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A spline approximation of the Abel transformation for use in optically-thick, cylindrically-symmetric plasmas
Radiat. Transfer Vol. 54, No. 6, pp. 1055-1058, 1995
J. Quanf. Specfrosc.
Copyright 0 1995 Elsevier Science Ltd Printed in Great Britain. All rightsreserved
00224073(95)00119-0
0022-4073/95$9.50+ 0.00
NOTE A SPLINE APPROXIMATION OF THE ABEL TRANSFORMATION FOR USE IN OPTICALLY-THICK, CYLINDRICALLY-SYMMETRIC PLASMAST WANG YI,$ DING PEIZHU,§ and MU YINGKUIq $Department of Mathematics and §Institute of Atomic and Molecular Physics, Jilin University, Changchun 130023, and IChangchun Finance Manager College, Changchun 130031,P.R. China (Received 10 February 1995)
Abstract-A spline approximation to obtain the radial emission coefficient for optically-thick, cylindrically-symmetric plasmas is proposed and an explicit formula to evaluate the values of the coefficient at nodes is deduced. Using the formula, the work to find approximate values of the coefficient is simplified very much. An example is evaluated using present spline approximation. It is shown that the spline approximation is precise and much easier to apply than both the Young’s iteration and the block-bi-quadratic interpolation.
1. INTRODUCTION
In many applications of optically-thick equation
and cylindrically-symmetric
plasmas, the generalized Abel
R S(z)
=
2
J(r)cosh G(z, r) J&
(0 < z < R)
sL
(1)
R
-In r(z) = 2
K(r) J&i
(O
(2)
s
must be solved.’ Here, R is the radius of zircular section of plasma; S(z) = N(z)/,&& and N(z) and T(Z) are, respectively, the measured transverse radiance and transmittance; J(r) and K(r) must be found and represent the radial emission and absorption coefficients, respectively. Also, G(z,r)=
:K(t+&
(O
(3)
s
The absorption coefficient K(r) may be found by using well known methods2.3 for optically-thin plasmas before G(z, r) is obtained from Eq. (3). Thus, the major remaining problem is to solve Eq. (1). In this paper, a spline approximation to solve the generalized Abel equation (1) is proposed and an explicit formula to evaluate the radial emission coefficients at nodes approximately is deduced. An example is evaluated using the explicit formula. It is shown that the spline approximation is precise and much easier to apply than both the Young’s iteration4 and the block-bi-quadratic interpolation.5 2. SPLINE
APPROXIMATION
We divide [0, R] into n subintervals O=z,,=r,,
-zi=rj+,-rj=Rln,
j=O,1,2
,...,
n-l
tsupported by National Science Foundation of China and State Commission of Science and Technology. 1055
1056
Note
and represent (zj, rk) as 0, k), S(Zj) as Sj, J(rk) as Jk, cash G(z, r) aS C(z, r), C(Zj, rk) aS Cjk. Because of J(r) and cash G(x, r) are sufficiently smooth and J’(0) = 0 and J(R) = 0, we choose
O
Q,(r) =
r, < r < R, I
rk
O
n-l c
F;(r)=
k = 2,3, . . . , n - 1
r,+,cr
and
JkCj,&(r),
j=o,1,2
,...,
n-1.
(5)
k=j
Obsviously G(O) = a; (0) = 0; &(rk)=l,
(,#k)
&(r[)=O
k,I=0,1,2
,...,
(6)
n-l
and then k=j,j-t-l,...,
Fjn(rk)=JkCjk=J(Tk)COShG(Zj,rk),
F;(R)
= J(R)cosh
G(zj, R) = 0;
n-l
j=O,1,2,
,n-I;
F;;‘(O) = 0.
Due to if J(r) and K(r) are sufficient smooth, we can also verify that
+ IS + IS 'k
I
[J(r)cosh G(zj, r) -F;(r)]
J&
I
(7)
= o(h2+(“2))
‘k
'k
I
k=j,j+l,...,
n-l;
[J(r)coshG(O,r)-F;f(r)]dr
=0(h3)
j=1,2
,...,
n-l;
I
(8)
k=l,2,...,n-1
‘k
”
[J(r)cosh
G(0, r) - F:(r)] dr = O(h’). i
After substituting F;(r)
for J(r)cosh n-l sj=2
c
R
Jkcjk
ak@)d!$-$
_i=1,2y...Fn-l;
s =i
k-j n-l
So=2
G(z, r) in Eq. (l), we have
J
(9)
R
c JkC,,
nk(r)
dr
0
k=O
From Eq. (3) obviously C’ = cash G(zj, rj) = 1, and then an explicit formula to evaluate the radial emission coefficients at nodes, r. = 0, r, , . . . , r,_ , , can be deduced as follows: J,-,
=
Ji =
1 hDn-,,n-,
&, I Sj1.1
sn-1;
h “2’
Dj*kCj,kJk
k-j+1
So-~C,,,-2h’~‘C,,J, k=2
j = 1,2,.
. . , n - 2;
(10)
1057
Note
where
s s R
D,,, = 2
:J
v+l(rj+,,r)rdr
?A*) J&
= i
dq
I,
s
=
h{Qj,j+ 1-j*(Lj,j+ I- L,j)} j=l,2,...,n-1;
1
R
Di,k = 2
+-I
- 2Qj.k+ Qj,k- I -j2(L,, +1 - 2Lj,,c + Lj,k- I)> =h{Q,.,+, k=j+lj+2 Q,.k= k,/m,
,...,
n-l;
j=l,2
,...,
n-2;
n-l;
j=l,2
,...,
n-l.
Lj,k = ln(k + ,/m)
k=j,j+l,...,
3. EXAMPLE
,
AND DISCUSSION
So as to check whether present spline approximation is precise and useful, we choose the example in Ref. 5. Verify easily that J(r) = 1 - r* is the exact solution of Eq. (1) with R = 1, cash G(.z, r) = 1 + r* - z*
and
S(z) = $1 - z*)“*(6 - z*),
(12)
i.e. of the generalized Abel equation A(1 -z*)“*(6-z*)=2
‘J(r)(l sZ
+r*-z*)
rdr
(O
(13)
J_
We have solved Eq. (13) numerically using our spline approximation and have compared the resulting values of radial emission coefficient at the nodes by the explicit formula (10) with exact values of J(r) = 1 - r* in Table 1. From Table 1, we can see that the resulting values converge rapidly to the exact solution, J(r) = 1 - r*, of Eq. (13). With an increase in the number of nodes n, the relative errors decrease and become < 5% when n = 10, 2.5% at n = 20 and 0.4% at n = 40. We have also solved numerically Eq. (13), respectively, with both Young’s iteration4 and the block-bi-quadric interpolation.5 And because formula (10) is explicit, the work to find approximate Table
1. Approximate
values
and
relative errors using our spline approximation; 0.075,. . ,0.925, 0.975 are omitted for n =40. n = 10
Node 0.00
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95
Exact value 1 0.9975 0.99 0.9775 0.96 0.9375 0.91 0.8775 0.84 0.7975 0.75 0.6975 0.64 0.5775 0.51 0.4375 0.36 0.2775 0.19 0.0975
Approximate value
for
simplicity,
n = 20 Relative error(%)
0.999476
-0.0524
0.989591
-0.0413
0.9598369
-0.017
0.9102374
0.0261
0.8407379
0.08785
0.7515315
0.2042
0.6421368
0.3339
0.5138795
0.7607
0.3633748
0.9374
0.1993348
4.9131
the nodes
0.025,
n=40
Approximate value
Relative error(%)
0.999853 0.9973589 0.9898734 0.9774006 0.9599348 0.9394768 0.9190272 0.8775881 0.840163 0.7977428 0.7503358 0.6979393 0.6405526 0.5781843 0.5108147 0.4384869 0.3611086 0.2789147 0.191253 0.09991522
-0.0147 -0.01414 -0.01278 -0.01016 -0.09679 -0.00247 0.00298 0.01003 0.0194 0.03044 0.04477 0.06298 0.08634 0.11849 0.15974 0.22557 0.30794 0.5098 0.65947 2.477
Approximate value 0.9999558 0.9974582 0.9899631 0.977469 0.9599808 0.9374898 0.9100022 0.8775161 0.8400351 0.797554 0.7500798 0.6976025 0.6401339 0.5776649 0.5101997 0.4377403 0.3602815 0.2778271 0.1903709 0.09786919
Relative error(%) - 0.00442 -0.00419 -0.00372 -0.00317 -0.002 -0.00108 0.00024 0.00183 0.00418 0.00677 0.01064 0.01469 0.02092 0.02855 0.03916 0.05493 0.07819 0.11787 0.19521 0.37866
1058
Note
values of the radial emission coefficient is simplified very much and computer time required is far less than both Young’s iteration4 and the block-bi-quadratic interpolation.’ REFERENCES 1. 2. 3. 4. 5.
S. Suckewer, C. H. Skinner, H. Milchberg, C. Keane, and D. Voorhees, Whys. Rev. Lett. 55, 1753 (1985). K. Bockasten, J. Opt. Sot. Am. 51, 943 (1961). Ding Peizhu and Pan Shoufu, Laser JL 7, 1 (1980). S. J. Young, JQSRT 25, 479 (1981). Ding Peizhu, Mu Yingkui, and Pan Shoufu, JQSRT 45, 115 (1991).
J. Quanf. Specfrosc.
Copyright 0 1995 Elsevier Science Ltd Printed in Great Britain. All rightsreserved
00224073(95)00119-0
0022-4073/95$9.50+ 0.00
NOTE A SPLINE APPROXIMATION OF THE ABEL TRANSFORMATION FOR USE IN OPTICALLY-THICK, CYLINDRICALLY-SYMMETRIC PLASMAST WANG YI,$ DING PEIZHU,§ and MU YINGKUIq $Department of Mathematics and §Institute of Atomic and Molecular Physics, Jilin University, Changchun 130023, and IChangchun Finance Manager College, Changchun 130031,P.R. China (Received 10 February 1995)
Abstract-A spline approximation to obtain the radial emission coefficient for optically-thick, cylindrically-symmetric plasmas is proposed and an explicit formula to evaluate the values of the coefficient at nodes is deduced. Using the formula, the work to find approximate values of the coefficient is simplified very much. An example is evaluated using present spline approximation. It is shown that the spline approximation is precise and much easier to apply than both the Young’s iteration and the block-bi-quadratic interpolation.
1. INTRODUCTION
In many applications of optically-thick equation
and cylindrically-symmetric
plasmas, the generalized Abel
R S(z)
=
2
J(r)cosh G(z, r) J&
(0 < z < R)
sL
(1)
R
-In r(z) = 2
K(r) J&i
(O
(2)
s
must be solved.’ Here, R is the radius of zircular section of plasma; S(z) = N(z)/,&& and N(z) and T(Z) are, respectively, the measured transverse radiance and transmittance; J(r) and K(r) must be found and represent the radial emission and absorption coefficients, respectively. Also, G(z,r)=
:K(t+&
(O
(3)
s
The absorption coefficient K(r) may be found by using well known methods2.3 for optically-thin plasmas before G(z, r) is obtained from Eq. (3). Thus, the major remaining problem is to solve Eq. (1). In this paper, a spline approximation to solve the generalized Abel equation (1) is proposed and an explicit formula to evaluate the radial emission coefficients at nodes approximately is deduced. An example is evaluated using the explicit formula. It is shown that the spline approximation is precise and much easier to apply than both the Young’s iteration4 and the block-bi-quadratic interpolation.5 2. SPLINE
APPROXIMATION
We divide [0, R] into n subintervals O=z,,=r,,
-zi=rj+,-rj=Rln,
j=O,1,2
,...,
n-l
tsupported by National Science Foundation of China and State Commission of Science and Technology. 1055
1056
Note
and represent (zj, rk) as 0, k), S(Zj) as Sj, J(rk) as Jk, cash G(z, r) aS C(z, r), C(Zj, rk) aS Cjk. Because of J(r) and cash G(x, r) are sufficiently smooth and J’(0) = 0 and J(R) = 0, we choose
O
Q,(r) =
r, < r < R, I
rk
O
n-l c
F;(r)=
k = 2,3, . . . , n - 1
r,+,cr
and
JkCj,&(r),
j=o,1,2
,...,
n-1.
(5)
k=j
Obsviously G(O) = a; (0) = 0; &(rk)=l,
(,#k)
&(r[)=O
k,I=0,1,2
,...,
(6)
n-l
and then k=j,j-t-l,...,
Fjn(rk)=JkCjk=J(Tk)COShG(Zj,rk),
F;(R)
= J(R)cosh
G(zj, R) = 0;
n-l
j=O,1,2,
,n-I;
F;;‘(O) = 0.
Due to if J(r) and K(r) are sufficient smooth, we can also verify that
+ IS + IS 'k
I
[J(r)cosh G(zj, r) -F;(r)]
J&
I
(7)
= o(h2+(“2))
‘k
'k
I
k=j,j+l,...,
n-l;
[J(r)coshG(O,r)-F;f(r)]dr
=0(h3)
j=1,2
,...,
n-l;
I
(8)
k=l,2,...,n-1
‘k
”
[J(r)cosh
G(0, r) - F:(r)] dr = O(h’). i
After substituting F;(r)
for J(r)cosh n-l sj=2
c
R
Jkcjk
ak@)d!$-$
_i=1,2y...Fn-l;
s =i
k-j n-l
So=2
G(z, r) in Eq. (l), we have
J
(9)
R
c JkC,,
nk(r)
dr
0
k=O
From Eq. (3) obviously C’ = cash G(zj, rj) = 1, and then an explicit formula to evaluate the radial emission coefficients at nodes, r. = 0, r, , . . . , r,_ , , can be deduced as follows: J,-,
=
Ji =
1 hDn-,,n-,
&, I Sj1.1
sn-1;
h “2’
Dj*kCj,kJk
k-j+1
So-~C,,,-2h’~‘C,,J, k=2
j = 1,2,.
. . , n - 2;
(10)
1057
Note
where
s s R
D,,, = 2
:J
v+l(rj+,,r)rdr
?A*) J&
= i
dq
I,
s
=
h{Qj,j+ 1-j*(Lj,j+ I- L,j)} j=l,2,...,n-1;
1
R
Di,k = 2
+-I
- 2Qj.k+ Qj,k- I -j2(L,, +1 - 2Lj,,c + Lj,k- I)> =h{Q,.,+, k=j+lj+2 Q,.k= k,/m,
,...,
n-l;
j=l,2
,...,
n-2;
n-l;
j=l,2
,...,
n-l.
Lj,k = ln(k + ,/m)
k=j,j+l,...,
3. EXAMPLE
,
AND DISCUSSION
So as to check whether present spline approximation is precise and useful, we choose the example in Ref. 5. Verify easily that J(r) = 1 - r* is the exact solution of Eq. (1) with R = 1, cash G(.z, r) = 1 + r* - z*
and
S(z) = $1 - z*)“*(6 - z*),
(12)
i.e. of the generalized Abel equation A(1 -z*)“*(6-z*)=2
‘J(r)(l sZ
+r*-z*)
rdr
(O
(13)
J_
We have solved Eq. (13) numerically using our spline approximation and have compared the resulting values of radial emission coefficient at the nodes by the explicit formula (10) with exact values of J(r) = 1 - r* in Table 1. From Table 1, we can see that the resulting values converge rapidly to the exact solution, J(r) = 1 - r*, of Eq. (13). With an increase in the number of nodes n, the relative errors decrease and become < 5% when n = 10, 2.5% at n = 20 and 0.4% at n = 40. We have also solved numerically Eq. (13), respectively, with both Young’s iteration4 and the block-bi-quadric interpolation.5 And because formula (10) is explicit, the work to find approximate Table
1. Approximate
values
and
relative errors using our spline approximation; 0.075,. . ,0.925, 0.975 are omitted for n =40. n = 10
Node 0.00
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95
Exact value 1 0.9975 0.99 0.9775 0.96 0.9375 0.91 0.8775 0.84 0.7975 0.75 0.6975 0.64 0.5775 0.51 0.4375 0.36 0.2775 0.19 0.0975
Approximate value
for
simplicity,
n = 20 Relative error(%)
0.999476
-0.0524
0.989591
-0.0413
0.9598369
-0.017
0.9102374
0.0261
0.8407379
0.08785
0.7515315
0.2042
0.6421368
0.3339
0.5138795
0.7607
0.3633748
0.9374
0.1993348
4.9131
the nodes
0.025,
n=40
Approximate value
Relative error(%)
0.999853 0.9973589 0.9898734 0.9774006 0.9599348 0.9394768 0.9190272 0.8775881 0.840163 0.7977428 0.7503358 0.6979393 0.6405526 0.5781843 0.5108147 0.4384869 0.3611086 0.2789147 0.191253 0.09991522
-0.0147 -0.01414 -0.01278 -0.01016 -0.09679 -0.00247 0.00298 0.01003 0.0194 0.03044 0.04477 0.06298 0.08634 0.11849 0.15974 0.22557 0.30794 0.5098 0.65947 2.477
Approximate value 0.9999558 0.9974582 0.9899631 0.977469 0.9599808 0.9374898 0.9100022 0.8775161 0.8400351 0.797554 0.7500798 0.6976025 0.6401339 0.5776649 0.5101997 0.4377403 0.3602815 0.2778271 0.1903709 0.09786919
Relative error(%) - 0.00442 -0.00419 -0.00372 -0.00317 -0.002 -0.00108 0.00024 0.00183 0.00418 0.00677 0.01064 0.01469 0.02092 0.02855 0.03916 0.05493 0.07819 0.11787 0.19521 0.37866
1058
Note
values of the radial emission coefficient is simplified very much and computer time required is far less than both Young’s iteration4 and the block-bi-quadratic interpolation.’ REFERENCES 1. 2. 3. 4. 5.
S. Suckewer, C. H. Skinner, H. Milchberg, C. Keane, and D. Voorhees, Whys. Rev. Lett. 55, 1753 (1985). K. Bockasten, J. Opt. Sot. Am. 51, 943 (1961). Ding Peizhu and Pan Shoufu, Laser JL 7, 1 (1980). S. J. Young, JQSRT 25, 479 (1981). Ding Peizhu, Mu Yingkui, and Pan Shoufu, JQSRT 45, 115 (1991).