A high accuracy approximation for half - space problems with anisotropic scattering

Similar documents
The Radiative Transfer Equation

Light Scattering in Inhomogeneous Atmospheres

Lecture 2 Solutions to the Transport Equation

Lesson 6: Diffusion Theory (cf. Transport), Applications

arxiv: v1 [astro-ph.sr] 17 Sep 2014

Finite element investigations for the construction of composite solutions of even-parity transport equation

An Analytical Solution of the Radiative Transfer Equation for Inhomogeneous Finite Medium with Fresnel Boundary Conditions

Monte Carlo method projective estimators for angular and temporal characteristics evaluation of polarized radiation

Extensions of the TEP Neutral Transport Methodology. Dingkang Zhang, John Mandrekas, Weston M. Stacey

Spectral analysis for a class of linear pencils arising in transport theory

Green's functions for the two-dimensional radiative transfer equation in bounded media

1 3 µ2 p ν (τ ν ) = I ν. s ν (τ ν ) + (1) µ dq ν. dτ ν. and. = Q ν {( 1 µ 2) p ν (τ ν ) }, (2)

d 1 µ 2 Θ = 0. (4.1) consider first the case of m = 0 where there is no azimuthal dependence on the angle φ.

Algebraic Properties of Solutions of Linear Systems

STOCHASTIC & DETERMINISTIC SOLVERS

Astro 305 Lecture Notes Wayne Hu

22.54 Neutron Interactions and Applications (Spring 2004) Chapter 7 (2/26/04) Neutron Elastic Scattering - Thermal Motion and Chemical Binding Effects

Reflectance imaging at superficial depths in strongly scattering media

Problem Set 8 Mar 5, 2004 Due Mar 10, 2004 ACM 95b/100b 3pm at Firestone 303 E. Sterl Phinney (2 pts) Include grading section number

Analysis of Scattering of Radiation in a Plane-Parallel Atmosphere. Stephanie M. Carney ES 299r May 23, 2007

The collision probability method in 1D part 2

Electrodynamics Qualifier Examination

(1 + 2y)y = x. ( x. The right-hand side is a standard integral, so in the end we have the implicit solution. y(x) + y 2 (x) = x2 2 +C.

Lesson 8: Slowing Down Spectra, p, Fermi Age

Diffusion on the half-line. The Dirichlet problem

22.54 Neutron Interactions and Applications (Spring 2004) Chapter 7 (2/26/04) Neutron Elastic Scattering - Thermal Motion and Chemical Binding Effects

p(θ,φ,θ,φ) = we have: Thus:

4 Power Series Solutions: Frobenius Method

TERRESTRIAL REDSHIFTS FROM A DIFFUSE LIGHT SOURCE. FOM Institute for Atomic and Molecular Physics, Kruislaan 407, 1098 SJ Amsterdam, The Netherlands

MCRT: L4 A Monte Carlo Scattering Code

PhD Qualifying Exam Nuclear Engineering Program. Part 1 Core Courses

The infinite square well in a reformulation of quantum mechanics without potential function

ULTRAFAST LASER PULSE TRAIN RADIATION TRANSFER IN A SCATTERING-ABSORBING 3D MEDIUM WITH AN INHOMOGENEITY

Problem 1: A 3-D Spherical Well(10 Points)

An Angular Multigrid Acceleration Method for S n Equations with Highly Forward-Peaked Scattering

Moveout approximation for P waves in a homogeneous VTI medium

Serco Assurance. Resonance Theory and Transport Theory in WIMSD J L Hutton

PHYS 404 Lecture 1: Legendre Functions

Notes: Most of the material presented in this chapter is taken from Jackson, Chap. 2, 3, and 4, and Di Bartolo, Chap. 2. 2π nx i a. ( ) = G n.

PHYS 771, Quantum Mechanics, Final Exam, Fall 2011 Instructor: Dr. A. G. Petukhov. Solutions

Hydrodynamic Limit with Geometric Correction in Kinetic Equations

To remain at 0 K heat absorbed by the medium must be removed in the amount of. dq dx = q(l) q(0) dx. Q = [1 2E 3 (τ L )] σ(t T 4 2 ).

Sound Propagation through Media. Nachiketa Tiwari Indian Institute of Technology Kanpur

In this method, one defines

Trefftz type method for 2D problems of electromagnetic scattering from inhomogeneous bodies.

Polarized light propagation and scattering in random media

CHAPTER-III GENERAL GROUP THEORETIC TRANSFORMATIONS FROM BOUNDARY VALUE TO INITIAL VALUE PROBLEMS

Donoghue, Golowich, Holstein Chapter 4, 6

6. LIGHT SCATTERING 6.1 The first Born approximation

SIMPLE RADIATIVE TRANSFER

Adaptability analysis of radiative transport diffusion approximation in planar-graded-index media

Comparative Analysis of Techniques for Source Radiation in Cylindrical EBG with and without Periodic Discontinuities

On the Analytical Solution of the S N. Radiative Transport Equation in a Slab for a Spacedependent. Journal of Physics: Conference Series

Analytic Assessment of Eigenvalues of the Neutron Transport Equation *

A successive order of scattering model for solving vector radiative transfer in the atmosphere

Preface to the Second Edition. Preface to the First Edition

Notes for Lecture 10

SOLVING TRANSIENT CONDUCTION AND RADIATION USING FINITE VOLUME METHOD

Before you begin read these instructions carefully:

UNIVERSITY OF MANITOBA

2. NOTES ON RADIATIVE TRANSFER The specific intensity I ν

Higher-order ordinary differential equations

8: Source-Sink Problems in 1 Energy Group

Optical Imaging Chapter 5 Light Scattering

Exact Solutions of the Einstein Equations

2 The Radiative Transfer Equation

An eigenvalue method using multiple frequency data for inverse scattering problems

An asymptotic preserving unified gas kinetic scheme for the grey radiative transfer equations

Beer-Lambert (cont.)

J10M.1 - Rod on a Rail (M93M.2)

H 2 -matrices with adaptive bases

Chemistry 432 Problem Set 4 Spring 2018 Solutions

Finite-dimensional spaces. C n is the space of n-tuples x = (x 1,..., x n ) of complex numbers. It is a Hilbert space with the inner product

Complex Potential Functions and Integro-Differential Equation in Elastic Media Problem in Presence of Heat

This ODE arises in many physical systems that we shall investigate. + ( + 1)u = 0. (λ + s)x λ + s + ( + 1) a λ. (s + 1)(s + 2) a 0

Notes on Quantum Mechanics

DIFFERENCE EQUATIONS

KEYWORDS: chord length sampling, random media, radiation transport, Monte Carlo method, chord length probability distribution function.

An Approximate Solution for Volterra Integral Equations of the Second Kind in Space with Weight Function

Columbia University Department of Physics QUALIFYING EXAMINATION

Two special equations: Bessel s and Legendre s equations. p Fourier-Bessel and Fourier-Legendre series. p

Rocking behaviour of a rigid foundation with an arbitrary embedment

Sheet 06.6: Curvilinear Integration, Matrices I

Scattering in one dimension

Quantum Physics III (8.06) Spring 2007 FINAL EXAMINATION Monday May 21, 9:00 am You have 3 hours.

Fundamental Stellar Parameters

On the Chord Length Sampling in 1-D Binary Stochastic Media

A far-field based T-matrix method for three dimensional acoustic scattering

arxiv: v2 [astro-ph.im] 9 Jan 2017

A Summary of the Black Hole Perturbation Theory. Steven Hochman

OPAC102. The Acoustic Wave Equation

Detectors in Nuclear Physics: Monte Carlo Methods. Dr. Andrea Mairani. Lectures I-II

You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.

Key words : Radiative transfer, Singular integral equation, Riemann Hilbert Problem, Cauchy Integral formulae.

5 Measure theory II. (or. lim. Prove the proposition. 5. For fixed F A and φ M define the restriction of φ on F by writing.

MATHEMATICAL TRIPOS PAPER 67 COSMOLOGY

Power Series Solutions to the Legendre Equation

Supplementary Figure S1: Numerical PSD simulation. Example numerical simulation of the power spectral density, S(f) from a trapped particle

Introduction to Finite Element Method

Transcription:

IOSR Journal of Applied Physics (IOSR-JAP e-iss: 78-486. Volume 4, Issue (May. - Jun., PP 7-46 A high accuracy approximation for half - space problems with anisotropic scattering M. H. Haggag,, H.F. Mashali (Physics Department, Faculty of Science, Taif University, Taif, K.S.A (Physics Department, Faculty of Science, Mansoura University, Mansoura, Egypt Abstract: An approximate model, which is developed previously, is extended to solve the half space problems in the case of extremely anisotropic scattering kernels. The scattering kernel is assumed to be a combination of isotropic plus a forward and backward leak. The transport equation is transformed into an equivalent fictitious one involving only multiple isotropic scattering, therefore permitting the application of the previously developed method for treating isotropic scattering. It has been shown that the method solves the albedo half space problem in a concise manner and leads to fast converging numerical results as shown in the Tables. For pure scattering and weakly absorbing medium the computations can be performed by hand with a pocket calculator Keywords: Approximate model, Anisotropic scattering, Albedo problem, Half space, Transport equation. I. Introduction The study of radiative transfer and neutron transport in anisotropically scattering media usually develops the scattering function in a series of Legendre polynomials of the scattering angle [-6]. This is useful when the number of terms needed is small. However, when the scattering has some strongly preferred directions, many terms must be used and the method will be cumbersome. There are some scattering functions for which a development in Legendre polynomials is not useful because of much stronger anisotropy may exist. In an attempt to overcome this problem, that is to estimate the effect of strong anisotropy on the solution,a special scattering model which combines backward scattering and forward scattering with an admixture of isotropic scattering in arbitrary proportions has been used [7-]. In the case of purely forward scattering with an isotropic component there are no particular problems in casting the transport equation into a suitable form for solution. On the other hand, when backward scattering is included it is necessary to employ a transformation due to Inönü [4] to cast the equation into an equation corresponding to isotropic scattering. In the present paper, we wish to generalize a method previously introduce to demonstrate the computational merits of it when applied to the anisotropic scattering model mentioned above. This method is a semi analytical approximate method and suitable for obtaining highly accurate solutions for one dimensional transport problems. It was introduced by the authors [5-7] for treating multiple isotropic scattering in a homogeneous half space problems. ow let us consider the one speed, time independent, homogeneous transport equation, μ Z ψ z, μ + ψ z, μ = ω Π(μ μ ψ z, μ dμ ( where ψ is the angular intensity, with distance measured in units of mean free path, is the direction cosine of the angle between the positive z-axis and the particle velocity vector, and ω is the single scattering albedo (mean number of secondaries per collision. Here Π μ μ represents the scattering law, which we write as [7], Π μ, μ = l δ μ + μ + m δ μ μ + n, ( where l and m are real constants in the range l, m and give the fraction of particles which emerge from a collision in the backward and forward directions, respectively. In addition n = l m and gives the fraction of particles which emerge isotropically from a collision. Using equation ( we can write equation ( as μ ψ z, μ + m ω z o ψ z, μ = l ω oψ z, μ + n ω o ψ( z, μ dμ ( In terms of the reduced optical variable 7 Page

equation ( becomes τ = m ω o z (4 μ τ ψ τ, μ + ψ τ, μ = α ψ τ, μ + β ψ(τ, μ dμ (5 where we have defined and α = β = l ω mω o (6 n ω m ω (7 Equation (5 shows that forward scattering(l = does not introduce any analytical complications since the form of the resulting equation of transport is no different from the usual equation. However, backward scattering does lead to an equation of transport considerably different from the usual one, because the arguments ±μ cause it to be a type of integro-differential functional equation. II. The Albedo Problem The albedo problem is the problem of obtaining the angular radiation or neutron density everywhere in a source free half space, if a prescribed angular density is incident on the surface z=o. ow, we seek solutions of the homogeneous transport equation (5 subject to the boundary conditions and that ψ((,μ = F(μ, μ >, (8a ψ τ, μ as τ (8b As discussed by Inönü, [4], we can transform the problem for anisotropic scattering into an equivalent equation with an isotropic scattering. Equation (5 reduces to the form valid for the purely isotropic scattering, where μ x Ψ(x, μ +Ψ(x, μ = ω Ψ(x, μ dμ (9 x = τ α ( ω = β (- α ( The relation between angular distributions of (5 and (9 is ψ τ, μ = B+ Ψ(x, μ B Ψ(x, - μ, ( with B = α + α ( ow, if Ψ(x, μ is a solution of (9 with ω given by (, then ψ(τ,μ, represented by ( and (, is a solution of (5. It is now straightforward to translate the boundary condition (8a on ψ into equivalent one on Ψ. We find Ψ, μ = R F μ + R Ψ, μ μ >, (4 8 Page

where R is given by A high accuracy approximation for half - space problems with anisotropic scattering R = B B + (5 The problem is thus reduced to that of solving (9 complemented by the reflected boundary condition (4 at the surface x = of the half space.. III. Analysis And Method Of Solution We now want to transform (9 into an integral equation suitable for numerical treatment. For this purpose we introduce the total intensity Φ(x associated with the angular intensity Ψ(x,μ appearing in (9. ow equations (9 and (4 can be formally solved, μ >, to give, Ψ(x, μ = ω μ x e μ (x-y Φ(y dy + ωr μ e μ (x+y Φ(ydy +(-RF(μe x μ, (6 Ψ(x, - μ = ω μ e μ (y-x x Φ(y dy (7 Integrating (6 and (7 over to obtain the following inhomogeneous integral equation forφ(x, Φ (x = S(x + ω H x,y Φ y dy, (8 where H(x, y = E x-y + RE x +y, (9 with E n (x denoting the exponential integral function of order n and the inhomogeneous term S(x is defined by S (x = - R F μ e x μ dμ ( To completely determine the angular distributions Ψ(x, ± μ as given by (6 and (7 we must solve Eq. (8 for. Φ(x. For this purpose, we introduce an appropriate parameters approximate expansion for Φ(x. A proper choice for the parametric expansion is of the form Φ(x = Ae x ν + n= A n E n+ (x (a where A and A n are expansion coefficients to be determined and is the positive root of ω ν = (b Using this expansion, the angular distributions everywhere in the half space can be calculated from,, + ν ln ν Ψ(x, μ = ( R F(μ e - x μ + ω ν e - x ν - e - x μ ν μ e- x μ +R ν + μ A + and ω Ψ(x, - μ = ω ν μ e- x μ n= η n+ (x, μ + R η n+ (, - μ A n ( e x ν A + ω e x μ η ν + μ μ n= n+ (, - μ - η n+ (x, - μ A n ( The functions η n (x, μ are defined by η n (x, μ = e y μ x E n (y dy (4 9 Page

and explicitly are given in [5 ]. Specializing ( to get Ψ(, - μ = ων A + ω ν+ μ μ n = η n + (,- μ A n (5 and from (, the exit angular distribution of the original problem is given by ψ, μ = R F μ + ( + RΨ(, μ (6 Another quantity, of physical interest, relevant to the present problem is the albedo which is defined by A = μ ψ, μ dμ μ F μ dμ (7 Using Eq. (6, A* can be expressed as A = R + ω + R μ F μ dμ ν ν ln + ν A + J,n+ n= A n, (8 where the constants J n, m are integrals evaluated to yield n + m J n,m = n+ n ln + k= k k + m+ m ln + k= k k (9 IV. The EXPASIO COEFFICIETS To solve for the expansion coefficients A and A n we substitute (a into (8 then, operate on the resulting expression first with the operator e x ν. dx, and second with E m+ (x.dx where m =,,,,. As a result of these two operations, one obtains (+ linear algebraic equations for (+ unknown expansion coefficients which are written as and T(ν A + T m (ν A + n = T n (ν A n = d (ν ( n = D mn A n = d m ( The analytic expressions for the integrals of T, T n and D mn are evaluated to give T ν = ν ν+ ω ν + R ln + ωr ν, < ω < ν ν+ R, ω = ( 4 T n ν = ω ν + Y n+ ν + RY n+ ν + R J,n+, < ω < R J,n+, ω = ( and for D mn one has D mn = J m+,n+ ω H m+,n+ (4 where - m + n H mn = R + - n+ + - m+ ψ m+n+ + R + m+ R + n+ m i= i i ψ m+n+ i + R n i= m i= n j= i+j ij ( m+ n i j i ψ i m+n+ i + (5 4 Page

In ( the numerical values of A high accuracy approximation for half - space problems with anisotropic scattering Y m can be calculated from the recurrence relation ± ± Y m ν = ±ν Y m ν + ln ± J m ν,m, m > (6 Where the values corresponding to m = are given by : Y + = ν ln + ν + π + n ν n ν+ n n=, (7 and Y ν = Y + ν ν ln + ν ln ν. (8 In equation (5 J and ~ are calculated from the expression n,n - n starting from ψ n+ = n ln + ln + - n n n - n n- ln + n- - i i= - n- i n ψ n, n > (9 ψ = ln + π 6 (4 V. umerical Results In order to illustrate the application of the foregoing analysis, we considered two cases for F(μ.In the first case F μ = whereas in the second F μ = δ(μ μ. In these two cases the expressions of d(ν and d m, in the algebraic system of equations, respectively, are, ν R ν ln( + ν, ω <, d ν = R, ω = (4 d m = R J,m+, (4 and d ν = ( R ν μ ν+ μ (4 d m ν = R η m+ (, μ (44 The physical aspects of the above cases are well documented, so we focus our attention on showing the convergence of the proposed method of solution. Some quantities of physical interest are calculated and compared with other available results. The calculated quantities are the medium albedo and the exist angular intensity at x = for various combinations of scattering parameters l, m, n and single scattering albedo ω. We keep all parameters except one fixed, and let the free parameter vary among the set of values under consideration. The numerical results are presented in Tables 6. From these calculations, it is seen that the numerical results obtained by the present method are converging even in the lowest order approximations. As physically expected, we observe a decrease of A * if the strength of the forward scattering increases. Similarly the value of A * increases if we enlarge the backward scattering part. 4 Page

Table.The albedo A * as calculated using the zero order approximation when F(µ = (a present results (b results of [7] (,m,n a =.5 b a =.7 b a =.9 b (,,.5985.5985.8.8.958.958 (,, (,,.7.4654.7.4654.9946.5656.9945.5656.498.478.498.478.5988.895.94 =.95.5988.895.94.78.9.5794 =.99.78.9.5795.48759.5966.5777 =.999.48759.5966.5777 (,,.4896.4896.6779.678.8864.8864 (,, (,,.59.59666.59.59667.7557.79456.7558.79456.9466.997.9466.997.6.6447.6744.6.6447.6744.79558.8959.875.79558.8959.875.9978.9884.9454.9978.9884.9454 Table. Shows the convergence of A * for various values of when F(µ = (,m,n =.5 4 6 [7] (,,.59846.59848.59848.59848.59848.5985.7.75.75.75.75.7 (,, (,,.46544.46544.46544.46544.46544.4654.5988.59884.59884.59884.59884.5988.8954.8956.8956.8956.8956.895.97.99.99.99.99.94 =.7 (,,.8.84.84.84.84.8 4 Page

.99459.99459.99459.99459.99459.9945 (,, (,,.56555.56557.56557.56557.56557.5656.787.788.788.788.788.78 (,,.87.87.87.87.87.9.57944.57945.57946.57946.57946.5795 =.9 (,, (,,.9577.9579.9579.9579.9579.958.498.4986.4986.4986.498.498 (,,.4789.4784.4784.4784.4784.478 (,,.48759.48759.48759.48759.48759.48759.59658.5966.5966.5966.5966.5966.5777.57774.57774.57774.57774.5777 Table. The albedo A * as calculated using the zero order approximation when F(µ = (. (a present results (b results of [].5. 7. 9 (,m,n a b a b a b (,,.4578.4578.9454.945.496.499,, (.87.86.594.594.4878.488 (,,.54.5.866.868.4489.4495.465.466.446,,,, (.6594.78.478 (.554.74.5975,,.95. 99. 999 (.575.5756.677.6778.85476,, (.478.4987.777.7777.8945 (,,.5548.5554.7569.7584.98,,,,,, (.55.756.9977 (.584.7795.94 (.688.79959.988 4 Page

(, m, n (,, A high accuracy approximation for half - space problems with anisotropic scattering Table 4. Shows the convergence of A * for various value of when F( (.5 4 6 [].45779.45776.45776.45776.45776.45776.867.858.857.857.857.858 (,, (,,.56.57.56.56.56.56.465.4648.4648.4648.4648.6594.659.659.659.659.559.555.555.555.555.7.9458.9458.9457.9457.9457.9457 (,,.594.5944.594.594.594.594 (,, (,,.866.868.868.868.868.868 (,,.466.465.4649.4649.4649.78.78.78.78.78.76.75.75.75.75.9.496.499.499.499.499.499,.48777.488.488.488.488.488, (,,.44887.44948.44947.44947.44947.44948 (, (, (, m, n.4457.4479.4479.4479.4479.4789.4785.4785.4785.4785.59747.59764.59764.59764.59764 Table 5. Shows the convergence of (, at 9 µ 4 6..97786.9796.979.979.979 (,,.4.46.476.475.476.476.6.77.8.8.9.9.8.6869.686.686.686.686 44 Page

..496.499.499.499.499..5974.55.54.5.5 (,,.4.46896.468.468.468.468.6.4676.46677.46678.46678.46678.8.7958.79569.79569.79569.79569..48777.488.488.488.488..5786.57897.57896.57895.57895.4.588.5865.5865.5865.5865.6.4988.49847.49847.49847.49847.8.46789.4678.4678.4678.4678..4457.4479.4479.4479.4479..66.69.69.69.69.4.5786.5787.5787.5787.5787.6.5647.55979.55979.55979.55585.8.5.5.5.5.5..4789.4785.4785.47854.47854..65884.65956.65956.65955.65955.4.686.68577.68577.68577.68577.6.57568.57565.57565.57565.57565.8.5596.559.559.559.559..59747.59764.59764.59764.59764 (, m, n (, (,,, Table 6. Shows the convergence of (, at. 9 when F( (. µ 4 6..5668.569.569796.5697.5697.4.55748.548.5445.5444.5444.6.4984.49688.49676.496766.496766.8.46789.4659.4655.46547.46547..48975.4977.4979.49796.49796..7746.775664.7767.77686.77688.4.7559.7496.749566.749586.749585.6.79.766.796.7.7.8.67679.6744.6746.674.674..64845.65949.664.669.669..497888.5854.56.554.554.4.474896.47489.4767.4764.4764.6.444.44.4448.445.445.8.44.4848.488.484.484..68577.699.694.6946.6946 45 Page

..779.75.7589.7575.7576.4.68764.685766.685965.685984.685984.6.65584.6549.6546.65465.65465.8.698.6.69965.6996.6996..79.7466.749.744.744..456.458.4546.4588.4588.4.496.4958.497.4976.4976.6.496.489.484.4845.4845.8.5746.74864.7486.748.748..6865.689.688.684.684 VI. Conclusion An efficient analytical approximation is proposed to solve radiative heat transfer or neutron transport problems in plane parallel semi infinite media in the presence of multiple synthetic scattering kernel.the unknown function in the integral form of the transport equation is approximated by a set of trial functions with unknown coefficients. The unknown coefficients in the trial functions are found to be solutions for a system of linear, inhomogeneous algebraic equations. These coefficients are used to get numerical results of the exit distribution ψ(, μ and the albedo A * from equations (6 and (8. As is seen from the Tables, the present numerical results are found to be in excellent agreement with the available results reported in the literatures References [] K. M. Case and P. F. Zweifel, Linear transport theory (Addison Wesley, Reading, Mass., 967 [] S. Chandrasekhar, Radiative transfer (Dover, ew York, 96 [] G. I. Bell and S. Glasstone, uclear reactor theory (ew York: Van ostrand Reinhold, 97 [4] G. Spiga, F. Santarelli and C. Stramigioli, Radiative transfer in an absorbing and anisotropically scattering slab with a reflecting boundary, Int. J. Heat Mass Transfer, (6, 98, 84-85. [5] M. Benassi, R. M. Cotta and C. E. Siewert, The P method for radiative transfer problems with reflective boundary conditions, J. Quant. Spectrosc. Radiat. Transfer, (6, 98, 547-55. [6] L. B. Barichello, R. D. M. Garcia and C. E. Siewert, A spherical harmonics solution for radiative transfer problems with reflecting boundaries and internal sources, J. Quant. Spectvosc. Radiat. Transfer, 6(, 998, 47-6. [7] C. E. Siewert and C. Devaus, The effect of forward and backward scattering on the law of darkening for the Milne problem and the spherical albedo, J. Quant. Spectrosc. Radiat. Transfer, (, 978, 9-. [8] A. Shafiq, H. E. De Meyer and C. C. Grosjean, The albedo problem in the case of multiple synthetic scattering taking place in a plane- symmetric slab I and II, Ann. ucl. Energy, (, 985, 9-4. [9] M. C. Gülecyüz and A. Kaskas, The Milne and the constant source problems for the FBIS kernel, Tr. J. of Phys., (6, 998, 46-468. [] C. Yildiz, Variation of the albedo and the transmission factor with forward and backward scattering in neutron transport theory the F method, Ann. ucl. Energy, 7(9,, 8-84. [] C. Tezcan, M. C. Gülecyüz, R. G. Türeci and A. Kaskas, The H method for half-space albedo and constant source problems for isotropic and anisotropic scattering kernels, J. Quant. Spectrosc. Radiat. Transfer, (, 7, 6-69. [] C. Tezcan and R. Sever, The critical slab with the backward-forward - isotropic scattering model, Ann. ucl. Energy, (, 985, 57-576. [] C. Tezcan, The Milne problem for isotropic scattering with a backward leak, Transport Theory & Stat. Phys., 6(&, 977, 9-99 [4] E.Inönü, A theorem on anisotropic scattering, Transport Theory & Stat. Phys. (&, 97, 7-46. [5] M. H. Haggag, A. R. Degheidy and A. El-Depsy, Efficient and accurate method for radiation transfer problems, J. Quant. Spectrosc. Radiate. Transfer 58(, 997, 9-. [6] H. F. Machali, A. R. Degheidy and A. El-Depsy, Accurate analytic approximation to Milne's problem for a scattering and absorbing medium, J. Quant. Spectrosc. Radiate. Transfer, 6(, 999, 5-5. [7] M. H. Haggag, A. El-Depsy and D. A. Gharbiea, Two-region Milne problem with anisotropic scattering,ann. ucl. Energy, 5(9, 8, 7-77. 46 Page

2024 © sciencedocbox.com Privacy Policy | Terms of Service | Feedback