkmn denote the eigenvalues and Ukmn(x) the corresponding vector-valued eigenfunctions of (9) and (10). The eigenfunctions Ukmn(x) can be written in the form
(17)Ukmn(x)=U1kmn(x)e1→+U2kmn(x)e2→+U3kmn(x)e3→,
and λkmn, Ujkmn(x), and j=1,2,3 satisfy
(18)ΔxUjkmn(x)+λkmnUjkmn(x)=0,x∈V;(19)(αj(x)∂Ujkmn(x)∂n+βj(x)Ujkmn(x))|x1=b1=0.
Applying the standard technique of the spectral theory (see, e.g., [18]) we find all eigenvalues and corresponding eigenfunctions of the problem (18) and (19) for each j as follows.Forj=1,
(20)λkmn=(kπb1)2+(mπb2)2+(nπb3)2,U1kmn(x)=Akmnsin(kπb1x1)cos⁡(mπb2x2)cos⁡(nπb3x3),k=1,2,…;  n,m=0,1,…  are eigenvalues and corresponding eigenfunctions.Forj=2,
(21)λkmn=(kπb1)2+(mπb2)2+(nπb3)2,U2kmn(x)=Bkmncos⁡(kπb1x1)sin(mπb2x2)cos⁡(nπb3x3),m=1,2,…; k,n=0,1,… are eigenvalues and corresponding eigenfunctions.Forj=3,
(22)λkmn=(kπb1)2+(mπb2)2+(nπb3)2,U3kmn(x)=Ckmncos⁡(kπb1x1)cos⁡(mπb2x2)sin(nπb3x3),n=1,2,…; k,m=0,1,… are eigenvalues and corresponding eigenfunctions.HereAkmn, Bkmn, and Ckmn are constants, such that
(23)∫0b3∫0b2∫0b1Ujkmn(x)·Ujk′m′n′(x)dx={0,if(k,m,n)≠(k′,m′,n′)1,ifk=k′,m=m′,n=n′.k,m,n=0,1,…; j=1,2,3. That is, Akmn=Bkmn=Ckmn=8/b1b2b3, Ak0n=Akm0=B0mn=Bkm0=C0mn=Ck0n=4/b1b2b3Ak00=B0m0=C00n=2/b1b2b3, and k=1,2,…; m=1,2,…; n=1,2,….Remark 5.
LetUjkmn(x), j=1,2,3, be eigenfunctions of (18) and (19) defined above. By the direct computation we can show that Ukmn(x)=(U1kmn(x),U2kmn(x),U3kmn(x))T satisfies the boundary conditions (10) for the rectangular parallelepiped V.Remark 6.
For eachj=1,2,3 the system of eigenfunctions {Ujkmn(x)} is orthogonal and complete in the space of the square integrable functions over V. That is, any scalar-valued function h(x)∈C0∞(V) can be written in the form of uniformly convergent series over V¯ (see, e.g., [18])
(24)h(x)=∑k=0∞∑m=0∞∑n=0∞hjkmnUjkmn(x),
where
(25)hjkmn=∭Vh(x)Ujkmn(x)dx.
The constants hjkmn defined in (25) are called the Fourier coefficients of h(x) relative to the basis functions {Ujkmn(x)}.Lemma 7 (see [18]).
Leth(x)∈C0∞(V). Then
(26)∭V|h(x)|2dx=∑k=0∞∑m=0∞∑n=0∞[hjkmn]2<∞
holds. Here hjkmn are the Fourier coefficients (25). The equality (26) is known as Parseval’s equality. The proof of this lemma can be found, for example, in [18].As a result, any vector functionΨ(x)=(ψ1(x),ψ2(x),ψ3(x))T with components from C0∞(V) can be presented in the form
(27)Ψ(x)=∑k=0∞∑m=0∞∑n=0∞(ψ3kmnU3kmn(x)e→3ψ1kmnU1kmn(x)e1→+ψ2kmnU2kmn(x)e2→hhhhhhihhhhh+ψ3kmnU3kmn(x)e3→),∀x∈V¯,
where
(28)ψjkmn=∭Vψj(x)Ujkmn(x)dx,j=1,2,3.
We note that if ψj(x)∈C0∞(V) then ∑k=0∞∑m=0∞∑n=0∞ψjkmnUjkmn(x); j=1,2,3, is uniformly convergent series over V¯.


## 4. Computation of Frequency-Dependent Magnetic Matrix Green’s Function

### 4.1. The Regularization ofcurlx[δ(x-x0)es→]

Lets=1, x=(x1,x2,x3)∈R3, x0=(x10,x20,x30)∈R3, δ(x-x0)=δ(x1-x10)δ(x2-x20)δ(x3-x30) be the Dirac delta distribution. The vector distribution -curlx[δ(x-x0)e1→]=curlx0[δ(x-x0)e1→] can be written as follows:
(29)-curlx[δ(x-x0)e1→]=curlx0[δ(x-x0)e1→]=0·e1→+∂δ(x-x0)∂x30e2→-∂δ(x-x0)∂x20e3→.
Using formulas (26) and (28) we find the Fourier series expansion of curlx0[δ(x-x0)e1→] in the form of (27), where
(30)ψ1kmn=0,ψ2kmn=∂U2kmn(x0)∂x30,ψ3kmn=-∂U3kmn(x0)∂x20.
(See, e.g., [19–22].)Lemma 8.
Letx0∈V be fixed, and h(x)∈C0∞(V). Then the series
(31)∑k=0∞∑m=1∞∑n=0∞∂U2kmn(x0)∂x30h2kmn,(32)∑k=0∞∑m=0∞∑n=1∞∂U3kmn(x0)∂x20h3kmn
are absolutely convergent. Here h2kmn and h3kmn are defined by (25) for the given function h(x).The proof of Lemma8 can be done by standard arguments of calculus (see Appendix A).Using Lemma8 the distributions ∂δ(x-x0)/∂x30 and ∂δ(x-x0)/∂x20 can be defined by
(33)〈∂δ(x-x0)∂x30,h(x)〉=∂h(x0)∂x30=∑k=0∞∑m=1∞∑n=0∞∂U2kmn(x0)∂x30h2kmn,〈∂δ(x-x0)∂x20,h(x)〉=∂h(x0)∂x20=∑k=0∞∑m=0∞∑n=1∞∂U3kmn(x0)∂x20h3kmn,
for any function h(x)∈C0∞(V).Let us consider the infinitely differentiable functionsψ21,N(x,x0), ψ31,N(x,x0) defined by
(34)ψ21,N(x,x0)=∑k=0N∑m=1N∑n=0N∂U2kmn(x0)∂x30U2kmn(x),(35)ψ31,N(x,x0)=-∑k=0N∑m=0N∑n=1N∂U3kmn(x0)∂x20U3kmn(x).Lemma 9.
Letx0∈V be fixed, and let functions ψ21,N(x,x0) and ψ31,N(x,x0) be defined by (34) and (35). Then for any h(x)∈C0∞(V)(36)lim⁡N→∞∭Vψ21,N(x,x0)h(x)dx=〈∂δ(x-x0)∂x30,h(x)〉,(37)lim⁡N→∞∭Vψ31,N(x,x0)h(x)dx=-〈∂δ(x-x0)∂x20,h(x)〉.The proof of Lemma7 follows from Lemma 4 and standard arguments of the theory of distributions (see, e.g., [19, 20]).Remark 10.
Equalities (36) and (37) mean that
(38)lim⁡N→∞ψ21,N(x,x0)=∂δ(x-x0)∂x30,lim⁡N→∞ψ31,N(x,x0)=-∂δ(x-x0)∂x20
in the space of distributions D′(V) (see, e.g., [19, 20]).Remark 11.
The functionsψ21,N(x,x0) and ψ31,N(x,x0) can be taken as regularizations (approximations) of the distributions ∂δ(x-x0)/∂x30 and -∂δ(x-x0)/∂x20, respectively. Here N is a parameter of the regularization (see, e.g., [19, 20]).Remark 12.
Let(39)Ψ1,N(x,x0)=(0,ψ21,N,ψ31,N)T=∑k=0N∑m=0N∑n=0N(∂U2kmn(x0)∂x30U2kmn(x)e2→hhhhhhhihhhhh-∂U3kmn(x0)∂x20U3kmn(x)e3→).
Using Lemma 9 and Remarks 10 and 11 we find that
(40)lim⁡N→∞Ψ1,N(x,x0)=curlx0[δ(x-x0)e1→]
in the space of distributions D′(V) and therefore the vector function Ψ1,N(x,x0), with infinitely differentiable components, can be taken as a regularization of the generalized vector function curlx0[δ(x-x0)e1→].
Similarly, we find that the vector functionsΨ2,N(x,x0) and Ψ3,N(x,x0), defined by
(41)Ψ2,N(x,x0)=∑k=0N∑m=0N∑n=0N(-∂U1kmn(x0)∂x30U1kmn(x)e1→hhhhhhhhhhhhh+∂U3kmn(x0)∂x10U3kmn(x)e3→),(42)Ψ3,N(x,x0)=∑k=0N∑m=0N∑n=0N(∂U1kmn(x0)∂x20U1kmn(x)e1→hhhhhhhhhihhh-∂U2kmn(x0)∂x10U2kmn(x)e2→),
can be taken as the regularizations of curlx0[δ(x-x0)e2→] and curlx0[δ(x-x0)e3→].


### 4.2. Solving (6) and (7) with the Regularized Inhomogeneous Term

Let us consider (6) and (7) for frequency-dependent magnetic Green’s function. We replace curlx[δ(x-x0)es→] by their regularizations defined by (39), (41), and (42) for s=1,2,3, respectively. Moreover, (6) and (7) are replaced by the following equations:
(43)ΔxI3Hs,N+ω2ϵμHs,N=Ψs,N(x,x0),(44)(curlxHs,N×n→)|Γ=0,(Hs,N·n→)|Γ=0.
We will find solutions of the problem (43) and (44) in the form
(45)Hs,N(x,ω,x0)=∑k=0N∑m=0N∑n=0N(Y1s,kmn(ω,x0)U1kmn(x)e1→hhhhhhhhhhhhh+Y2s,kmn(ω,x0)U2kmn(x)e2→hhhhhhhhhhhh+Y3s,kmn(ω,x0)U3kmn(x)e3→),
where x∈V; Ujkmn(x) are eigenfunctions of the eigenvalue-eigenfunction problem (18) and (19); Yjs,kmn(ω,x0) and j=1,2,3, are the unknown Fourier coefficients.Substituting (45) into (43) and using orthogonality of Ujkmn(x), we find
(46)(ω2ϵμ-λkmn)Yjs,kmn(ω,x0)=ψjs,kmn(x0),
for j=1,2,3; s=1,2,3; k=0,1,…, m=0,1,…, and n=0,1,….Letω2≠λkmn/ϵμ; then Yjs,kmn(ω,x0) is uniquely determined by
(47)Yjs,kmn(ω,x0)=ψjs,kmn(x0)ω2ϵμ-λkmn,
where
(48)ψjs,kmn(x0)=∫∫∫VΨjs(x,x0)Ujkmn(x)dx,k,m,n=0,1,…. As a result we obtain the following explicit formula:
(49)H1,N(x,ω,x0)=∑k=0N∑m=0N∑n=0N1ω2ϵμ-λkmnhhhhhhhhhhh×(∂U2kmn(x0)∂x30U2kmn(x)e2→hhhhhhhhhhhhhh-∂U3kmn(x0)∂x20U3kmn(x)e3→),
for all x∈V, x0∈V, and all parameter ω satisfying ω2ϵμ≠λkmn   (k=0,1,…; m=0,1,…; n=0,1,…).Similarly, the solutions of (43) and (44) for s=2 and s=3 are obtained in the form
(50)H2,N(x,ω,x0)=∑k=0N∑m=0N∑n=0N1ω2ϵμ-λkmnhhhhhhhhhhh×(∂U1kmn(x0)∂x30U1kmn(x)e1→hhhhhhhhhhhhhh-∂U3kmn(x0)∂x10U3kmn(x)e3→),(51)H3,N(x,ω,x0)=∑k=0N∑m=0N∑n=0N1ω2ϵμ-λkmnhhhhhhhhhhhh×(∂U1kmn(x0)∂x20U1kmn(x)e1→hhhhhhhhhhhhhhh-∂U2kmn(x0)∂x10U2kmn(x)e2→).
Using the explicit formulas for Hs,N(x,ω,x0) and Ujkmn(x) we can check directly that Hs,N(x,ω,x0) satisfy the boundary condition (44).Remark 13.
By direct computation we can show that the vector functionsHs,N(x,ω,x0), defined by, (49)–(51), satisfy
(52)divxHs,N(x,ω,x0)=0,s=1,2,3.


### 4.3. Frequency-Dependent Magnetic Green’s Function and Its Regularization

In this section we show thatlim⁡N→∞Hjs,N(x,ω,x0); (s=1,2,3;j=1,2,3) exists in the space of distributions D′(V), and the distributions Hjs=lim⁡N→∞Hjs,N are the entries of matrix-valued Green’s function GH(x,ω,x0).Let us consider for example,s=1, j=2 and an arbitrary function h(x)∈C0∞(V) of the form (24) and (25). The integrable over V function Hjs,N(x,t,x0) defines a regular functional (distribution) by
(53)〈H21,N(x,ω,x0),h(x)〉=∑k=0N∑m=1N∑n=0N1ω2ϵμ-λkmn∂U2kmn(x0)∂x30h2kmn.
Using Lemma 8 we find that ∑k=0∞∑m=1∞∑n=0∞(1/(ω2ϵμ-λkmn))(∂U2kmn(x0)/∂x30)h2kmn is absolutely convergent for any h(x)∈C0∞(V). This means that lim⁡N→∞〈H21,N(x,ω,x0),h(x)〉 exists for any h(x)∈C0∞(V) and defines the linear continuous functional H21,N(x,ω,x0) over C0∞(V).Similarly, we can findHjs(x,ω,x0) as lim⁡N→∞Hjs,N(x,ω,x0) in the complete space D′(V) (see, e.g., [19, 20]) for s=1,2,3 and j=1,2,3.Moreover, sinceHs,N(x,ω,x0)=(H1s,N(x,ω,x0),H2s,N(x,ω,x0),H3s,N(x,tω,x0))T satisfies equalities (43) and (44) then Hs(x,ω,x0)=(H1s(x,ω,x0),H2s(x,ω,x0),H3s(x,ω,x0))T satisfies the following equalities:
(54)ω2ϵμHs-curlxcurlxHs=-curlx[δ(x-x0)es→],(curlxHs×n→)|Γ=0,(Hs·n→)|Γ=0,
that is, equalities (6) and (7). This means that Hs(x,ω,x0) is the s-column of frequency-dependent magnetic matrix Green’s function.


## 4.1. The Regularization ofcurlx[δ(x-x0)es→]

Lets=1, x=(x1,x2,x3)∈R3, x0=(x10,x20,x30)∈R3, δ(x-x0)=δ(x1-x10)δ(x2-x20)δ(x3-x30) be the Dirac delta distribution. The vector distribution -curlx[δ(x-x0)e1→]=curlx0[δ(x-x0)e1→] can be written as follows:
(29)-curlx[δ(x-x0)e1→]=curlx0[δ(x-x0)e1→]=0·e1→+∂δ(x-x0)∂x30e2→-∂δ(x-x0)∂x20e3→.
Using formulas (26) and (28) we find the Fourier series expansion of curlx0[δ(x-x0)e1→] in the form of (27), where
(30)ψ1kmn=0,ψ2kmn=∂U2kmn(x0)∂x30,ψ3kmn=-∂U3kmn(x0)∂x20.
(See, e.g., [19–22].)Lemma 8.
Letx0∈V be fixed, and h(x)∈C0∞(V). Then the series
(31)∑k=0∞∑m=1∞∑n=0∞∂U2kmn(x0)∂x30h2kmn,(32)∑k=0∞∑m=0∞∑n=1∞∂U3kmn(x0)∂x20h3kmn
are absolutely convergent. Here h2kmn and h3kmn are defined by (25) for the given function h(x).The proof of Lemma8 can be done by standard arguments of calculus (see Appendix A).Using Lemma8 the distributions ∂δ(x-x0)/∂x30 and ∂δ(x-x0)/∂x20 can be defined by
(33)〈∂δ(x-x0)∂x30,h(x)〉=∂h(x0)∂x30=∑k=0∞∑m=1∞∑n=0∞∂U2kmn(x0)∂x30h2kmn,〈∂δ(x-x0)∂x20,h(x)〉=∂h(x0)∂x20=∑k=0∞∑m=0∞∑n=1∞∂U3kmn(x0)∂x20h3kmn,
for any function h(x)∈C0∞(V).Let us consider the infinitely differentiable functionsψ21,N(x,x0), ψ31,N(x,x0) defined by
(34)ψ21,N(x,x0)=∑k=0N∑m=1N∑n=0N∂U2kmn(x0)∂x30U2kmn(x),(35)ψ31,N(x,x0)=-∑k=0N∑m=0N∑n=1N∂U3kmn(x0)∂x20U3kmn(x).Lemma 9.
Letx0∈V be fixed, and let functions ψ21,N(x,x0) and ψ31,N(x,x0) be defined by (34) and (35). Then for any h(x)∈C0∞(V)(36)lim⁡N→∞∭Vψ21,N(x,x0)h(x)dx=〈∂δ(x-x0)∂x30,h(x)〉,(37)lim⁡N→∞∭Vψ31,N(x,x0)h(x)dx=-〈∂δ(x-x0)∂x20,h(x)〉.The proof of Lemma7 follows from Lemma 4 and standard arguments of the theory of distributions (see, e.g., [19, 20]).Remark 10.
Equalities (36) and (37) mean that
(38)lim⁡N→∞ψ21,N(x,x0)=∂δ(x-x0)∂x30,lim⁡N→∞ψ31,N(x,x0)=-∂δ(x-x0)∂x20
in the space of distributions D′(V) (see, e.g., [19, 20]).Remark 11.
The functionsψ21,N(x,x0) and ψ31,N(x,x0) can be taken as regularizations (approximations) of the distributions ∂δ(x-x0)/∂x30 and -∂δ(x-x0)/∂x20, respectively. Here N is a parameter of the regularization (see, e.g., [19, 20]).Remark 12.
Let(39)Ψ1,N(x,x0)=(0,ψ21,N,ψ31,N)T=∑k=0N∑m=0N∑n=0N(∂U2kmn(x0)∂x30U2kmn(x)e2→hhhhhhhihhhhh-∂U3kmn(x0)∂x20U3kmn(x)e3→).
Using Lemma 9 and Remarks 10 and 11 we find that
(40)lim⁡N→∞Ψ1,N(x,x0)=curlx0[δ(x-x0)e1→]
in the space of distributions D′(V) and therefore the vector function Ψ1,N(x,x0), with infinitely differentiable components, can be taken as a regularization of the generalized vector function curlx0[δ(x-x0)e1→].
Similarly, we find that the vector functionsΨ2,N(x,x0) and Ψ3,N(x,x0), defined by
(41)Ψ2,N(x,x0)=∑k=0N∑m=0N∑n=0N(-∂U1kmn(x0)∂x30U1kmn(x)e1→hhhhhhhhhhhhh+∂U3kmn(x0)∂x10U3kmn(x)e3→),(42)Ψ3,N(x,x0)=∑k=0N∑m=0N∑n=0N(∂U1kmn(x0)∂x20U1kmn(x)e1→hhhhhhhhhihhh-∂U2kmn(x0)∂x10U2kmn(x)e2→),
can be taken as the regularizations of curlx0[δ(x-x0)e2→] and curlx0[δ(x-x0)e3→].


## 4.2. Solving (6) and (7) with the Regularized Inhomogeneous Term

Let us consider (6) and (7) for frequency-dependent magnetic Green’s function. We replace curlx[δ(x-x0)es→] by their regularizations defined by (39), (41), and (42) for s=1,2,3, respectively. Moreover, (6) and (7) are replaced by the following equations:
(43)ΔxI3Hs,N+ω2ϵμHs,N=Ψs,N(x,x0),(44)(curlxHs,N×n→)|Γ=0,(Hs,N·n→)|Γ=0.
We will find solutions of the problem (43) and (44) in the form
(45)Hs,N(x,ω,x0)=∑k=0N∑m=0N∑n=0N(Y1s,kmn(ω,x0)U1kmn(x)e1→hhhhhhhhhhhhh+Y2s,kmn(ω,x0)U2kmn(x)e2→hhhhhhhhhhhh+Y3s,kmn(ω,x0)U3kmn(x)e3→),
where x∈V; Ujkmn(x) are eigenfunctions of the eigenvalue-eigenfunction problem (18) and (19); Yjs,kmn(ω,x0) and j=1,2,3, are the unknown Fourier coefficients.Substituting (45) into (43) and using orthogonality of Ujkmn(x), we find
(46)(ω2ϵμ-λkmn)Yjs,kmn(ω,x0)=ψjs,kmn(x0),
for j=1,2,3; s=1,2,3; k=0,1,…, m=0,1,…, and n=0,1,….Letω2≠λkmn/ϵμ; then Yjs,kmn(ω,x0) is uniquely determined by
(47)Yjs,kmn(ω,x0)=ψjs,kmn(x0)ω2ϵμ-λkmn,
where
(48)ψjs,kmn(x0)=∫∫∫VΨjs(x,x0)Ujkmn(x)dx,k,m,n=0,1,…. As a result we obtain the following explicit formula:
(49)H1,N(x,ω,x0)=∑k=0N∑m=0N∑n=0N1ω2ϵμ-λkmnhhhhhhhhhhh×(∂U2kmn(x0)∂x30U2kmn(x)e2→hhhhhhhhhhhhhh-∂U3kmn(x0)∂x20U3kmn(x)e3→),
for all x∈V, x0∈V, and all parameter ω satisfying ω2ϵμ≠λkmn   (k=0,1,…; m=0,1,…; n=0,1,…).Similarly, the solutions of (43) and (44) for s=2 and s=3 are obtained in the form
(50)H2,N(x,ω,x0)=∑k=0N∑m=0N∑n=0N1ω2ϵμ-λkmnhhhhhhhhhhh×(∂U1kmn(x0)∂x30U1kmn(x)e1→hhhhhhhhhhhhhh-∂U3kmn(x0)∂x10U3kmn(x)e3→),(51)H3,N(x,ω,x0)=∑k=0N∑m=0N∑n=0N1ω2ϵμ-λkmnhhhhhhhhhhhh×(∂U1kmn(x0)∂x20U1kmn(x)e1→hhhhhhhhhhhhhhh-∂U2kmn(x0)∂x10U2kmn(x)e2→).
Using the explicit formulas for Hs,N(x,ω,x0) and Ujkmn(x) we can check directly that Hs,N(x,ω,x0) satisfy the boundary condition (44).Remark 13.
By direct computation we can show that the vector functionsHs,N(x,ω,x0), defined by, (49)–(51), satisfy
(52)divxHs,N(x,ω,x0)=0,s=1,2,3.


## 4.3. Frequency-Dependent Magnetic Green’s Function and Its Regularization

In this section we show thatlim⁡N→∞Hjs,N(x,ω,x0); (s=1,2,3;j=1,2,3) exists in the space of distributions D′(V), and the distributions Hjs=lim⁡N→∞Hjs,N are the entries of matrix-valued Green’s function GH(x,ω,x0).Let us consider for example,s=1, j=2 and an arbitrary function h(x)∈C0∞(V) of the form (24) and (25). The integrable over V function Hjs,N(x,t,x0) defines a regular functional (distribution) by
(53)〈H21,N(x,ω,x0),h(x)〉=∑k=0N∑m=1N∑n=0N1ω2ϵμ-λkmn∂U2kmn(x0)∂x30h2kmn.
Using Lemma 8 we find that ∑k=0∞∑m=1∞∑n=0∞(1/(ω2ϵμ-λkmn))(∂U2kmn(x0)/∂x30)h2kmn is absolutely convergent for any h(x)∈C0∞(V). This means that lim⁡N→∞〈H21,N(x,ω,x0),h(x)〉 exists for any h(x)∈C0∞(V) and defines the linear continuous functional H21,N(x,ω,x0) over C0∞(V).Similarly, we can findHjs(x,ω,x0) as lim⁡N→∞Hjs,N(x,ω,x0) in the complete space D′(V) (see, e.g., [19, 20]) for s=1,2,3 and j=1,2,3.Moreover, sinceHs,N(x,ω,x0)=(H1s,N(x,ω,x0),H2s,N(x,ω,x0),H3s,N(x,tω,x0))T satisfies equalities (43) and (44) then Hs(x,ω,x0)=(H1s(x,ω,x0),H2s(x,ω,x0),H3s(x,ω,x0))T satisfies the following equalities:
(54)ω2ϵμHs-curlxcurlxHs=-curlx[δ(x-x0)es→],(curlxHs×n→)|Γ=0,(Hs·n→)|Γ=0,
that is, equalities (6) and (7). This means that Hs(x,ω,x0) is the s-column of frequency-dependent magnetic matrix Green’s function.


## 5. Approximate Solution of Electric Green’s Function

Let columns of magnetic Green’s function be derived approximately; that is, let solutionsHs,N, s=1,2,3 of the problem (43) and (44) be computed by formulae (49), (50), and (51). For the approximate derivation of Es(x,ω,x0), satisfying (8), we replace curlxHs(x,ω,x0) by curlxHs,N(x,ω,x0) and δ(x-x0)es→ by its regularization ∑k=0N∑m=0N∑n=0NUskmn(x0)Uskmn(x)es→ in (8) for s=1,2,3, respectively, and find an explicit formula for Es,N(x,ω,x0) as follows:
(55)Es,N(x,ω,x0)=iωϵ[∑k=0N∑m=0N∑n=0NUskmn(x0)Uskmn(x)e→scurlxHs,N(x,ω,x0)hhhhhhh-∑k=0N∑m=0N∑n=0NUskmn(x0)Uskmn(x)es→],
for all x∈V, x0∈V, and all parameter ω satisfying ω2ϵμ≠λkmn.


### 5.1. The Regularization ofEs(x,ω,x0)

Let us consider the differentiable functionE22,N(x,ω,x0), given by (55). This function defines a linear continuous regular functional over C0∞(V) by the following relation (see, e.g., [19, 20]):
(56)〈E22,N,h(x)〉=-iωϵ∑k=0N∑m=0N∑n=0N1ω2ϵμ-λkmnhhhhhhhhhhhhhhh×[∂U1kmn(x0)∂x30hhhhhhhhhhhihhhhhhh×∭VU1kmn(x)∂h(x)∂x3dxhhhhhhhhhhhhhhhihhh+∂U3kmn(x0)∂x10hhhhhhhhhhhhhhhhhhh×∭VU3kmn(x)∂h(x)∂x1dx]+iωϵ∑k=0N∑m=0N∑n=0NU2kmn(x0)h2kmn,
for an arbitrary h(x)∈C0∞(V). The series in the right-hand side of the last equality is absolutely convergent for any h(x)∈C0∞(V) (see, Appendix B). This means that lim⁡N→∞⁡〈E22,N(x,ω,x0),h(x)〉 exists for any h(x)∈C0∞(V) and defines a linear continuous regular functional over C0∞(V). We denote this functional as E22(x,ω,x0); that is, E22(x,ω,x0)=lim⁡N→∞E22,N(x,ω,x0) in the complete space D′(V). Similarly, we can find Ejs(x,ω,x0) as lim⁡N→∞Ejs,N(x,ω,x0) in the space D′(V) for s=1,2,3 and j=1,2,3.Thes-column of electric Green’s function is
(57)Es(x,ω,x0)=iωϵ[curlxHs(x,ω,x0)-δ(x-x0)es→].
Let now Es(x,ω,x0) = (E1s(x,ω,x0), E2s(x,ω,x0), E3s(x,ω,x0))T be the generalized vector function, whose components are distributions defined as Ejs=lim⁡N→∞Ejs,N, j=1,2,3. Since Es,N(x,ω,x0)=(E1s,N(x,ω,x0),E2s,N(x,ω,x0),E3s,N(x,ω,x0))T satisfies (55), then the generalized vector function Es(x,ω,x0) satisfies (4).Remark 14.
We note that the generalized vector functionsHs(x,ω,x0), Es(x,ω,x0), defined in Sections 4.3 and 5.1, satisfy (6), (7), and (8). It follows that Hs(x,ω,x0) and Es(x,ω,x0) satisfy (3)–(5), and therefore, Hs(x,ω,x0) and Es(x,ω,x0) are s-columns of frequency-dependent electric and magnetic Green’s functions GE(x,ω,x0) and GH(x,ω,x0), respectively.


## 5.1. The Regularization ofEs(x,ω,x0)

Let us consider the differentiable functionE22,N(x,ω,x0), given by (55). This function defines a linear continuous regular functional over C0∞(V) by the following relation (see, e.g., [19, 20]):
(56)〈E22,N,h(x)〉=-iωϵ∑k=0N∑m=0N∑n=0N1ω2ϵμ-λkmnhhhhhhhhhhhhhhh×[∂U1kmn(x0)∂x30hhhhhhhhhhhihhhhhhh×∭VU1kmn(x)∂h(x)∂x3dxhhhhhhhhhhhhhhhihhh+∂U3kmn(x0)∂x10hhhhhhhhhhhhhhhhhhh×∭VU3kmn(x)∂h(x)∂x1dx]+iωϵ∑k=0N∑m=0N∑n=0NU2kmn(x0)h2kmn,
for an arbitrary h(x)∈C0∞(V). The series in the right-hand side of the last equality is absolutely convergent for any h(x)∈C0∞(V) (see, Appendix B). This means that lim⁡N→∞⁡〈E22,N(x,ω,x0),h(x)〉 exists for any h(x)∈C0∞(V) and defines a linear continuous regular functional over C0∞(V). We denote this functional as E22(x,ω,x0); that is, E22(x,ω,x0)=lim⁡N→∞E22,N(x,ω,x0) in the complete space D′(V). Similarly, we can find Ejs(x,ω,x0) as lim⁡N→∞Ejs,N(x,ω,x0) in the space D′(V) for s=1,2,3 and j=1,2,3.Thes-column of electric Green’s function is
(57)Es(x,ω,x0)=iωϵ[curlxHs(x,ω,x0)-δ(x-x0)es→].
Let now Es(x,ω,x0) = (E1s(x,ω,x0), E2s(x,ω,x0), E3s(x,ω,x0))T be the generalized vector function, whose components are distributions defined as Ejs=lim⁡N→∞Ejs,N, j=1,2,3. Since Es,N(x,ω,x0)=(E1s,N(x,ω,x0),E2s,N(x,ω,x0),E3s,N(x,ω,x0))T satisfies (55), then the generalized vector function Es(x,ω,x0) satisfies (4).Remark 14.
We note that the generalized vector functionsHs(x,ω,x0), Es(x,ω,x0), defined in Sections 4.3 and 5.1, satisfy (6), (7), and (8). It follows that Hs(x,ω,x0) and Es(x,ω,x0) satisfy (3)–(5), and therefore, Hs(x,ω,x0) and Es(x,ω,x0) are s-columns of frequency-dependent electric and magnetic Green’s functions GE(x,ω,x0) and GH(x,ω,x0), respectively.


## 6. Computational Experiments

In this section, we present computational experiments to show applicability of the approximate computation of frequency-dependent magnetic and electric matrix Green’s functions in the rectangular parallelepiped. Here we consider the parallelepiped(58)V={x∈R3:0<x1<3,0<x2<4,0<x3<2.5}
which contains a dielectric material characterized by the electric permittivity ϵ and the magnetic permeability μ. We choose ϵ and μ as c=1/ϵμ   (c≈3·108 m/s is the speed of light) and ω is equal to the value of c. The boundary of V, denoted as Γ, is a perfect conducting boundary on which the tangential components of the electric field component and normal component of the magnetic field component vanish. There is an electric current J(x,ω) inside V which produces the electric and magnetic fields. In our experiments we take Js(x,ω), s=1,2,3, as
(59)Js(x,ω)=δ(x-x0)es→,wherees→ are the basis vectors from R3; x0=(x10,x20,x30) is the 3D parameter from V and chosen as x10=2, x20=3, x30=1.5; δ(x-x0)=δ(x1-x10)δ(x2-x20)δ(x3-x30), δ(xj-xj0) is the Dirac delta distribution at xj=xj0, j=1,2,3.Two types of computational experiments will be presented. The first one is related to convergence and error analysis of our method; the second one is the computation of the elements of matrix magnetic and electric Green’s functions in the rectangular parallelepipedV and the visualization of the computed magnetic and electric fields components by MATLAB graphic tools.


### 6.1. Convergence Analysis

Green’s function of the magnetic field is3×3 matrix whose elements are distributions. Generally speaking, distributions do not have values at fixed points [19, 20]. The elements of regularized (approximate) matrix Green’s function, which we have computed by our method, are classical differentiable and integrable functions over V having the values at fixed points. These classical functions define the regular distributions by the standard rule (see, [19, 20]). The comparison of the distributions can be made by the comparison of their integral characteristics. Some integral characteristics of the elements of matrix Green’s function of the magnetic field can be found explicitly. For example, let H21(x,ω,x0) be a solution of (6) and let (7) for s=1 and j=2 and let h21(x3,ω,x0) be defined by
(60)h21(x3,ω,x0)=∫0b1∫0b2H21(x,ω,x0)cos⁡(πx1b1)sin(πx2b2)dx2dx1.
We can show that h21=(x3,ω,x0) satisfies
(61)∂2h21∂x32+k2h21=-∂∂x3δ(x3-x30)cos⁡(πx10b1)sin(πx20b2),hhhhhhhhhhhhhhhhhhhhhhh0<x3<b3,(62)dh21dx3|x3=0=0,dh21dx3|x3=b3=0,
where k2=ω2ϵμ-(π/b1)2-(π/b2)2. A solution of (61) and (62) can be written in the form
(63)h21(x3,ω,x0)={sin(k(b3-x30))sin(kb3)cos⁡(kx3)×cos⁡(πx10b1)sin(πx20b2),0<x3<x30,-sin(kx30)sin(kb3)cos⁡(k(b3-x3))×cos⁡(πx10b1)sin(πx20b2),x30<x3<b3.
We can compare the values of h21(x3,ω,x0), as the values of the integral characteristics of H21, with the values of the integral characteristic
(64)h21,N(x3,ω,x0)=∫0b1∫0b2H21,N(x,ω,x0)cos⁡(πx1b1)sin(πx2b2)dx2dx1=∑n=1N‍-1k2-(nπ/b3)22nπ(b3)2hhhhhh×cos⁡(πx10b1)sin(πx20b2)sin(nπx30b3)cos⁡(nπx3b3)
of H21,N. Here H21,N is the element of approximate matrix Green’s function computed by our method and N is the parameter of the approximation corresponding to the number of the terms in each series of the right hand side of (64).In Figures1, 2, 3, and 4 and Table 1 the values of h21,N are compared with the values of h21 for fixed x10=2, x20=3, x30=1.5, and ϵ and μ such that c=1/ϵμ and ω=c and different numbers N. The maximum of relative error rN(x3)=|h21-h21,N|/|h21| at the point x3=1.3 of the interval [0,2.5] for N=15,50,100,150 is r15=0.1806, r50=0.0625, r100=0.0316, and r150=0.0211, respectively. Making the error analysis we have concluded that the biggest N corresponds to the smallest error between values of h21,N and h21. We have taken N=100 as a reasonable value between accuracy and the time for our computation in MATLAB.Table 1
The accuracy of the calculation of the magnetic field forx10=2; x20=3; x30=1.5;  rN=|h21-h21,N|/|h21|; N=15, 50, 100, and 150.
x
3
r
15
r
50
r
100
r
150
0.5
0.0705
0.0213
0.0107
0.0071
0.7
0.0597
0.0232
0.0116
0.0078
0.9
0.0208
0.0271
0.0136
0.0090
1.1
0.0492
0.0356
0.0179
0.0119
1.3
0.1806
0.0625
0.0316
0.0211
1.7
0.1525
0.0531
0.0268
0.0179
1.9
0.0417
0.0317
0.0159
0.0106
2.1
0.0213
0.0251
0.0126
0.0084
2.2
0.0645
0.0235
0.0118
0.0078
2.4
0.0233
0.0220
0.0110
0.0073Figure 1
The graphs ofh21 and h21,N for N=15, x10=2, x20=3, and x30=1.5.Figure 2
The graphs ofh21 and h21,N for N=50, x10=2, x20=3, and x30=1.5.Figure 3
The graphs ofh21 and h21,N for N=100, x10=2, x20=3, and x30=1.5.Figure 4
The graphs ofh21 and h21,N for N=150, x10=2, x20=3, and x30=1.5.


### 6.2. Approximate Computation and Visualization of the Elements of Matrix Magnetic and Electric Green’s Functions

The method described in Sections4 and 5 has been applied for the approximate computation of frequency-dependent magnetic and electric Green’s matrices in the rectangular parallelepiped V={x∈R3:0<x1<b1,0<x2<b2,0<x3<b3}, respectively. We note that each column of magnetic and electric frequency-dependent Green’s matrices is the magnetic and electric fields arising from the pulse dipole polarized in the direction of the basis vector of R3. For computation we take b1=3, b2=4, and b3=2.5 and ε and μ such that c=1/εμ, ω=c (c is the value of the light velocity), x10=2, x20=3, and x30=1.5. Computation of Hjs,N and Ejs,N has been implemented in MATLAB and their graphs are presented. Figures 5(a), 5(b), 6(a), and 6(b) present the second and third components of the first column of the regularized magnetic matrix, respectively, H21,100(x,ω,x0) and H31,100(x,ω,x0). Figures 7(a)-7(b), 8(a)-8(b), and 9(a)-9(b) present each component of the first column of the regularized electric matrix, respectively, E11,100(x,ω,x0), E21,100(x,ω,x0), and E31,100(x,ω,x0). The views of the surfaces z=H21,N, z=H31,N, z=E11,N, z=E21,N, and z=E31,N (N=100) from the top of z-axis are presented in Figures 5(a), 6(a), 7(a), 8(a), and 9(a), where x1-axis is vertical and x2-axis is a horizontal one. Figures 5(b), 6(b), 7(b), 8(b), and 9(b) are 3D images of the surfaces H21,100(x,ω,x0), H31,100(x,ω,x0), E11,100(x,ω,x0), E21,100(x,ω,x0), and E31,100(x,ω,x0) for x3=1.5, respectively. Here, x1 and x2 are horizontal axes and z-axis is the magnitude.The second component of the first column of regularized magnetic Green’s function.
(a)
H
2
1
,
N
(
x
1
,
x
2
,
1.5
,
ω
,
x
0
), N=100
(b)
H
2
1
,
N
(
x
1
,
x
2
,
1.5
,
ω
,
x
0
), N=100The third component of the first column of regularized magnetic Green’s function.
(a)
H
3
1
,
N
(
x
1
,
x
2
,
1.5
,
ω
,
x
0
), N=100
(b)
H
3
1
,
N
(
x
1
,
x
2
,
1.5
,
ω
,
x
0
), N=100The first component of the first column of regularized electric Green’s function.
(a)
E
1
1
,
N
(
x
1
,
x
2
,
1.5
,
ω
,
x
0
), N=100
(b)
E
1
1
,
N
(
x
1
,
x
2
,
1.5
,
ω
,
x
0
), N=100The second component of the first column of regularized electric Green’s function.
(a)
E
2
1
,
N
(
x
1
,
x
2
,
1.5
,
ω
,
x
0
), N=100
(b)
E
2
1
,
N
(
x
1
,
x
2
,
1.5
,
ω
,
x
0
), N=100The third component of the first column of regularized electric Green’s function.
(a)
E
3
1
,
N
(
x
1
,
x
2
,
1.5
,
ω
,
x
0
), N=100
(b)
E
3
1
,
N
(
x
1
,
x
2
,
1.5
,
ω
,
x
0
), N=100


## 6.1. Convergence Analysis

Green’s function of the magnetic field is3×3 matrix whose elements are distributions. Generally speaking, distributions do not have values at fixed points [19, 20]. The elements of regularized (approximate) matrix Green’s function, which we have computed by our method, are classical differentiable and integrable functions over V having the values at fixed points. These classical functions define the regular distributions by the standard rule (see, [19, 20]). The comparison of the distributions can be made by the comparison of their integral characteristics. Some integral characteristics of the elements of matrix Green’s function of the magnetic field can be found explicitly. For example, let H21(x,ω,x0) be a solution of (6) and let (7) for s=1 and j=2 and let h21(x3,ω,x0) be defined by
(60)h21(x3,ω,x0)=∫0b1∫0b2H21(x,ω,x0)cos⁡(πx1b1)sin(πx2b2)dx2dx1.
We can show that h21=(x3,ω,x0) satisfies
(61)∂2h21∂x32+k2h21=-∂∂x3δ(x3-x30)cos⁡(πx10b1)sin(πx20b2),hhhhhhhhhhhhhhhhhhhhhhh0<x3<b3,(62)dh21dx3|x3=0=0,dh21dx3|x3=b3=0,
where k2=ω2ϵμ-(π/b1)2-(π/b2)2. A solution of (61) and (62) can be written in the form
(63)h21(x3,ω,x0)={sin(k(b3-x30))sin(kb3)cos⁡(kx3)×cos⁡(πx10b1)sin(πx20b2),0<x3<x30,-sin(kx30)sin(kb3)cos⁡(k(b3-x3))×cos⁡(πx10b1)sin(πx20b2),x30<x3<b3.
We can compare the values of h21(x3,ω,x0), as the values of the integral characteristics of H21, with the values of the integral characteristic
(64)h21,N(x3,ω,x0)=∫0b1∫0b2H21,N(x,ω,x0)cos⁡(πx1b1)sin(πx2b2)dx2dx1=∑n=1N‍-1k2-(nπ/b3)22nπ(b3)2hhhhhh×cos⁡(πx10b1)sin(πx20b2)sin(nπx30b3)cos⁡(nπx3b3)
of H21,N. Here H21,N is the element of approximate matrix Green’s function computed by our method and N is the parameter of the approximation corresponding to the number of the terms in each series of the right hand side of (64).In Figures1, 2, 3, and 4 and Table 1 the values of h21,N are compared with the values of h21 for fixed x10=2, x20=3, x30=1.5, and ϵ and μ such that c=1/ϵμ and ω=c and different numbers N. The maximum of relative error rN(x3)=|h21-h21,N|/|h21| at the point x3=1.3 of the interval [0,2.5] for N=15,50,100,150 is r15=0.1806, r50=0.0625, r100=0.0316, and r150=0.0211, respectively. Making the error analysis we have concluded that the biggest N corresponds to the smallest error between values of h21,N and h21. We have taken N=100 as a reasonable value between accuracy and the time for our computation in MATLAB.Table 1
The accuracy of the calculation of the magnetic field forx10=2; x20=3; x30=1.5;  rN=|h21-h21,N|/|h21|; N=15, 50, 100, and 150.
x
3
r
15
r
50
r
100
r
150
0.5
0.0705
0.0213
0.0107
0.0071
0.7
0.0597
0.0232
0.0116
0.0078
0.9
0.0208
0.0271
0.0136
0.0090
1.1
0.0492
0.0356
0.0179
0.0119
1.3
0.1806
0.0625
0.0316
0.0211
1.7
0.1525
0.0531
0.0268
0.0179
1.9
0.0417
0.0317
0.0159
0.0106
2.1
0.0213
0.0251
0.0126
0.0084
2.2
0.0645
0.0235
0.0118
0.0078
2.4
0.0233
0.0220
0.0110
0.0073Figure 1
The graphs ofh21 and h21,N for N=15, x10=2, x20=3, and x30=1.5.Figure 2
The graphs ofh21 and h21,N for N=50, x10=2, x20=3, and x30=1.5.Figure 3
The graphs ofh21 and h21,N for N=100, x10=2, x20=3, and x30=1.5.Figure 4
The graphs ofh21 and h21,N for N=150, x10=2, x20=3, and x30=1.5.


## 6.2. Approximate Computation and Visualization of the Elements of Matrix Magnetic and Electric Green’s Functions

The method described in Sections4 and 5 has been applied for the approximate computation of frequency-dependent magnetic and electric Green’s matrices in the rectangular parallelepiped V={x∈R3:0<x1<b1,0<x2<b2,0<x3<b3}, respectively. We note that each column of magnetic and electric frequency-dependent Green’s matrices is the magnetic and electric fields arising from the pulse dipole polarized in the direction of the basis vector of R3. For computation we take b1=3, b2=4, and b3=2.5 and ε and μ such that c=1/εμ, ω=c (c is the value of the light velocity), x10=2, x20=3, and x30=1.5. Computation of Hjs,N and Ejs,N has been implemented in MATLAB and their graphs are presented. Figures 5(a), 5(b), 6(a), and 6(b) present the second and third components of the first column of the regularized magnetic matrix, respectively, H21,100(x,ω,x0) and H31,100(x,ω,x0). Figures 7(a)-7(b), 8(a)-8(b), and 9(a)-9(b) present each component of the first column of the regularized electric matrix, respectively, E11,100(x,ω,x0), E21,100(x,ω,x0), and E31,100(x,ω,x0). The views of the surfaces z=H21,N, z=H31,N, z=E11,N, z=E21,N, and z=E31,N (N=100) from the top of z-axis are presented in Figures 5(a), 6(a), 7(a), 8(a), and 9(a), where x1-axis is vertical and x2-axis is a horizontal one. Figures 5(b), 6(b), 7(b), 8(b), and 9(b) are 3D images of the surfaces H21,100(x,ω,x0), H31,100(x,ω,x0), E11,100(x,ω,x0), E21,100(x,ω,x0), and E31,100(x,ω,x0) for x3=1.5, respectively. Here, x1 and x2 are horizontal axes and z-axis is the magnitude.The second component of the first column of regularized magnetic Green’s function.
(a)
H
2
1
,
N
(
x
1
,
x
2
,
1.5
,
ω
,
x
0
), N=100
(b)
H
2
1
,
N
(
x
1
,
x
2
,
1.5
,
ω
,
x
0
), N=100The third component of the first column of regularized magnetic Green’s function.
(a)
H
3
1
,
N
(
x
1
,
x
2
,
1.5
,
ω
,
x
0
), N=100
(b)
H
3
1
,
N
(
x
1
,
x
2
,
1.5
,
ω
,
x
0
), N=100The first component of the first column of regularized electric Green’s function.
(a)
E
1
1
,
N
(
x
1
,
x
2
,
1.5
,
ω
,
x
0
), N=100
(b)
E
1
1
,
N
(
x
1
,
x
2
,
1.5
,
ω
,
x
0
), N=100The second component of the first column of regularized electric Green’s function.
(a)
E
2
1
,
N
(
x
1
,
x
2
,
1.5
,
ω
,
x
0
), N=100
(b)
E
2
1
,
N
(
x
1
,
x
2
,
1.5
,
ω
,
x
0
), N=100The third component of the first column of regularized electric Green’s function.
(a)
E
3
1
,
N
(
x
1
,
x
2
,
1.5
,
ω
,
x
0
), N=100
(b)
E
3
1
,
N
(
x
1
,
x
2
,
1.5
,
ω
,
x
0
), N=100


## 7. Conclusion

The new method for the approximate computation of frequency-dependent magnetic and electric Green’s matrices in the rectangular parallelepiped with a perfect conducting boundary has been developed. This method has the following steps.The first step is determination of the eigenvalues and corresponding eigenfunctions of the vector Laplace differential operator of the formΔxI3 (where Δ is the Laplace operator and I3 is the 3×3 identity matrix) in a parallelepiped subject to some boundary conditions. The components of these vector eigenfunctions form an orthogonal set which is complete in the space of the square integrable functions over the considered parallelepiped.The second step is an approximation (regularization) of the generalized vector functions (distributions)curlx[δ(x-x0)es→]; s=1,2,3, appear in the differential equations for the columns of magnetic Green’s matrix as the free terms. Here es→ are the basis vectors of R3(e1→=(1,0,0), e2→=(0,1,0), e3→=(0,0,1)); x0=(x10,x20,x30) is a 3D parameter from the considered parallelepiped, δ(x-x0)=δ(x1-x10)δ(x2-x20)δ(x3-x30), and δ(xj-xj0) is the Dirac delta function considered at xj=xj0 and j=1,2,3. We find the Fourier coefficients of these free terms relative to the obtained set of eigenfunctions and approximate them by the Fourier series with a finite number of terms. We note that the Dirac delta function is very often used for modeling the point source in physics and engineering (see, [19, 20, 23]). The Dirac delta function does not have point-wise values and can not be drawn. For this reason, a regularization (approximation) of the Dirac delta function in the form of the classical function which has point-wise values is usually used for drawing and computation. For the Dirac delta function, as a singular generalized function, there are different types of approximation (see, e.g., [19, 20, 23]). Unfortunately, the known approximations of the Dirac delta function are not convenient for our approach. For this reason we have suggested a new approximation which is very useful for the computation of considered Green’s function.The third step is an approximate computation of elements of Green’s matrix in the form of the Fourier series with a finite number of terms relative to the considered set of eigenfunctions. All terms of this series contain known functions which are given explicitly. The simple implementation of our method for computing magnetic Green’s function in a rectangular parallelepiped is based on the obtained presentation and does not contain any type of discretization.Using explicit formulas for the approximate elements of magnetic Green’s matrix we derive explicitly the approximate elements of electric Green’s matrix. Finally, we have obtained the columns of magnetic and electric matrix Green’s function in the space of generalized functions (distributions) and their regularizations (approximations). The method has been implemented in MATLAB. The computational experiments have confirmed its robustness. We can note also that application, for example, finite elements or finite difference approaches can be done if we approximate the generalized vector functionscurlx[δ(x-x0)es→] by vector functions with components which are classical functions. We note here that numerical computations of the values of magnetic Green’s function by equations, where curlx[δ(x-x0)es→] are replaced by their approximation, will contain the error depending on the grid of discretization. For this reason our method of the computation of the magnetic Green function has the accuracy which is significantly better than other methods based on any type of discretization.



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*Source: 586370-2014-05-28.xml*