Abstract

We study light propagation in tissues using the theory of radiative transport. In particular, we study the case in which there is both forward-peaked and large-angle scattering. Because this combination of the forward-peaked and large-angle scattering makes it difficult to solve the radiative transport equation, we present a method to construct approximations to study this problem. The delta–Eddington and Fokker–Planck approximations are special cases of this general framework. Using this approximation method, we derive two new approximations: the Fokker–Planck–Eddington approximation and the generalized Fokker–Planck–Eddington approximation. By computing the transmittance and reflectance of light by a slab we study the performance of these approximations.

© 2008 Optical Society of America

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References

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  1. A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1996).
  2. A. D. Kim and J. B. Keller, “Light propagation in biological tissue,” J. Opt. Soc. Am. A 20, 92-98 (2003).
    [CrossRef]
  3. J. H. Joseph, W. J. Wiscombe, and J. A. Wienman, “The delta-Eddington approximation for radiative flux transfer,” J. Atmos. Sci. 33, 2452-2459 (1976).
    [CrossRef]
  4. G. C. Pomraning, “The Fokker-Planck operator as an asymptotic limit,” Math. Models Meth. Appl. Sci. 2, 21-36 (1992).
  5. K. Przybylski and J. Ligou, “Numerical analysis of the Boltzmann equation including Fokker-Planck terms,” Nucl. Sci. Eng. 81, 92-109 (1982).
  6. M. Caro and J. Ligou, “Treatment of scattering anisotropy of neutrons through the Boltzmann-Fokker-Planck equation,” Nucl. Sci. Eng. 83, 242-250 (1983).
  7. L. Henyey and J. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70-83 (1941).
    [CrossRef]
  8. J. R. Mourant, J. P. Freyer, A. H. Hielscher, A. A. Eick, D. Shen, and T. M. Johnson, “Mechanisms of light scattering from biological cells relevant to noninvasive optical-tissue diagnostics,” Appl. Opt. 37, 3586-3593 (1998).
  9. L. O. Reynolds and N. J. McCormick, “Approximate two-parameter phase function for light scattering,” J. Opt. Soc. Am. 70, 1206-1212 (1980).
  10. R. Marchesini, A. Bertoni, S. Andreola, E. Melloni, and A. E. Sichirollo, “Extinction and absorption coefficients and scattering phase functions of human tissues in vitro,” Appl. Opt. 28, 2318-2324 (1989).
  11. W. M. Cornette and J. G. Shanks, “Physically reasonable analytical expression for the single-scattering phase function,” Appl. Opt. 31, 3152-3160 (1992).
  12. D. Toublanc, “Henyey-Greenstein and Mie phase functions in Monte Carlo radiative transfer computations,” Appl. Opt. 35, 3270-3274 (1996).
  13. A. Kienle, F. K. Forster, and R. Hibst, “Influence of the phase function on determination of the optical properties of biological tissue by spatially resolved reflectance,” Opt. Lett. 26, 1571-1573 (2001).
    [CrossRef]
  14. S. K. Sharma and S. Banerjee, “Role of approximate phase functions in Monte Carlo simulation of light propagation in tissues,” J. Opt. A 5, 294-302 (2003).
    [CrossRef]
  15. E. W. Larsen, “The linear Boltzmann equation in optically thick systems with forward-peaked scattering,” Prog. Nucl. Energy 34, 413-423 (1999).
    [CrossRef]
  16. A. D. Kim and M. Moscoso, “Beam propagation in sharply peaked forward scattering media,” J. Opt. Soc. Am. A 21, 797-803 (2004).
    [CrossRef]
  17. G. C. Pomraning, “Higher order Fokker-Planck operators,” Nucl. Sci. Eng. 124, 390-397 (1996).
  18. A. K. Prinja and G. C. Pomraning, “A generalized Fokker-Planck model for transport of collimated beams,” Nucl. Sci. Eng. 137, 227-235 (2001).
  19. C. L. Leakeas and E. W. Larsen, “Generalized Fokker-Planck approximations of particle transport with highly forward-peaked scattering,” Nucl. Sci. Eng. 137, 236-250 (2001).
  20. E. W. Larsen and L. Liang, “The atomic mix approximation for charged particle transport,” SIAM J. Appl. Math. 68, 43-58 (2007).
    [CrossRef]
  21. R. Sanchez and N. J. McCormick, “Solutions to inverse problems for the Boltzmann-Fokker-Planck equation,” Transp. Theory Stat. Phys. 12, 129-155 (1983).
    [CrossRef]
  22. J. E. Morel, “An improved Fokker-Planck angular differencing scheme,” Nucl. Sci. Eng. 89, 131-136 (1985).

2007 (1)

E. W. Larsen and L. Liang, “The atomic mix approximation for charged particle transport,” SIAM J. Appl. Math. 68, 43-58 (2007).
[CrossRef]

2004 (1)

2003 (2)

S. K. Sharma and S. Banerjee, “Role of approximate phase functions in Monte Carlo simulation of light propagation in tissues,” J. Opt. A 5, 294-302 (2003).
[CrossRef]

A. D. Kim and J. B. Keller, “Light propagation in biological tissue,” J. Opt. Soc. Am. A 20, 92-98 (2003).
[CrossRef]

2001 (3)

A. Kienle, F. K. Forster, and R. Hibst, “Influence of the phase function on determination of the optical properties of biological tissue by spatially resolved reflectance,” Opt. Lett. 26, 1571-1573 (2001).
[CrossRef]

A. K. Prinja and G. C. Pomraning, “A generalized Fokker-Planck model for transport of collimated beams,” Nucl. Sci. Eng. 137, 227-235 (2001).

C. L. Leakeas and E. W. Larsen, “Generalized Fokker-Planck approximations of particle transport with highly forward-peaked scattering,” Nucl. Sci. Eng. 137, 236-250 (2001).

1999 (1)

E. W. Larsen, “The linear Boltzmann equation in optically thick systems with forward-peaked scattering,” Prog. Nucl. Energy 34, 413-423 (1999).
[CrossRef]

1998 (1)

1996 (2)

G. C. Pomraning, “Higher order Fokker-Planck operators,” Nucl. Sci. Eng. 124, 390-397 (1996).

D. Toublanc, “Henyey-Greenstein and Mie phase functions in Monte Carlo radiative transfer computations,” Appl. Opt. 35, 3270-3274 (1996).

1992 (2)

W. M. Cornette and J. G. Shanks, “Physically reasonable analytical expression for the single-scattering phase function,” Appl. Opt. 31, 3152-3160 (1992).

G. C. Pomraning, “The Fokker-Planck operator as an asymptotic limit,” Math. Models Meth. Appl. Sci. 2, 21-36 (1992).

1989 (1)

1985 (1)

J. E. Morel, “An improved Fokker-Planck angular differencing scheme,” Nucl. Sci. Eng. 89, 131-136 (1985).

1983 (2)

R. Sanchez and N. J. McCormick, “Solutions to inverse problems for the Boltzmann-Fokker-Planck equation,” Transp. Theory Stat. Phys. 12, 129-155 (1983).
[CrossRef]

M. Caro and J. Ligou, “Treatment of scattering anisotropy of neutrons through the Boltzmann-Fokker-Planck equation,” Nucl. Sci. Eng. 83, 242-250 (1983).

1982 (1)

K. Przybylski and J. Ligou, “Numerical analysis of the Boltzmann equation including Fokker-Planck terms,” Nucl. Sci. Eng. 81, 92-109 (1982).

1980 (1)

1976 (1)

J. H. Joseph, W. J. Wiscombe, and J. A. Wienman, “The delta-Eddington approximation for radiative flux transfer,” J. Atmos. Sci. 33, 2452-2459 (1976).
[CrossRef]

1941 (1)

L. Henyey and J. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70-83 (1941).
[CrossRef]

Andreola, S.

Banerjee, S.

S. K. Sharma and S. Banerjee, “Role of approximate phase functions in Monte Carlo simulation of light propagation in tissues,” J. Opt. A 5, 294-302 (2003).
[CrossRef]

Bertoni, A.

Caro, M.

M. Caro and J. Ligou, “Treatment of scattering anisotropy of neutrons through the Boltzmann-Fokker-Planck equation,” Nucl. Sci. Eng. 83, 242-250 (1983).

Cornette, W. M.

Eick, A. A.

Forster, F. K.

Freyer, J. P.

Greenstein, J.

L. Henyey and J. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70-83 (1941).
[CrossRef]

Henyey, L.

L. Henyey and J. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70-83 (1941).
[CrossRef]

Hibst, R.

Hielscher, A. H.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1996).

Johnson, T. M.

Joseph, J. H.

J. H. Joseph, W. J. Wiscombe, and J. A. Wienman, “The delta-Eddington approximation for radiative flux transfer,” J. Atmos. Sci. 33, 2452-2459 (1976).
[CrossRef]

Keller, J. B.

Kienle, A.

Kim, A. D.

Larsen, E. W.

E. W. Larsen and L. Liang, “The atomic mix approximation for charged particle transport,” SIAM J. Appl. Math. 68, 43-58 (2007).
[CrossRef]

C. L. Leakeas and E. W. Larsen, “Generalized Fokker-Planck approximations of particle transport with highly forward-peaked scattering,” Nucl. Sci. Eng. 137, 236-250 (2001).

E. W. Larsen, “The linear Boltzmann equation in optically thick systems with forward-peaked scattering,” Prog. Nucl. Energy 34, 413-423 (1999).
[CrossRef]

Leakeas, C. L.

C. L. Leakeas and E. W. Larsen, “Generalized Fokker-Planck approximations of particle transport with highly forward-peaked scattering,” Nucl. Sci. Eng. 137, 236-250 (2001).

Liang, L.

E. W. Larsen and L. Liang, “The atomic mix approximation for charged particle transport,” SIAM J. Appl. Math. 68, 43-58 (2007).
[CrossRef]

Ligou, J.

M. Caro and J. Ligou, “Treatment of scattering anisotropy of neutrons through the Boltzmann-Fokker-Planck equation,” Nucl. Sci. Eng. 83, 242-250 (1983).

K. Przybylski and J. Ligou, “Numerical analysis of the Boltzmann equation including Fokker-Planck terms,” Nucl. Sci. Eng. 81, 92-109 (1982).

Marchesini, R.

McCormick, N. J.

R. Sanchez and N. J. McCormick, “Solutions to inverse problems for the Boltzmann-Fokker-Planck equation,” Transp. Theory Stat. Phys. 12, 129-155 (1983).
[CrossRef]

L. O. Reynolds and N. J. McCormick, “Approximate two-parameter phase function for light scattering,” J. Opt. Soc. Am. 70, 1206-1212 (1980).

Melloni, E.

Morel, J. E.

J. E. Morel, “An improved Fokker-Planck angular differencing scheme,” Nucl. Sci. Eng. 89, 131-136 (1985).

Moscoso, M.

Mourant, J. R.

Pomraning, G. C.

A. K. Prinja and G. C. Pomraning, “A generalized Fokker-Planck model for transport of collimated beams,” Nucl. Sci. Eng. 137, 227-235 (2001).

G. C. Pomraning, “Higher order Fokker-Planck operators,” Nucl. Sci. Eng. 124, 390-397 (1996).

G. C. Pomraning, “The Fokker-Planck operator as an asymptotic limit,” Math. Models Meth. Appl. Sci. 2, 21-36 (1992).

Prinja, A. K.

A. K. Prinja and G. C. Pomraning, “A generalized Fokker-Planck model for transport of collimated beams,” Nucl. Sci. Eng. 137, 227-235 (2001).

Przybylski, K.

K. Przybylski and J. Ligou, “Numerical analysis of the Boltzmann equation including Fokker-Planck terms,” Nucl. Sci. Eng. 81, 92-109 (1982).

Reynolds, L. O.

Sanchez, R.

R. Sanchez and N. J. McCormick, “Solutions to inverse problems for the Boltzmann-Fokker-Planck equation,” Transp. Theory Stat. Phys. 12, 129-155 (1983).
[CrossRef]

Shanks, J. G.

Sharma, S. K.

S. K. Sharma and S. Banerjee, “Role of approximate phase functions in Monte Carlo simulation of light propagation in tissues,” J. Opt. A 5, 294-302 (2003).
[CrossRef]

Shen, D.

Sichirollo, A. E.

Toublanc, D.

Wienman, J. A.

J. H. Joseph, W. J. Wiscombe, and J. A. Wienman, “The delta-Eddington approximation for radiative flux transfer,” J. Atmos. Sci. 33, 2452-2459 (1976).
[CrossRef]

Wiscombe, W. J.

J. H. Joseph, W. J. Wiscombe, and J. A. Wienman, “The delta-Eddington approximation for radiative flux transfer,” J. Atmos. Sci. 33, 2452-2459 (1976).
[CrossRef]

Appl. Opt. (4)

Astrophys. J. (1)

L. Henyey and J. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70-83 (1941).
[CrossRef]

J. Atmos. Sci. (1)

J. H. Joseph, W. J. Wiscombe, and J. A. Wienman, “The delta-Eddington approximation for radiative flux transfer,” J. Atmos. Sci. 33, 2452-2459 (1976).
[CrossRef]

J. Opt. A (1)

S. K. Sharma and S. Banerjee, “Role of approximate phase functions in Monte Carlo simulation of light propagation in tissues,” J. Opt. A 5, 294-302 (2003).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Math. Models Meth. Appl. Sci. (1)

G. C. Pomraning, “The Fokker-Planck operator as an asymptotic limit,” Math. Models Meth. Appl. Sci. 2, 21-36 (1992).

Nucl. Sci. Eng. (6)

K. Przybylski and J. Ligou, “Numerical analysis of the Boltzmann equation including Fokker-Planck terms,” Nucl. Sci. Eng. 81, 92-109 (1982).

M. Caro and J. Ligou, “Treatment of scattering anisotropy of neutrons through the Boltzmann-Fokker-Planck equation,” Nucl. Sci. Eng. 83, 242-250 (1983).

G. C. Pomraning, “Higher order Fokker-Planck operators,” Nucl. Sci. Eng. 124, 390-397 (1996).

A. K. Prinja and G. C. Pomraning, “A generalized Fokker-Planck model for transport of collimated beams,” Nucl. Sci. Eng. 137, 227-235 (2001).

C. L. Leakeas and E. W. Larsen, “Generalized Fokker-Planck approximations of particle transport with highly forward-peaked scattering,” Nucl. Sci. Eng. 137, 236-250 (2001).

J. E. Morel, “An improved Fokker-Planck angular differencing scheme,” Nucl. Sci. Eng. 89, 131-136 (1985).

Opt. Lett. (1)

Prog. Nucl. Energy (1)

E. W. Larsen, “The linear Boltzmann equation in optically thick systems with forward-peaked scattering,” Prog. Nucl. Energy 34, 413-423 (1999).
[CrossRef]

SIAM J. Appl. Math. (1)

E. W. Larsen and L. Liang, “The atomic mix approximation for charged particle transport,” SIAM J. Appl. Math. 68, 43-58 (2007).
[CrossRef]

Transp. Theory Stat. Phys. (1)

R. Sanchez and N. J. McCormick, “Solutions to inverse problems for the Boltzmann-Fokker-Planck equation,” Transp. Theory Stat. Phys. 12, 129-155 (1983).
[CrossRef]

Other (1)

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1996).

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Figures (4)

Fig. 1
Fig. 1

Transmittance due to a plane wave incident normally on a slab of thickness 2 mm with μ a = 0.01 mm 1 and μ s = 50.0 mm 1 . Scattering is governed by the Henyey–Greenstein scattering phase function with g = 0.98 . In both (a) and (b) the solid curve corresponds to the solution of the transport equation with the Henyey–Greenstein scattering phase function (HG). In (a) the circles correspond to the delta–Eddington approximation ( δ E ), the triangles correspond to the Fokker–Planck approximation (FP), and the squares correspond to the generalized Fokker–Planck approximation (gFP). In (b) the circles correspond to the Fokker–Planck–Eddington approximation (FPE) and the pluses correspond to the generalized Fokker–Planck–Eddington approximation (gFPE).

Fig. 2
Fig. 2

Reflectance for the same problem shown in Fig. 1.

Fig. 3
Fig. 3

Transmittance (plotted on a logarithmic scale) due to a plane wave incident normally on a slab of thickness 2 mm with μ a = 0.01 mm 1 and μ s = 3.1586 mm 1 . Scattering is governed by the double Henyey–Greenstein scattering phase function with g 1 = 0.85 , g 2 = 0.34 , and β = 0.86 . In both (a) and (b) the solid curve corresponds to the solution of the transport equation with the double Henyey–Greenstein scattering phase function (HG2). In (a) the circles correspond to the delta–Eddington approximation ( δ E ), the triangles correspond to the Fokker–Planck approximation (FP), and the squares correspond to the generalized Fokker–Planck approximation (gFP). In (b) the circles correspond to the Fokker–Planck–Eddington approximation (FPE) and the pluses correspond to the generalized Fokker–Planck–Eddington approximation (gFPE).

Fig. 4
Fig. 4

Reflectance for the same problem as for Fig. 3.

Equations (82)

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Ω · Ψ + μ a Ψ = μ s L Ψ .
L Ψ = Ψ + S 2 f ( Ω · Ω ) Ψ ( Ω , r ) d Ω .
Ψ ( Ω , r b ) = b ( Ω , r b ) , Ω · n ( r b ) < 0 , r b D .
f ( Ω · Ω ) = n = 0 2 n + 1 4 π f n P n ( Ω · Ω ) ,
f n = 2 π 1 1 f ( Ω · Ω ) P n ( Ω · Ω ) d ( Ω · Ω ) .
P n ( Ω · Ω ) = 4 π 2 n + 1 m = n n Y n m ( Ω ) Y n m ( Ω ) ¯ , n 0 ,
f ( Ω · Ω ) = n = 0 f n m = n n Y n m ( Ω ) Y n m ( Ω ) ¯ .
S 2 Y n m ( Ω ) Y n m ( Ω ) ¯ d Ω = δ n , n δ m , m .
L Y n m ( Ω ) = ( 1 f n ) Y n m ( Ω ) .
f HG ( Ω · Ω ; g ) = 1 4 π 1 g 2 ( 1 + g 2 2 g Ω · Ω ) 3 / 2 .
λ n HG = 1 g n , n = 0 , 1 , 2 , .
f HG 2 ( Ω · Ω ) = β f HG ( Ω · Ω ; g 1 ) + ( 1 β ) f HG ( Ω · Ω ; g 2 ) .
λ n HG 2 = 1 β g 1 n ( 1 β ) g 2 n , n = 0 , 1 , 2 , .
f ( Ω · Ω ) α δ ( Ω Ω ) + ( 1 α ) 4 π [ P 0 ( Ω · Ω ) + 3 g P 1 ( Ω · Ω ) ] .
S 2 δ ( Ω Ω ) u ( Ω ) d Ω = u ( Ω ) ,
λ n δ E = ( 1 α ) ( 1 δ n , 0 g δ n , 1 ) .
L Ψ L FP Ψ = 1 2 ( 1 f 1 ) Δ Ω Ψ .
f ( Ω · Ω ) a 0 δ ( Ω Ω ) + a 1 δ ( Ω Ω ) .
S 2 δ ( Ω Ω ) u ( Ω ) d Ω = Δ Ω u ( Ω ) .
λ n FP = 1 a 0 + a 1 n ( n + 1 ) .
f ( Ω · Ω ) a 0 δ ( Ω Ω ) + a 1 δ ( Ω Ω ) + a 2 δ ( i v ) ( Ω Ω ) + ,
L gFP Ψ = a 1 Δ Ω ( I a 2 Δ Ω ) 1 Ψ ,
λ n gFP = a 1 n ( n + 1 ) 1 + a 2 n ( n + 1 ) .
f ( Ω · Ω ) m = 0 M a m δ ( 2 m ) ( Ω Ω ) + n = 0 N 2 n + 1 4 π b n P n ( Ω · Ω ) .
f FPE ( Ω · Ω ) = a 0 δ ( Ω Ω ) + a 1 δ ( Ω Ω ) + 1 4 π [ b 0 P 0 ( Ω · Ω ) + 3 b 1 P 1 ( Ω · Ω ) ] .
L FPE Ψ = ( 1 a 0 ) Ψ + a 1 Δ Ω Ψ + 1 4 π S 2 [ b 0 P 0 ( Ω · Ω ) + 3 b 1 P 1 ( Ω · Ω ) ] Ψ ( Ω , r ) d Ω .
λ n FPE = ( 1 a 0 ) + a 1 n ( n + 1 ) b 0 δ n , 0 b 1 δ n , 1 .
a 0 + b 0 = 1 ,
a 0 2 a 1 + b 1 = f 1 ,
a 0 6 a 1 = f 2 ,
a 0 12 a 1 = f 3 .
a 0 = 2 f 2 f 3 ,
a 1 = ( f 2 f 3 ) / 6 ,
b 0 = 1 2 f 2 + f 3 ,
b 1 = f 1 5 f 2 / 3 + 2 f 3 / 3.
L gFPE Ψ = ( 1 a 0 ) Ψ + a 1 Δ Ω ( 1 a 2 Δ Ω ) 1 Ψ + 1 4 π S 2 [ b 0 P 0 ( Ω · Ω ) + 3 b 1 ( Ω · Ω ) ] Ψ ( Ω , r ) d Ω .
λ n gFPE = ( 1 a 0 ) a 1 n ( n + 1 ) 1 + a 2 n ( n + 1 ) b 0 δ n , 0 b 1 δ n , 1 .
a 0 + b 0 = 1 ,
a 0 2 a 1 1 + 2 a 2 + b 1 = f 1 ,
a 0 6 a 1 1 + 6 a 2 = f 2 ,
a 0 12 a 1 1 + 12 a 2 = f 3 ,
a 0 20 a 1 1 + 20 a 2 = f 4 ,
a 0 = 7 f 2 f 4 5 f 2 f 3 2 f 3 f 4 2 f 2 7 f 3 + 5 f 4 ,
a 1 = 7 6 f 2 2 f 3 f 2 2 f 4 f 2 f 3 2 + f 2 f 4 2 + f 3 2 f 4 f 3 f 4 2 ( 2 f 2 7 f 3 + 5 f 4 ) 2 ,
a 2 = 1 12 7 f 3 4 f 2 3 f 4 2 f 2 7 f 3 + 5 f 4 ,
b 0 = 5 f 2 f 3 7 f 2 f 4 + 2 f 3 f 4 + 2 f 2 7 f 3 + 5 f 4 2 f 2 7 f 3 + 5 f 4 ,
b 1 = 27 f 2 f 3 35 f 2 f 4 + 8 f 3 f 4 + 8 f 1 f 2 35 f 1 f 3 + 27 f 1 f 4 8 f 2 35 f 3 + 27 f 4 .
μ Ψ z + μ a Ψ = μ s L ˜ Ψ ,
L ˜ Ψ = Ψ + 1 1 h ( μ , μ ) Ψ ( μ , z ) d μ .
Ω · Ω = μ μ + 1 μ 2 1 μ 2 cos ( φ φ ) .
h ( μ , μ ) = 0 2 π f [ μ μ + 1 μ 2 1 μ 2 cos ( φ φ ) ] d ( φ φ ) .
Ψ ( μ , 0 ) = F δ ( μ 1 ) , 0 < μ 1.
Ψ ( μ , Z 0 ) = 0 , 1 μ < 0.
h ( μ , μ ) = 2 ( 1 g 2 ) π ( a b ) ( a + b ) 1 / 2 E ( 2 b a + b ) ,
L ˜ δ E Ψ = ( 1 α ) Ψ + 1 α 2 1 1 ( 1 + 3 g μ μ ) Ψ ( μ , z ) d μ .
L ˜ FP Ψ = 1 2 ( 1 f 1 ) L Ψ = 1 2 ( 1 f 1 ) μ [ ( 1 μ 2 ) Ψ μ ] .
L ˜ gFP Ψ = a 1 L ( I a 2 L ) 1 Ψ ,
a 1 = ( 1 f 1 ) ( 1 + a 2 ) / 2 , a 2 = 2 3 f 1 + f 2 6 ( f 2 f 3 ) .
L ˜ FPE Ψ = ( 1 a 0 ) Ψ + a 1 L Ψ + 1 2 1 1 ( b 0 + b 1 μ μ ) Ψ ( μ , z ) d μ ,
L gFPE Ψ = ( 1 a 0 ) Ψ + a 1 L ( I a 2 L ) 1 Ψ + 1 2 1 1 ( b 0 + b 1 μ μ ) Ψ ( μ , z ) d μ ,
R ( μ ) = F 1 Ψ ( μ , 0 ) , 1 μ < 0 ,
T ( μ ) = F 1 Ψ ( μ , Z 0 ) , 0 < μ 1 ,
1 1 f ( μ ) d μ j = 1 2 N f ( μ j ) w j ,
μ j Ψ j z + μ a Ψ j = μ s L ˜ 2 N Ψ j , j = 1 , , 2 N .
M Ψ z + μ a Ψ = μ s L ˜ 2 N Ψ .
λ M v = ( μ s L ˜ 2 N μ a I 2 N ) v ,
V j n = V ˜ 2 N j + 1 , n , j = 1 , , 2 N .
Ψ j ( z ) = n = 1 N [ a n e λ n ( z Z 0 ) V j n + b n e λ n z V 2 N j + 1 , n ] .
Ψ j ( 0 ) = n = 1 N [ a n e λ n Z 0 V j n + b n V 2 N j + 1 , n ] = F d j , j = N + 1 , , 2 N ,
Ψ j ( Z 0 ) = n = 1 N [ a n V j n + b n e λ n Z 0 V 2 N j + 1 , n ] = 0 , j = 1 , , N .
[ L ˜ 2 N ] i j = δ i j + h ( μ i , μ j ) w j ,
[ L ˜ 2 N ] i j = ( 1 g 2 ) δ i j + 1 g 2 2 [ 1 + 3 g 1 + g μ i μ j ] w j .
[ L 2 N ] i i = ( β i + 1 / 2 + β i 1 / 2 ) , i = 1 , , 2 N ,
[ L 2 N ] i , i + 1 = β i + 1 / 2 , i = 1 , , 2 N 1 ,
[ L 2 N ] i + 1 , i = β i 1 / 2 , i = 1 , , 2 N 1 ,
β i + 1 / 2 = γ i + 1 / 2 w i 1 μ i + 1 μ i , β i 1 / 2 = γ i 1 / 2 w i 1 μ i μ i 1 .
γ i + 1 / 2 = γ i 1 / 2 + 2 μ i w i , i = 1 , , 2 N .
[ L ˜ 2 N ] i j = 1 2 ( 1 f 1 ) [ L 2 N ] i j .
( I a 2 L 2 N ) A = I .
L ˜ 2 N = a 1 L 2 N A .
[ L ˜ 2 N ] i j = ( 1 a 0 ) δ i j + a 1 [ L 2 N ] i j + 1 2 [ b 0 + b 1 μ i μ j ] w j .
[ L ˜ 2 N ] i j = ( 1 a 0 ) δ i j + a 1 [ L 2 N A ] i j + 1 2 [ b 0 + b 1 μ i μ j ] w j ,

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