<p />

7adb049a-7da4-4df5-9328-7844be310250 127 4 5 decimal

&[DATE] &[TIME] - &[FILENAME]

"Original" Bessel function, first kind, of order 0

"Original" Bessel function, first kind, of order 1

x J.0_Bessel 1 - (k^14 / x 2^* (k^k! (2^/ * (k 017 sum :

x J.1_Bessel 1 - (x J.0_Bessel (x diff * :

Checking zeros:

x J.0_Bessel x 0.1 11 solve 2.404826 5.520078 8.653728 31 mat

x J.1_Bessel x 0.1 13 solve 3.831706 7.015586 10.173477 31 mat

Asymptotic Expansions of the Bessel Function acc. to Hankel, based on Paper by John Harris

x J.n 2 π x * / sqrt x n 2 / 14 / + (π * - cos x P.n * x n 2 / 14 / + (π * - sin x Q.n * - (* :

x Y.n 2 π x * / sqrt x n 2 / - 14 / + sin π * (* x P.n * x n 2 / - 14 / + cos π * (x Q.n * - :

For calculating the sum terms of Pn(x):n ->order, m -> term no.

n m kP 122 m *^m! (* / 4 n 2^* 2 i * 1 - (2^- (i 1 m product * :

n m sP 1 - (m^n 2 m * k * 2 x * (2 m *^/ :

For calculating the sum terms of Qn(x):n ->order, m -> term no.

n m kQ 122 m *^m! (* / 4 n 2^* 2 i * 1 - (2^- (i 1 m product * :

n m SQ 1 - (m^n 2 m * 1 + k1 * 2 x * (2 m * 1 +^/ :

x P.0 19128 x 2^* / - 367532768 x 4^* / + 24012454194304 x 6^* / - 130439058752147483648 x 8^* / + 23921325984953217432719360 x 10^* / - :

x Q.0 18 x * / - 751024 x 3^* / + 59535262144 x 5^* / - 5797291533554432 x 7^* / + 1187451971330634870492913664 x 9^* / - :

Experimental: Generating Pn (x) automatically n mmax P.n n m k 122 m *^m! (* / 4 n 2^* 2 i * 1 - (2^- (i 1 m product * : n m S 1 - (m^n 2 m * k * 2 x * (2 m *^/ : m 0 mmax range term m 1 + el n m S : 11 line for sum term s el s 1 mmax 1 + sum : sum 51 line :

x J.0 2 π x * / sqrt x π 4 / - cos x P.0 * x π 4 / - sin x Q.0 * - (* (:

x J.0 x 233 solve 2.411988 5.52008 8.653728 11.791534 14.930918 18.071064 21.211637 24.352472 27.493479 30.634606 101 mat

Checking zeros:

Another Approx Bessel Function, first kind,J0-order x J.0_2 2 π x * / sqrt 1316 x 2^* / + 53512 x 4^* / - (* x π 4 / - 18 x * / + 25384 x 3^* / - cos * :

looks quite agreeable

The output should be correct but is not beautiful :-(

<p />

x P.1 915128 x 2^* / + 472532768 x 4^* / - 28378354194304 x 6^* / + 147830933252147483648 x 8^* / - 7403020883764396088116142080 x 10^* / + 132690473926380000004018156653772810 x 12^* / - :

x Q.1 38 x * / 1051024 x 3^* / - 72765262144 x 5^* / + 6689182533554432 x 7^* / - 331788050812971217623228416 x 9^* / + 574417636044929951268147200 x 11^* / - :

x J.1 2 π x * / sqrt x 12 / 14 / + (π * - cos x P.1 * x 12 / 14 / + (π * - sin x Q.1 * - (* :

scale_factor 5.520078 J.1 5.520078 J.1_Bessel / :

5.520078 J.1 3.0564 -

5.520078 J.1_Bessel 0.3403 -

x J.1 x J.1 scale_factor / :

x J.1 x 233 solve 3.916539 7.062716 10.206105 13.348652 16.490837 19.632833 22.774718 25.916532 29.058298 32.200031 101 mat

Checking zeros:

does not look so good!

so we chose this approach:

x J.1_new 1 - (x J.0 (x diff * :

x J.1_new x 233 solve 3.832077 7.015587 10.173468 13.323691 16.47063 19.615858 22.760084 25.903672 29.046829 32.18968 101 mat

Checking zeros again:

looks much better!

<p />

Plotting the "original" J0(x) and the approximation J0_Bessel(x)of 0-order

x J.0_Bessel x J.0 2 1 sys

Plotting the "original" J1(x) and the approximation J1_Bessel(x)of 1-order

x J.1_Bessel x J.1_new 2 1 sys

Now first we combine the "original" J0_Bessel and the approximation J0(x) first for 0-order:

x J.0_Bessel_modified x J.0_Bessel x 11.791534 ≤ (* x J.0 x 11.791534 > (* + 11 line :

Let's plot the source functions and the new function J0_Bessel_modified

x J.0_Bessel x J.0 ( x J.0_Bessel_modified 3 1 sys

Now let's check out the zeros of JO_Bessel_modified

x J.0_Bessel_modified x 0.1 75 solve 2.404823 5.52008 8.653728 11.79152 14.930918 18.071064 21.211637 24.352472 27.49348 30.634606 33.77582 36.917099 40.058426 43.199792 46.341188 49.48261 52.624052 55.765511 58.906984 62.048469 65.189965 68.33147 71.472982 74.614501 241 mat

looks agreeable!

Let's combine the "original" J1_Bessel and the approximation J1_new(x) first for 1-order:

x J.1_Bessel_modified x J.1_Bessel x 10.173468 ≤ (* x J.1_new x 10.173468 > (* + 11 line :

Let's plot the source functions and the new function J1_Bessel_modified

x J.1_Bessel x J.1_new ( x J.1_Bessel_modified 3 1 sys

Now let's check out the zeros of JO_Bessel_modified

x J.1_Bessel_modified x 0.1 75 solve 3.831706 7.015588 10.173468 13.323693 16.47063 19.615859 22.760084 25.903672 29.046829 32.18968 35.332308 38.474766 41.617094 44.759319 47.901461 51.043535 54.185554 57.327525 60.469458 63.611357 66.753227 69.895072 73.036895 231 mat

beautiful!

Back to my original Problem: This is the trancendent equation which needs to be solved

Biot 10 :

1 Biot / 0.1

x μ x x J.1_Bessel_modified * x J.0_Bessel_modified / Biot - :

The approx Bessel function curve μ (x)and the zeros look good

x J.0_Bessel_modified x J.1_Bessel_modified x μ 3 1 sys

Quick check, there will be some "false zeros", precision ok For cleaning up we need a spacing (about π) for searching zeros:

Reciproc_Biot Check Biot 1 Reciproc_Biot / : a x x J.1_Bessel_modified * x J.0_Bessel_modified / Biot (- x 0.1 15 solve : μ a transpose : μ 41 line :

This is an excerpt from a publication by Prof Grigull. The values for δ had been calculated by computers from Zuse and Siemens in the year 1964 by means of the Newton method. Dr.-Ing. Ulrich Grigull Professor fuer Technische Thermodynamik an der Technischen Hochschule Muenchen Temperaturausgleich in einfachen Koerpern Ebene Platte, Zylinder, Kugel halbunendlicher Koerper Springer Verlag Berlin / Goettingen / Heidelberg 1964 Library of Congress Catalog Card Number 64-15144

Here are the results: Precisionlooks very good, but there are some "fake zeros" due to the solve function. This issue still has to be addresed and solved.

0.1 Check 2.179495 3.330313 5.033209 7.956316 7.956879 10.648191 10.936351 12.29185 13.958027 19 mat

0.2 Check 1.989811 3.011953 4.713142 7.552164 7.617708 10.163465 10.622305 12.249796 13.678558 19 mat

0.5 Check 1.599445 2.879485 4.290958 7.125173 7.288389 9.913478 10.365832 12.226626 13.471882 19 mat

0.8 Check 1.365882 2.851812 4.136083 7.014361 7.189813 9.857451 10.295009 12.221061 13.416913 19 mat

1 Check 1.255782 2.843017 4.079478 6.978118 7.155799 9.839356 10.270986 12.219225 13.398397 19 mat

2 Check 0.940769 2.826044 3.959371 6.907089 7.086381 9.804029 10.222458 12.215583 13.361149 19 mat

5 Check 0.616975 2.816242 3.883506 6.865552 7.044029 9.78338 10.193105 12.213417 13.338693 19 mat

8 Check 0.492285 2.813835 3.864179 6.855307 7.033379 9.778282 10.185747 12.212877 13.33307 19 mat

10 Check 0.441679 2.813036 3.85771 6.851904 7.029825 9.776588 10.183292 12.212698 13.331195 19 mat

20 Check 0.314261 2.811445 3.844732 6.845119 7.02271 9.773209 10.178381 12.212339 13.327444 19 mat

50 Check 0.199497 2.810494 3.836922 6.84106 7.018437 9.771187 10.175434 12.212123 13.325193 19 mat

80 Check 0.157867 2.810257 3.834966 6.840047 7.017368 9.770682 10.174697 12.21207 13.32463 19 mat

100 Check 0.141239 2.810178 3.834314 6.83971 7.017012 9.770514 10.174451 12.212052 13.324442 19 mat