GENERALIZED POLYNOMIAL CHAOS AND DISPERSIVE ...sites.science.oregonstate.edu/~math_reu/proceedings/REU...Generalized Polynomial Chaos and Dispersive Dielectric Media 21 free electric

GENERALIZED POLYNOMIAL CHAOS AND DISPERSIVE DIELECTRIC MEDIA

ERIN BELA AND ERIK HORTSCH

ADVISOR: PROFESSOR GIBSONOREGON STATE UNIVERSITY

ABSTRACT. We investigate polynomial chaos as a method to improve the accuracy of the Debyemodel and allow for easy simulation using the Finite Difference Time Domain (FDTD) method. Wedevelop a framework for incorporating distributions of relaxation times using polynomial chaos anddemonstrate that the resultant system adds no physical conditions. We conclude by demonstratingthe exponential convergence of the system under a correct choice of polynomials and show how aninverse problem may be formulated to determine parameters of the distribution from experimentaldata or comparison with another model.

1. INTRODUCTION

Many modern technologies rely on the transmission of electromagnetic pulses. Often thesepulses must travel through dispersive dielectric materials. One potential application of these resultsis the use of ultrawideband(UWB) microwave pulses in cancer detection. Such an approach hasthe advantages of being able to detect smaller tumors and being less harmful than exposing apatient to X-ray radiation. Also of interest is studying the effects of terahertz radiation, used inairport security screeners, on human tissue.While most of these materials are well understood,there are still some holes in current numerical methods. Of greatest interest is the relaxationtimes of these media. In past models, relaxation time was often assumed to be constant, however,anomaly dispersive media may be better described using a distribution of relaxation times [4]. Forthe current work we focus on creating a framework by which simulations can be carried out usinga first order ODE model for polarization with different distributions of relaxation times such asuniform, beta, gamma and possibly log-normal. We begin by introducing Maxwell’s equationsfor dispersive media. We then introduce polynomial chaos as a way of including distributions ofparameters before finally developing a FDTD method for simulation. We perform a preliminarystability analysis on the resultant system and demonstrate its exponential convergence. Finally,we conclude by formulating an inverse problem , which may then be used obtain a distribution ofrelaxation times from experimental data.

1.1. Maxwell’s Equations. We begin by introducing Maxwell’s equations, which govern the be-havior of electromagnetic fields. Our initial goal is to simulate the behavior of such fields in adispersive dielectric media, namely a Debye media. The following gives the differential form ofMaxwell’s equations, where E and H are the electric and magnetic fields, respectively, D and Bare the electric and magnetic flux densities, respectively, and the scalar ρ represents the density of

Date: August 13, 2010.This work was done during the Summer 2010 REU program in Mathematics at Oregon State University.

20

Generalized Polynomial Chaos and Dispersive Dielectric Media 21

free electric charges unaccounted for in the electric polarization. The conduction current densityis given by J. Maxwell’s equations are

(1)∂D∂t

+J = ∇×H

(2)∂B∂t

=−∇×E

(3) ∇ ·D = ρ

(4) ∇ ·B = 0.The Constitutive Laws describe the response of the medium to the electromagnetic field

(5) D = ε0ε∞E+P

(6) B = µH+M

(7) J = σE+Js.

In the equations above, P refers to the macroscopic polarization, M the magnetization, and Js thesource current density. The parameters ε0 and ε∞ are the permittivity of free space and the relativepermittivity, respectively, while µ is the magnetic permeability and σ is the electric conductivity.By combining equations (1), (5) and (7) while neglecting the source term for simplicity we get,

(8) ε0ε∞∂E∂t

=−σE− ∂P∂t

+∇×H.

Combining equations (2) and (6) under the assumptions that there is no magnetization and thatµ = µ0 we get,

(9) µ0∂H∂t

=−∇×E.

For the purposes of the simulations to follow we will restrict to one dimension, however, themethods described below apply directly to three dimensions. We assume that all waves travel inthe z direction, have an electric field oscillating in the x direction and a magnetic field oscillatingin the y direction. Taking this into consideration equations (8) and (9) become,

(10) ε̃∞∂E∂t

=−∂H∂z−σE− ∂P

∂t

(11)∂H∂t

=− 1µ0

∂E∂z

where ε̃∞ := ε∞ε0, E := Ex, H := Hy, P := Px.

22 E Bela and E Hortsch

1.2. Polarization Models. For the purposes of this paper we are concerned with the way in whichelectromagnetic waves propagate through Debye media. These media have a polarization whichcan be expressed in convolution form as:

(12) P(t,x) = g?E(t,x) =∫ t

0g(t− s,x;q)E(s,x)ds

where in Debye q = {εs,ε∞,τ}. The function g is the dielectric response function (DRF) whichcan be thought of as representing the memory effect caused by the dielectric. For Debye materialsthis DRF is,

(13) g(t,x) =ε̃dτ

e−t/τ

where ε̃d := ε0(εs− ε∞) and τ is the relaxation time, which is typically taken to be a scalar. Sub-stituting equation (12) into equation (5) and taking the Laplace transform in time we get,

(14) D̂ = ε(ω) Ê

where ω is the angular frequency and ε(ω) is the complex permittivity. For the Debye model thecomplex permittivity is given by,

(15) ε(ω) = ε∞ +εd

1− iωτ .

More realistically τ it can be modeled as random variable with a probability distribution. Whenthis is the case we will refer to (15) as the stochastic complex permittivity.

1.3. Distributions. In previous years, numerical methods to approximate solutions to Maxwell’sEquations have relied on the assumption that certain parameters, such as relaxation time, of themedium can be approximated by their mean values. This was later improved upon by assuminguniform distributions of these parameters. This, however, is still inaccurate as relaxation timesappear to follow a log-normal distribution [4], which could be approximated more closely by abeta or gamma distribution. In order to incorporate these distributions we must first restate ourpolarization term to include distributions of τ. Borrowing from Banks and Gibson [2] we definethe stochastic polarization as,

(16) P (t,z) =∫ τbτa

P(t,z;τ)dF(τ)

where F(τ) is the density function for τ. This can be expressed as the solution to an ordinarydifferential equation using τ as a random variable. For the Debye model the stochastic ordinarydifferential equation (SODE) is,

(17) τ∂P∂t

+P = ε̃dE.

The stochastic polarization is the microscopically chaotic polarization that is influenced by distri-butions of τ. We take the macroscopic polarization, which appears in Maxwell’s equations and theconstitutive laws to be the expected value of the stochastic polarization.


2. POLYNOMIAL CHAOS

Generalized polynomial chaos is a method for expressing stochastic solutions in terms of trun-cated expansions of orthogonal polynomials of the random inputs [18]. Different types of orthog-onal polynomials may be chosen to achieve better, sometimes exponential, convergence. First webegin by expressing stochastic polarization at each point in space as an expansion in orthogonalpolynomials [11]:

(18) P (ξ) =∞

∑i=0

αi(t)φi(ξ)

where φi(ξ) is the ith orthogonal polynomial spanning our random space and ξ is a random variablewith some standard distribution (e.g. U[-1,1] ). We may assume that the stochastic relaxation isgiven by τ= rξ+m where m,r > 0 Plugging the solution (18) into equation (17) results in,

(19) (rξ+m)∞

∑i=0

α̇i(t)φi(ξ)+∞

∑i=0

αi(t)φi(ξ) = ε̃dE

where m and r are shift and scaling parameters, respectively.All orthogonal polynomials have a recurrence relation which can be expressed in the form[18],

(20) ξφn(ξ) = anφn+1(ξ)+bnφn(ξ)+ cnφn−1(ξ).

We substitute (20) into equation (19) in order to remove the explicit dependence on the randomvariable ξ. This gives

(21) r∞

∑i=0

α̇i(t) [aiφi+1(ξ)+biφi(ξ)+ ciφi−1(ξ)]+∞

∑i=0

[αi(t)+mα̇i(t)]φi(ξ) = ε̃dE.

Projecting onto the finite dimensional random space spanned by {φ j}pj=0 by taking the weightedinner product with each basis function produces:

rp

∑i=0

α̇i(t) [ai〈φi+1,φ j〉w +bi〈φi,φ j〉w + ci〈φi−1,φ j〉w]

+p

∑i=0

[αi(t)+mα̇i(t)]〈φi,φ j〉w = ε̃dE〈φ0,φ j〉w.(22)

In the above, 〈φi,φ j〉w is the weighted inner product of φi and φ j and is defined to be:

〈φi,φ j〉w :=∫

Ωφi(ξ)φ j(ξ)W (ξ)dξ(23)

=δi j√

hih j

where δi j is the Kronecker delta function, W (ξ) is chosen to be the weighting function with respectto which polynomials are orthogonal over the domain Ω, and hi := 〈φi,φi〉w.

This produces a system of equations which can be expressed in matrix form as:

(24) A~̇α(t)+~α(t) = ~f

where A = rM+mI and


M =

b0 c1 0 · · · 0a0 b1 c2

...

0 . . . . . . . . . 0... ap−2 bp−1 cp0 · · · 0 ap−1 bp

and ~f =

ε̃dE0...0

where ai, bi and ci are the recursion coefficients. Note that the deterministic value ~f forces thesystem and is dependent the electric field, which itself depends on the expected value value P =E [P ]. This can be approximated by α0 [18]. If the chosen set of orthogonal polynomials areorthogonal with respect to the density function of the random variable ξ then the error in theprojection, and in the mean value, converges exponentially with p. We replace the SODE (17) withthe deterministic system of ODEs (24).

2.1. Scaling the polynomials. In order for the orthogonal polynomials to be applied to a distri-bution of relaxation times they must be scaled so that they are orthogonal over the correct interval.This can be done linearly and does not affect they recurrence relation. The polynomials may bescaled in this manner so that they are defined over the arbitrary interval from a to b:

(25) φn(ξ) = φn(

2τ−ab−a −1

)thus

ξ = 2τ−ab−a −1(26)

τ=b−a

2ξ+

b+a2

.

Since τ := rξ+m

(27) r =b−a

2and m =

b+a2

This is a fairly intuitive result, m is the midpoint between a and b, the center of the distribution,and r is the distance from the midpoint to either edge of the distribution (radius). Note that formany distributions the midpoint is not the mean.

2.2. Legendre Polynomials. The Legendre Polynomials are a set of orthogonal polynomialswhich are orthogonal with respect to a constant over the domain Ω = [−1,1]. This means thatthey are ideally suited for characterizing uniformly distributed random variables. The constantsfor their recurrence relation are [18]:

an =n+1

2n+1bn = 0

cn =n

2n+1


It should be noted that the Legendre polynomials are a special case of the Jacobi Polynomialswhere α and β are zero. The matrix A for the case when p = 2 is

A =

m 13r 0r m 25r0 23r m

.

2.3. Jacobi Polynomials. The Jacobi Polynomials are orthogonal with respect to the followingweighting function [18]:

w(ξ) = (ξ−1)α(1+ξ)β

This closely resembles the probability density function of the beta distribution:

P(x) =(1− x)αxβ

B(α+1,β+1).

However, in order for us to use these polynomials, both must be scaled so that they are valid overthe interval from a to b. This is done by a simple linear substitution and produces:

w(τ) =(

2b−a

)α+β(τ−a)α(τ−b)β

P(τ) =(b−τ)α (τ−a)β

B(α+1,β+1)(b−a)α+β.

We can ignore all of the terms which are constants with respect to τ since they can simply befactored into the hi term produced when taking the inner product, and this term cancels. Thereforethe scaled polynomials are orthogonal with respect to the scaled Beta distribution.

The recurrence relation for Jacobi Polynomials has the following coefficients[18]:

an =2(n+1)(n+α+β+1)

(2n+α+β+1)(2n+α+β+2)

bn =β2−α2

(2n+α+β)(2n+α+β+2)

cn =2(n+α)(n+β)

(2n+α+β)(2n+α+β+1)

(28)

This recurrence relation holds true for any linear scaling of the random input parameter. As anexample, if we choose α = 2, β = 5, and p = 2 we obtain the following system for the stochasticpolarization model:

A =

13r+m

25r 0

29r

733r+m

1433r

0 1855r21143r+m

.


3. DISCRETIZATION

Combining (10), (11), and (24) we obtain a model for electromagnetic waves propagatingthrough a dispersive dielectric media, which accounts for a distribution of relaxation times τ. Wesubstitute the expected value of the stochastic polarization, E(P )≈ α0, for the macroscopic polar-ization in P in (10).The polynomial chaos model now has the following form:

∂H∂t

=− 1µ0

∂E∂z

ε̃∞∂E∂t

=−∂H∂z−σE− ∂α0

∂t

A~̇α+~α = ~f .

(29)

We can discretize these equations according to the one-dimensional Yee Scheme[19], addingan additional central difference approximation to the polynomial chaos system. This method usesa staggered grid, with both the electric field, E, and the stochastic polarization, ~α, evaluated athalf-steps in time and integer steps in space, and the magnetic field, H, evaluated at integer stepsin time and half-steps in space. In our discretization, we choose j to represent the spatial step andn to represent the time step.

j j

FIGURE 1. Yee Scheme in 1-Dimension

The discretized form of (29) becomes

(30)Hn+1

j+ 12−Hn

j+ 12∆t

=− 1µ0

En+ 12j+1 −E

n+ 12j

∆z

(31) ε̃∞E

n+ 12j −E

n− 12j

∆t=−

Hnj+ 12−Hn

j− 12∆z

−σE

n+ 12j +E

n− 12j

2−

αn+12

0, j −αn− 120, j

∆t


(32) A~αn+

12

j −~αn− 12j

∆t+~αn+

12

j +~αn− 12j

2=

~fn+ 12j +

~fn− 12j

2.

Multiplying both sides of the discretized polynomial chaos system (32) by 2∆t and rearranging theterms, we see that we can write the system as the matrix equation

(33) (2A+∆tI)~αn+12

j = (2A−∆tI)~αn− 12j +∆tI(~f

n+ 12j +

~fn− 12j ).

To derive useful update equations which can be used in programming our simulation, we need

to solve for Hn+1j+ 12

, En+ 12j , and~α

n+ 12 explicitly in terms of already known values. Since,

~fn+ 12j =

ε̃dE

n+ 12j

0...0

,we see that equations(31) and (33) are coupled. In order to obtain sequential update equations, we

solve (31) for En+ 12j and substitute this into the polynomial chaos system. Using the simplifications

θ := 2ε̃∞ +σ∆t and θ∗ := 2ε̃∞−σ∆t, we obtain,

(34) En+ 12j =

1θ[θ∗En−

12

j −2∆t∆z

(Hnj+ 12−Hnj− 12 )−2(α

n+ 120, j −α

n− 120, j )].

Substituting this result into (33) and collecting all αn+12

i, j terms on the left-hand side, and all αn− 12i, j

terms on the right-hand side, the first row of (33) becomes

[2(rb0 +m+κ)+∆t]αn+ 120, j +(2rc1)α

n+ 121, j = [2(rb0 +m+κ)−∆t]α

n− 120, j +(2rc1)α

n− 121, j

+(ε̃d∆t +κθ∗)En− 12j −

2κ∆t∆z

(Hnj+ 12−Hnj− 12 )

where κ := ε̃d∆tθ . Borrowing the representation from (33) we can write this system as

(35) (2Ã+∆tI)~αn+12 = (2Ã−∆tI)~αn− 12 + f̃ n,

where Ã is the matrix A with κ added to the (1,1) entry and f̃ n is the vector,

f̃ n =

(ε̃d∆t +κθ)E

n− 12j − 2κ∆t∆z (Hnj+ 12 −H

nj− 12

)

0...0

.Lastly, the expression for the magnetic field update is given by

(36) Hn+1j+ 12

=− ∆tµ0∆z

[En+ 12j+1 −E

n+ 12j ]+H

nj+ 12

.


The final form of our sequential update equations is:

(2Ã+∆tI)~αn+12

j = (2Ã−∆tI)~αn− 12j + f̃

n

En+ 12j =

1θ+

[θ∗En−12

j −2∆t∆z

(Hnj+ 12−Hnj− 12 )−2(α

n+ 120, j −α

n− 120, j )]

Hn+1j+ 12

=− ∆tµ0∆z

[En+ 12j+1 −E

n+ 12j ]+H

nj+ 12

.

(37)

3.1. The Invertibility of A. In order for our method (37) to have a solution we must be able toinvert the matrix (2Ã+∆tI).Theorem 3.1. For the Jacobi and Legendre polynomials the matrix (2Ã + ∆tI) is invertible ifm > r.

Proof. We start by taking the transpose of the matrix, if this matrix is strictly diagonally dominantit must be non-singular and therefor the original matrix must also be invertible. A matrix is strictlydiagonally dominant if

(38) |aii|> ∑j 6=i|ai j| for all i.

Since bi ≥ 0 for all i then (2Ã+∆tI)T has all non-negative entries along the main diagonal, wemay ignore the absolute value along these entries. The ∆tI and κ (which was added to the (1,1)number) may also be dropped, as they only add to the main diagonal and we would like to have asolution for arbitrarily small ∆t. Therefore, if AT is diagonally dominant then (2Ã+∆tI)T is alsodiagonally dominant. This yields the following conditions on A

b0r+m > |a0|rbir+m > (|ci|+ |ai|)r for 0 < i < p(39)bpr+m > |cp|r.

Solving for m, these conditions become:

m > (|a0|−b0)rm > (|ci|+ |ai|−bi)r for 0 < i < p(40)m > (|cp|−bp)r.

Since the middle condition is always increasing for i > 0, only the endpoint needs to be checkedfor the middle conditions. This means that although there are p+1 conditions, at most, only 3need to be checked. For Legendre and Jacobi polynomials these conditions converge m > r as pgoes to infinity. A similar procedure may be applied to ensure stability when working with otherpolynomials. Table 3.1 demonstrates this convergence for Legendre Polynomials, the same maybe shown for Jacobi Polynomials. �

While A may still be non-singular for some values of r ≥ m, these would imply the possibilityof negative relaxation times which is unphysical.


p largest eigenvalue m > max(|ci|+ |ai|−bi)r m > (|cp|−bp)r1 m = 0.577r m > r m > 13r

2 m = 0.775r m > r m > 25r4 m = 0.906r m > r m > 49r8 m = 0.968r m > r m > 817r

16 m = 0.991r m > r m > 1633r32 m = 0.997r m > r m > 3265r

TABLE 1. Maximum eigenvalue compared to diagonally dominant conditions forLegendre Polynomials

4. STABILITY ANALYSIS

We wish to determine the stability properties of the above numerical method (37). A finitedifference scheme is stable if the errors made at each time step of the calculation do not causethe errors to increase without bound as the computations are continued. We also require that oursolution be convergent. A convergent scheme is one that better approximates Maxwell’s equationsas the spatial and temporal step sizes are decreased. Finally, a consistent scheme is one that differsfrom the PDE point-wise by factors that go to zero as ∆z and ∆t go to zero. The following theoremcombines the ideas of convergence, consistency, and stability.

Theorem 4.1. (Lax-Richtmyer Equivalence Theorem) [14]A consistent finite difference scheme for a partial differential equation for which the initial valueproblem is well-posed is convergent if and only if it is stable.

It is known that the local truncation error (LTE) of the Yee-Scheme in a non-dispersive mediais O(∆t2)+O(∆z2)[19]. The central difference approximation in the discretization of polarizationadds at most O(∆t2) to the LTE. The finite-dimensional projection also contributes to the LTE,however, due to the exponential convergence we may assume, provided p is sufficiently large, thatthis error is less than O(∆t2)+O(∆z2). Thus we assume the numerical method (37) is consistentand is in fact second order in space and time. Therefore establishing the stability of the numericalmethod is sufficient to show convergence.

Plane wave solutions of (37) are of the form:

(41)

EnjHnjαn0, j

...αnp, j

=

ẼH̃α̃0...

α̃p

ζneik j∆z.

Here, i =√−1, k = ωc is the wave number, where c is the speed of light and ω = 2π f is the angular

frequency, ζ is the complex time eigenvalue, and ~x = [Ẽ, H̃, α̃0, · · · , α̃p]T is the corresponding


eigenvector. In order for our method to be stable we require that |ζ| ≤ 1. We proceed by substituting(41) into our discretized update equations (37).

We note that our system (2Ã+∆tI)~αn+12

j = (2Ã−∆tI)~αn− 12j + f̃

n corresponds to the followingsystem of p equations:

for i = 0

[2(rb0 +m+κ)+∆t]αn+ 120, j +(2rc1)α

n+ 121, j = [2(rb0 +m+κ)−∆t]α

n− 120, j +(2rc1)α

n− 121, j

+(ε̃d∆t +κθ∗)En− 12j −2κ

∆t∆z

(Hnj+ 12−Hnj− 12 ),

(42)

for 0 < i < p

(2rai−1)αn+ 12i−1, j +[2(rbi +m)+∆t]α

n+ 12i, j +(2rci+1)α

n+ 12i+1, j

= (2rai−1)αn− 12i−1, j +[2(rbi +m)−∆t]α

n− 12i, j +(2rci+1)α

n− 12i+1, j,

(43)

and for i = p

(2rap−1)αn+ 12p−1, j+[2(rbp +m)+∆t]α

n+ 12p, j = (2rap−1)α

n− 12p−1, j +[2(rbp +m)−∆t]α

n− 12p, j .(44)

We substitute each component of (41) into the first equation and cancel ζn−12 eik j∆z from both sides

to obtain

[2(rb0 +m+κ)+∆t]ζα̃0 +(2rc1)ζα̃1 = [2(rb0 +m+κ)−∆t]α̃0 +(2rc1)α̃1

+(ε̃d∆t +κθ∗)Ẽ−2κ∆t∆z

ζ12 (e

ik∆z2 − e− ik∆z2 )H̃.

Using Euler’s formula, eiθ = cosθ+ i sinθ, it follows that eik∆z

2 − e− ik∆z2 = 2isin(k∆z2 ). Substitutingin this identity and simplifying further results in the following equation:

[2κ(ζ−1)+(2rb0 +2m+∆t)ζ− (2rb0 +2m−∆t)]α̃0 +(2rc1)(ζ−1)α̃1

− (ε̃d∆t +κθ∗)Ẽ +4iκ∆t

∆zζ

12 sin(

k∆z2

)H̃ = 0.(45)

The remaining equations can be derived in a similar fashion and end up being considerably simpler.For our polynomial chaos system, we obtain the following equations:for 0 < i < p

(2rai−1)(ζ−1)α̃i−1 +[(2rbi +2m+∆t)ζ− (2rbi +2m−∆t)]α̃i+(2rci+1)(ζ−1)α̃i+1 = 0,

and for i = p

(2rap−1)(ζ−1)α̃p−1 +[(2rbp +2m+∆t)ζ− (2rbp +2m−∆t)]α̃p = 0.


Applying the same approach, we obtain the following for the electric and magnetic field updates,

(46) (θζ−θ∗)Ẽ + 4i∆t∆z

ζ12 sin(

k∆z2

)H̃ +2(ζ−1)α̃0 = 0.

(47) (ζ−1)H̃ + 2i∆tµ0∆z

ζ12 sin(

k∆z2

)Ẽ = 0

We can write our system as S~x = 0, where~x = [Ẽ, H̃, α̃0, · · · , α̃p]T ,

S =

θζ−θ∗ 4iρc∞ ζ12 2(ζ−1)

2iρµ0c∞

ζ12 ζ−1 0 0

−(ε̃d∆t +κθ∗) 4iκρc∞ ζ12 2κ(ζ−1)

0 0

+

0 00 0 0

0 (2A+∆tI)ζ− (2A−∆tI)

ν := c∞∆t∆z , c∞ :=

1√µ0ε̃∞

, and ρ := νsin(k∆z2 ).

This system has a non-trivial solution if and only if detB = 0. We obtain a characteristic poly-nomial is of the form

p+3

∑k=0

qkζp−k = 0

4.1. The Characteristic Polynomial when p=1. Neglecting conductivity (letting σ = 0), we ob-tain a polynomial of the following form for p = 1:

(48) q0ζ4 +q1ζ3 +q2ζ2 +q3ζ+q4 = 0,

with coefficients

q0 = 4|A|+[(rb0 +m)+ εq(rb1 +m)]∆t + εq∆t2

q1 = 4ρ2∆t2 +2∆t[4tr(A)ρ2− (rb0 +m)− εq(rb1 +m)]−16|A|(1−ρ2)q2 = 2∆t2(4ρ2− εq)−8|A|(4ρ2−3)q3 = 4ρ2∆t2−2∆t[4tr(A)ρ2− (rb0 +m)− εq(rb1 +m)]−16|A|(1−ρ2)q4 = 4|A|− [(rb0 +m)+ εq(rb1 +m)]∆t + εq∆t2.

(49)

Note that this is valid for any choice of polynomials. For Jacobi Polynomials, which correspond toa beta distribution of relaxation times we have

|A|= m2 + 2(β−α)α+β+4

rm+[

(β−α)2(α+β+3)(α+β+4)

− 1α+β+3

]m2

tr(A) = 2m+2(β−α)α+β+4

r,


where b0 and b1 defined as in (28). If we let α = 2 and β = 5 the above expressions simplify to

|A|= m2 + 611

rm− 155

m2

tr(A) = 2m− 611

r.

We also note b0 = 13 and b1 =733 . Finally, for Legendre polynomials these coefficients become

q0 = 4(m2−13

r2)+2m∆t(εq +1)+ εq∆t2

q1 = 16(m2−13

r2)(ρ2−1)+4m∆t[4ρ2− εq−1

]+4ρ2∆t2

q2 = 2∆t2(4ρ2− εq)−8(m2−13

r2)(4ρ2−3)

q3 = 16(m2−13

r2)(ρ2−1)−4m∆t[4ρ2− εq−1

]+4ρ2∆t2

q4 = 4(m2−13

r2)−2m∆t(εq +1)+ εq∆t2.

(50)

4.2. Routh-Hurwitz Stability Criterion. The Routh-Hurwitz Criterion [12] establishes that thepolynomial,

f (z) =N

∑k=0

bkzN−k, b0 > 0

where bk are arbitrary constant real coefficients, has no roots in the right-half of the complex-planeonly if all the entries of the first column of the Routh Table are non-negative. We construct theRouth Table as follows. The coefficients bk are arranged in two rows, noting that the leadingcoefficient always appears in the upper left corner:

c1,k = b2k, for k = 0, ...,bN2c

c2,k = b2k+1, for k = 0, ...,bN−1

2c

The remaining entries in the the table are obtained using the formula:

c j,k =−1

c j−1,0

∣∣∣∣ c j−2,0 c j−2,k+1c j−1,0 c j−1,k+1∣∣∣∣

For example, the Routh Table for f (z) = b0z3 +b1z2 +b2z+b3 isb0 b2b1 b3b? 0b4 0

and the Routh Table for f (z) = b0z4 +b1z3 +b2z2 +b3z+b4 is,


b0 b2 b4b1 b3 0b? b4 0b?? 0 0b4 0 0

where b? = b1b2−b0b3b1 and b?? = b

?b3−b1b4b? . We note there is also a special case where we can still

have roots in the right-half plane even if all of the entries in the first column of the Routh table arenonnegative. This occurs only when there is a row of zeros in our Routh table.

We can use the Routh-Hurwitz criterion to determine if any of the roots of our stability polyno-mial lie outside of the unit circle. The transformation

(51) ζ =z+1z−1

maps the outside of the unit circle to the right half plane. Thus if we apply (51) to our characteristicpolynomial, f (ζ), and the resulting polynomial f (z) has no roots in the right-half plane, we canconclude that f (ζ) has no roots outside of the unit circle and our solution is stable. We consider thestability of our polynomial chaos scheme using Legendre polynomials. We start by considering thecase p = 1, as the p = 0 case corresponds to the traditional single-pole Debye and is outlined byPereda. [12] The following theorem gives the exact conditions under which our method is stablein this case.

Theorem 4.2. The numerical polynomial chaos scheme (37) is stable for Legendre polynomialswith p = 1 if and only if the following conditions hold

m≥ 0ν≤ 1

εs ≥ ε∞.

Proof. Let p = 1 and suppose we choose to use Legendre polynomials in our polynomial chaossystem. The characteristic polynomial of the form given in (48) with coefficients given by (50). Ifwe let ζ = z+1z−1 and simplify by multiplying through by a factor of

(z−1)424 we obtain the polynomial

(52) q̂0z4 + q̂1z3 + q̂2z2 + q̂3z+ q̂4 = 0,

where

q̂0 = ρ2∆t2

q̂1 = 4mρ2∆t

q̂2 = 4|A|ρ2 +(εq−ρ2)∆t2

q̂3 = 2m∆t(εq +1−2ρ2)q̂4 = 4|A|(1−ρ2)

ρ = νsin(k∆z2 ) and εq =εsε∞ , and |A|= m

2− 13r2. The Routh Table is therefore


q̂0 q̂2 q̂4q̂1 q̂3 0

q? = q̂1q̂2−q̂0q̂3q̂1 q̂4 0

q?? = q?q̂3−q̂1q̂4

q? 0 0

q̂4 0 0

We need the entires in the first column to be nonnegative. We note that q̂0 is always nonnegative.Requiring q̂1 to be nonnegative leads to the stability condition m≥ 0. Using our condition for theinvertibility of A, m≥ r, we see

|A|= m2− 13

r2 ≥ 23

m2 ≥ 0

Since |A| is positive we can conclude q̂4 is positive provided ρ2 = ν2 sin2(k∆z2 ) ≤ 1. Using theworst case, sin2(k∆z2 ) = 1, we obtain the C.F.L condition ν≤ 1. Computing q? we obtain:

q? = 4|A|ρ2 + 12(εq−1)∆t2

We note that 0 ≤ 4|A|ρ2 ≤ 4m2. Thus 12(εq− 1)(∆t)2 is a lower bound for q?. Requiring (εq−1)(∆t)2 to be positive will thus insure that q? is positive. This leads to the condition

εq−1≥ 0⇒ εq ≥ 1⇒ εs ≥ ε∞

since εq := εsε∞ . We notice that q?? will be positive if and only if its numerator is positive (since q?

must be positive). Computing the numerator we obtain:

q?q̂3− q̂1q̂4 = m∆t(εq−1)[(εq +1−2ρ2)∆t2 +8|A|ρ2]since m∆t(εq− 1) is positive given the conditions above, we only need to verify that the term insquare brackets is positive. However,

(εq +1−2ρ2)∆t2 +8|A|ρ2 ≥ (2−2ρ2)∆t2 +8|A|ρ2 since εq ≥ 1= 2(1−ρ2)∆t2 +8|A|ρ2

Since 0≤ ρ2 ≤ 1,|A|= m2− 13r2 ≥ 0, and the sum of two nonnegative terms is nonnegative we canconclude that q?? ≥ 0. Therefore we have shown that the entries in the first column of the RouthTable are nonnegative if and only if m ≥ 0,ν ≤ 1, and εs ≥ ε∞. Furthermore these conditions aresufficient to establish the stability of out method in the case p = 1. �

This approach can be used to establish the stability of the polynomial chaos model with otherchoices of orthogonal polynomials as well as for larger values of p. It is worth noting that theonly distinction between these results and that of the analysis of the single-pole Debye is thereplacement of the condition τ̄≥ 0 by m≥ 0. We defined m to be the midpoint of the distributionof relaxation times, while τ̄ in the Debye model analyzed by Pereda is the average relaxation time.


In the case of Legendre chaos, which corresponds to a uniform distribution of τ, the mean andmidpoint are the same, but for many other distributions (including beta distributions in general)this is not the case.

Additionally, suppose our distribution is defined on the interval [a,b]. Without loss of generalitywe can assume a < b or a = b. The second case we wish to avoid since this represents a deltadistribution of τ, which would imply that τ has only a single value, the solution to which is wellunderstood. In addition, empirical data shows that materials such as dry skin represent a wide rangeof relaxation times, so this is not realistic as well. SInce r = b−a2 , it follows r must be nonnegative.Combining this with the invertibility condition we require 0 < r < m for our polynomial chaosmodel.

4.3. Numerical Stability Analysis. We use MAPLE to plot the maximum roots of our character-istic polynomial (for the p=1 case) as a function of k∆z for 0≤ k∆z≤ π. We begin by consideringa uniform distribution of relaxation times from τ2 to

3τ2 taking τ = 8.13× 10−12 seconds as done

by Petropoulos [13]. From this we see m = τ and r = τ2 . Additional parameters are εs = 78.2,ε∞ = 1, and σ = 5×10−4. We first consider the case ν = 1 for three temporal different resolutions:h = 0.1, h = 0.01, and h = 0.001, where h := ∆tτ . These correspond to 10, 100, and 1000 time-stepsper mean relaxation time τ respectively. It is evident from the plot that the maximum value of |ζ|is always less than or equal to 1, which numerically verifies the stability of our method proved inthe preceding section.

h=0.1h=0.01h=0.001

k∆z0 1 2 3

max z

0.90

0.95

1.00

1.05Legendre (p=1): nu=1

FIGURE 2. The maximum eigenvalue at three different temporal resolutions forthe Polynomial Chaos finite-difference scheme as a function of k∆z using LegendrePolynomials with p = 1 and ν = 1.

We provide as comparison a plot of the maximum eigenvalues for h = 0.1 and ν = 0.5,0.75, and1. Since ν := c∞∆t∆z and ∆t = hτ it follows that ∆z =

c∞hτν . If we fix h and increase the value of ν we


are therefore decreasing the spatial step size.The plot of ν = 0.5 is of particular interest to us as itis the value utilized in our simulation. Once again we see that the method is stable for all of thesechoices of ν.

nu=1nu=0.75nu=0.5

k∆z0 1 2 3

max z

0.90

0.95

1.00

1.05Legendre: p=1, h=0.1

FIGURE 3. The maximum eigenvalue at three different spatial resolutions for thePolynomial Chaos finite-difference scheme as a function of k∆z using LegendrePolynomials with p = 1 and h = 0.1.

For values of ν ≥ 1 our method becomes unstable. We consider the plot of ν = 1.1 at threeresolutions, h = 0.1, h = 0.01, and h = 0.001. As we can see, this is a rather tight bound, as thethe maximum roots now far exceed 1 on the right side of the plot.

Finally we compare the roots of largest magnitude, max|ζ| for the cases p = 1, through p = 5.Note that in practice a value of p = 2 or p = 3 is probably sufficient. We will show in a subsequentsection that due to computational limitations the accuracy does not actually improve beyond p = 6.

As a comparison we replicate our plots for Jacobi polynomials choosing α = 2 and β = 5. Onceagain we let m = τ and r = τ2 . Note that we have arbitrarily picked these values as they do notnecessarily correspond to a distribution of relaxation times in a material of interest. Using the samematerial parameters, we obtain the following plots. It is of interest to note that there is minimaldifference between the plots for Legendre polynomials (Jacobi polynomials with α = β = 0) andthe Jacobi polynomials with α = 2 and β = 5. This is important, as it suggests that the choice ofbeta distribution itself, provided we have 0< r


h = 0.1h=0.01h=0.001

k∆z0 1 2 3

max z

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5Legendre (p=1): nu=1.1

FIGURE 4. The maximum eigenvalue at three different temporal resolutions forthe Polynomial Chaos finite difference scheme as a function of k∆z using LegendrePolynomials with p = 1 and ν = 1.1. This violates our CFL condition of ν≤ 1 andgives an example of an unstable solution since max|ζ| exceeds 1.

p= 1p=2p=3p=4p=5

k∆z0 1 2 3

max |ζ|

0.90

0.92

0.94

0.96

0.98

1.00Legendre: nu=1, h=0.1

FIGURE 5. The maximum eigenvalue for the Polynomial Chaos finite-differencescheme as a function of k∆z using Legendre Polynomials with ν = 1 and h = 0.1.The first five values of p are shown.


h=0.1h=0.01h=0.001

k∆z0 1 2 3

max z

0.90

0.95

1.00

1.05Jacobi (p=1) : nu=1

FIGURE 6. The maximum eigenvalue at three different temporal resolutions forthe Polynomial Chaos finite-difference scheme as a function of k∆z using JacobiPolynomials (α = 2, β = 5) with p = 1 and ν = 1.

nu=1nu=0.75nu=0.5

k∆z0 1 2 3

max |ζ|

0.90

0.95

1.00

1.05Jacobi: p=1, h=0.1

FIGURE 7. The maximum eigenvalue at three different spatial resolutions forthe Polynomial Chaos finite-difference scheme as a function of k∆z using JacobiPolynomials(α = 2, β = 5) with p = 1 and h = 0.1.


h=0.1h=0.01h=0.001

k∆z0 1 2 3

max z

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5Jacobi (p=1) : nu=1.1

FIGURE 8. The maximum eigenvalue at three different temporal resolutions forthe Polynomial Chaos finite difference scheme as a function of k∆z using JacobiPolynomials (α= 2, β= 5) with p= 1 and ν= 1.1. This violates our CFL conditionof ν≤ 1 and gives an example of an unstable solution since max|ζ| exceeds 1.

p=1p=2p=3p=4

k∆z0 1 2 3

max z

0.90

0.92

0.94

0.96

0.98

1.00Jacobi: nu=1, h=0.1

FIGURE 9. The maximum eigenvalue for the Polynomial Chaos finite-differencescheme as a function of k∆z using Jacobi Polynomials (α = 2, β = 5) with ν = 1and h = 0.1. The first four values of p are shown.


5. SIMULATION

To simplify we allowed τ to be uniformly distributed from 12τ to32τ for the purpose of our

simulations. The following values were also used in the simulation unless otherwise noted,

Physical Constants:

ε0 = 8.85419×10−12

µ0 = 4π×10−7

c = 3×108 = 1√µ0ε0

(the speed of light in a vacuum)

Material Parameters:εs = 80.35ε∞ = 1

τ= 8.13×10−12

r =12τ

m = τσ = 0

Simulation Parameters:f = 10GHz (the frequency of the simulated pulse)h = 0.01ν = 0.5 (the CFL condition)

∆t = h×τ (the time step)

∆z =c∆tν

(the spacial step)

5.1. Convergence. In order for this method to be useful it must converge to an accurate solutionfor small values of p. In other words, our method will only be useful if we have to solve a systemof only a few equations instead of hundreds. Fortunately this is the case, our method displaysexponential convergence and can arrive at a solution accurate to approximately 10−12 using onlyp = 6 and m = 12s with 16 digits of precision, which corresponds to a seven equation system. Itshould also be noted that the accuracy can go beyond 10−12 for larger value of p, but this wouldalso necessitate more than 16 digits of precision. It should also be noted that changing r affects therate of convergence, with values closer to 0 causing more rapid convergence, and values closer tom resulting in slower convergence.

The error displayed in Figure 10 is calculated by running a simulation for 0.75×10−9 secondsfor various values of p ranging from 0 to 12. Then we assume that the result calculated when p is 12is the most accurate and subtract that data from the results for the other simulations. This gives usa rough estimate of how quickly the method converges. To plot these results we take the absolutevalue and add machine epsilon

(22−52 ≈ 2.22×10−16

)so that the results display properly on a


0 1 2 3 4 5 6 7

x 10−10

10−16

10−14

10−12

10−10

10−8

10−6

10−4

10−2

100

t

Error

Log-difference displaying exponential convergence of polynomial chaos r = 0.5 × τ

p = 0

p = 1

p = 2

p = 4

p = 6

p = 8

p = 10

FIGURE 10. Error Calculated for various values of p. r = 12τ and m = τ.

log-scale graph. Since this graph can be difficult to read, we also take the maximum value for eachvalue of p and graph that as Figure 11.

When compared with the same data for different values of r this also shows how increasing rleads to slower convergence. The plot for r = 0 is not displayed here since it is 0 always. This isbecause in case when r = 0 our matrix which corresponds to our system of equations, becomes ascalar multiple of the identity matrix. Therefore the solution does not change for α0 as p increases,and the remaining αi terms are zero.

Changing other parameters also affects the convergence of the method. Since this method isbased on FDTD it should converge to an exact solution as ∆t goes to 0. In order to demonstratethis we first define the value of ∆t as

(53) ∆t := hτ.

We will modify the value of ∆t by changing our parameter value of h. As Figure 12 shows, ourmethod converges as h and thus ∆t tend to 0. Counter intuitively, although our method convergesto a more accurate solution as h goes to 0, the noise floor of our method actually rises as displayedin Figure 13. The reason for this increased noise could be any number of things. However, currentassumption is that the increased number of time steps could be increasing the number of LocalTruncation Errors faster than these errors tend to zero. This implies that there is a sweet spot for∆t, values above or below that would actually decrease the precision of the method.


0 1 2 3 4 5 6 7 8 9 1010

−12

10−10

10−8

10−6

10−4

10−2

100

102

Maximum Difference Calculated for different values of p and r

p

Maxim

um

Err

or

r = 1.00τ

r = 0.75τ

r = 0.50τ

r = 0.25τ

FIGURE 11. Maximum Error for various values of p and r.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.1410

−3

10−2

10−1

100

101

Grid norm−1 error as h goes to 0

Grid n

orm

−1 E

rror

h

FIGURE 12. The average error for different values of ∆t


0 1 2 3 4 5 6 7 8 9 1010

−12

10−10

10−8

10−6

10−4

10−2

100

Log plot displaying the increased noise floor as h decreases

p

Maxim

um

Error

h= 0.1h= 0.02h= 0.005h= 0.001

FIGURE 13. The increased noise floor for smaller values of ∆t

6. INVERSE PROBLEM

One may use an inverse problem to determine the correct parameters of τ for a two pole Debyemodel with uniform distributions. Borrowing from their work the procedure is as follows. Webegin by defining τm to be distributed as,

(54) τm ∼U [amτm,(am +bm)τm]where am and bm are distribution parameters of τ for which we are concerned. We let q be the set ofall parameters in the representation of the distribution. This definition is for a uniform distribution,however, any other distribution could be used in its place. Next we define the stochastic complexpermittivity as a function of ω, the angular frequency,

(55) ε(ω) = ε∞ +∑m

εd,m1+ iωτm

The expected value of each pole can be found by integrating with respect to the density functionof the distribution being used,

(56) E [εm (ω)] =∫ am+bm

am

εd,m1+ iωτm

f (τm)dτ

For a uniform distribution this becomes,

E [εm (ω)] =1

bm

[∫ (am+bm)τmamτm

εd,m1−ω2τ2 dτ+ i

∫ (am+bm)τamτm

−εd,mωτ1−ω2τ2 dτ

](57)

=1

bm

εd,mω

[arctan(ωτ)− i

2ln(

1+(ωτ)2)](am+bm)τm

amτm(58)

While this integral (57) can be evaluated analytically for the uniform distribution, this is not nec-essarily the case for all distributions. When it may not be exactly evaluated, numerical integration


will suffice. We can evaluate the accuracy of the parameters by evaluating ε(ω) across a vector ofimportant frequencies and then subtracting from a target solution, be it real world data or anothermodel such as the Cole-Cole. This subtraction produces a vector of residuals, R(ω), which can beused to determine the cost, J, defined as,

(59) J(ω) := RT RMinimizing this cost function will produce the most accurate approximation to the real distributionof τ for each pole.

7. CONCLUSION

We have shown we can improve the traditional Debye model by replacing the average relax-ation time τ̄ with a distribution of relaxation times. The method of Generalized Polynomial Chaosprovides us with a convenient means of representing the stochastic polarization, P as a linear com-bination of orthogonal polynomials. By projecting into finite random space, we are able to replacea stochastic ordinary differential equation with a system of deterministic ODEs. Combining thesewith Maxwell’s and Faraday’s Law and noting that the electric field E depends only on the macro-scopic polarization E(P)≈ α0, we obtain the polynomial chaos model (24) for an electromagneticfield propagating through a dispersive dielectric media.

This model lends itself naturally to discretization using the Yee-Scheme with an additional cen-tral difference approximation for the polynomial chaos system. From the discretized equationswe are able to obtain a series of sequential update equations (37). These can then be used inperforming a MATLAB simulation.

We have proved that this method is stable for Legendre polynomials with p = 1 if m≥ 0, ν≤ 1,and εs ≥ ε∞. This method can be extended to other types of orthogonal polynomials and for largervalues of p. As a future endevor we would like to establish this result for Jacobi polynomials andobtain a beta distribution of relaxation times that more accurately fits our data. Also of interest isbeing able to generalize our result for any value of p. In the meantime we have numerically verifiedthe stability of Jacobi and Legendre polynomials for various spatial and temporal resolutions andvalues of p up to 5. In practice, we do not gain accuracy above p = 6 due to current limitations oncomputational precision, and a value of p = 2 or p = 3 will probably suffice.

REFERENCES

[1] HT Banks, VA Bokil, and NL Gibson. Analysis of stability and dispersion in a finite element method for Debyeand Lorentz dispersive media. Numerical Methods for Partial Differential Equations, 25(4), 2009.

[2] H.T. Banks and N.L. Gibson. Electromagnetic inverse problems involving distributions of dielectric mechanismsand parameters. Quarterly of Applied Mathematics, 64(4):749–795, 2006.

[3] K. Barrese and N. Chugh. Approximating dispersive mechanisms using the debye model with distributed dielec-tric parameters, 2008.

[4] CJF Böttcher, OC Van Belle, P. Bordewijk, A. Rip, and D.D. Yue. Theory of electric polarization. Journal of TheElectrochemical Society, 121:211C, 1974.

[5] K.S. Cole and R.H. Cole. Dispersion and absorption in dielectrics I. Alternating current characteristics. TheJournal of Chemical Physics, 9:341, 1941.

[6] Bert J. Debusschere, Habib N. Najm, Philippe P. Pébay, Omar M. Knio, Roger G. Ghanem, and Olivier P. LeMaı̂tre. Numerical challenges in the use of polynomial chaos representations for stochastic processes. SIAMJournal on Scientific Computing, 26(2):698–719, 2004.

[7] P. Debye. Polar Molecules. Dover, 1929.


[8] R.S. Edwards, A.C. Marvin, and S.J. Porter. Uncertainty Analyses in the Finite-Difference Time-DomainMethod. IEEE Transactions on Electromagnetic Compatibility, 52(1):155–163, 2010.

[9] Kira Heater. The fdtd method: Computation and analysis. Master’s thesis, University of Montana, 2005.[10] B.H. Jung and T.K. Sarkar. Solving time domain Helmholtz wave equation with MOD-FDM. Progress In Elec-

tromagnetics Research, 79:339–352, 2008.[11] D. Lucor, C.H. Su, and GE Karniadakis. Generalized polynomial chaos and random oscillators. International

Journal for Numerical Methods in Engineering, 60(3), 2004.[12] A. Pereda, L.A. Vielva, A. Vegas, and A. Prieto. Analyzing the stability of the fdtd technique by combining the

von neumann method with the routh-hurwitz criterion. Microwave Theory and Techniques, IEEE Transactionson, 49(2):377 –381, feb 2001.

[13] P.G. Petropoulos. Stability and phase error analysis of FD-TD in dispersive dielectrics. IEEE transactions onantennas and propagation, 42(1):62–69, 1994.

[14] John C. Strikwerda. Finite difference schemes and partial differential equations. SIAM, 2004.[15] Gábor Szegö. Orthogonal Polynomials. American Mathematical Society, 1975.[16] D. Webbe and M. Milne. Simulating polydisperse materials with distributions of the debye model, 2009.[17] D. Xiu and G.E. Karniadakis. The Wiener–Askey Polynomial Chaos for Stochastic Differential Equations. SIAM

Journal on Scientific Computing, 24(2):619–644, 2003.[18] Dongbin Xiu. Numerical Methods for Stochastic Computations. Princeton University Press, 2010.[19] Kane Yee. Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic

media. Antennas and Propagation, IEEE Transactions on, 14(3):302 –307, may 1966.

APPENDIX A. VARIABLES AND CONSTANTS

an,bn,cn - Coefficients from the recurrence relation for orthogonal polynomialsc - speed of light(free space) c = 1√µ0ε0 ; also c∞ =

c√ε∞

e - the number e~f = [ε̃dE,0, · · · ,0]T ; the vector found in the system A~̇α(t)+~α(t) = ~ff̃ n, found in modified α system (23)h = ∆tτi -index for orthogonal polynomialsi =√−1

j - spatial step, also used as an index for our polynomials in section 2k = ωc - the wave numberm - midpoint between a and b in the distribution of the relaxation time τ n -time stepp - number of polynomials used in the polynomial chaos expansionqk - coefficients of the characteristic polynomial (as a function of ζ)q̂k - coefficients of the transformed characteristic polynomial (as a function of z)r - distance from midpoint to the edge of the distribution of τt - timex, y, z - spatial dimensions

A - matrix from the polynomial chaos system, A~̇α(t)+~α(t) = ~f . Note A = rM+mI.Ã - modified form of A; add κ to (1,1) entryB - Magnetic flux density (vector)D - Electric flux density (vector)E - Electric Field vector, E := ExH - Magnetic Field vector, H := Hy


I - Identity MatrixJ -conduction current density, J := JxJs - source current densityM - matrix of recurrence relation coefficients obtained in polynomial chaos expansionM -magnetizationP -macroscopic polarizationP - stochastic polarizationS - system obtained during stability analysisW - weighting function

αi - coefficients from the polynomial chaos expansion of the stochastic polarization (18)α and β - Beta distribution parametersδi j -Kronecker delta functionε0 - permittivity of free spaceεs -static permitttivityε∞ -relative permittivity or infinite frequency permittivity, ε̃∞ = ε0ε∞εd = εs− ε∞; ε̃d = ε0(εs− ε∞)εq = εsε∞ζ - complex time eigenvalueθ = 2ε∞ +σ∆tθ∗ = 2ε∞−σ∆tκ - a simplification equal to ε̃d∆tθλ - wavelengthµ - magnetic permeabilityµ0 - magnetic permeability of free spaceν = c∞∆t∆z , courant numberξ - our random variableπ - for the number piρ - in section 1.1 the density of free electric charges unaccounted for in the electric polarization;elsewhere ρ = νsin(k∆z2 )σ - electric conductivityτ - relaxation timeφi - the ith orthogonal polynomialω - angular frequency

Mathematical Operators Σ, ∆B - Beta FunctionΓ - Gamma FunctionΩ - the domain of the distribution


CHAPMAN UNIVERSITYE-mail address: [email protected]

OREGON STATE UNIVERSITYE-mail address: [email protected]

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