Evaluation of Bessel Functions Using a Computer Program

Transcription

1 Evaluatio of Bessel Fuctios Usig a Coputer Progra P. S. Yeh, Ph.D. Abstract I cylidrical coordiate, there are two types of Bessel fuctios. These fuctios are the Bessel fuctio ad the odified Bessel fuctio. Both fuctios are expressed atheatically by ifiite power series, ad each oe cosists of differet orders, begiig with the zero-order, ad the the first order, the secod order, ad so o. The Bessel fuctio is the solutio of the Bessel differetial equatio, which is a liear, secod-order ordiary differetial equatio. Siilarly, the odified Bessel fuctio is the solutio of odified Bessel differetial equatio. The differece betwee these two differetial equatios is the sigs of the o-differetial ters. The applicatios of Bessel fuctios are i the scietific areas of elasticity, electrical field theory, aerodyaic flutter aalysis, fluid flow, ad heat trasfer by coductio. A executable coputer progra has bee developed i this study for the uerical coputatio of the Bessel fuctio ad the odified Bessel fuctio. This progra is called BESSEL.EXE. It is distributed free by cotactig the author through his e-ail address. Keyword: Bessel fuctios, odified Bessel fuctios, coputer progra. INTRODUCTION The Bessel differetial equatio is a ordiary liear differetial equatio give by the followig for [Broshtei,], [Carslaw, ], [Morse, 3]: r d T dr r dt + + ( r ) T = ( ) dr Where is assued to be a real ad positive iteger costat. T is the depedet variable such as the teperature, ad r is the idepedet variable such as the radial coordiate i the cylidrical syste. The first idepedet solutio of Equatio is give by the followig equatio, which is kow as the Bessel fuctio of the first kid of order [Broshtei, ], [Carslaw, ], [Morse, 3], [Scheider, 4]: J r + ( ) ( ) ( r) = =,,, 3, ( )!( + )! The secod idepedet solutio of Equatio (), which is kow as the Bessel fuctio of the secod kid of order, is give as follows [Broshtei, ], [Carslaw, ], [Morse, 3], [Scheider, 4], [Yeh, 5]: Professor Eeritus of Egieerig, Jacksoville State Uiversity, 5 Maco Drive SE, Jacksoville, AL E-ail: psyeh@jsu.edu

2 r Y ( r) = { J ( r)[l( ) ] π r + ( ) ( ) + r p p!( )! [ ] ( ) + ( )! + } +! + Where = 3 ad for =,,,,..., replace p + p by p ( 3 ) The secod type of Bessel differetial equatio, kow as the odified Bessel differetial equatio, is show i the followig [Broshtei, ], [Carslaw, ], [Morse, 3], [Scheider, 4], [Yeh, 5], [Hilderbrad, 7]: r d T dr r dt + ( r + ) T = ( 4) dr The two idepedet solutios to this differetial equatio are kow as the odified Bessel fuctio of the first kid of order, ad the odified Bessel fuctio of the secod kid of order, respectively. These two solutios are give below [Broshtei, ], [Carslaw, ], [Morse, 3], [Scheider, 4], [Yeh, 5], [Duffy, 7], [Hilderbrabd, 8]: The odified Bessel fuctio of the first kid of order : I r + ( ) ( r) = =,,, 3, ( 5)!( + )! The odified Bessel fuctio of the secod kid of order : r r ( ) ( ) K( r) = ( ) I( r)[l( ) +. ] p p!( + )! [ ] + ( ) r + ( + )! ( )! =,,, 3,... ( 6) + Where for =, replace p + p by p Notice that all solutios to the Bessel ad odified Bessel differetial equatios, as represeted by Equatios (), (3), (5) ad (6), are expressed i ters of ifiite power series. O the applicatios of the Bessel fuctios ad the odified Bessel fuctios, ay techical books have preseted these fuctios either i graphical forat or i uerical tables, or i both optios. However, o record o the availability of a coputer progra ca be foud by the preset author. A suary of iforatio o the Bessel fuctios ad the odified Bessel fuctios is give i Table.

3 Table. Copariso of Graphs ad Tables for Bessel ad Modified Bessel Fuctios Coputer Progra The geeralized flowchart of the coputer progra, which was writte i FORTRAN [Yeh, 5], [Yeh, 6], is show i the followig as Figure : Figure. Flowchart for the Coputer Progra Note that a total of five files are geerated fro the coputer progra. These files are stored i the C: drive, that is, the hard disk drive. Two of the five files, aely, BesselRG.FOR ad BesselMBS.FOR, are i the failiar for of uerical tables. However, copared to ost of the existig tables i ay of the published books, the upper rage of the idepedet variable r (or R) i the preset study is higher, which is 3. for the Bessel fuctios, ad 3. for the odified Bessel fuctios. Through the use of a statistical or a graphical software, such as the Microsoft Excel, these tables ca be preseted i graphical fors, as it will be show i a later sectio. Presetatio of Results

4 A. The cotet of the coputer file BesselPR.FOR is show i Figure. It provides a siple descriptio of the progra, ad a istructio o how to use the progra iteractively. This progra calculates Bessel Fuctios of the First kid ad the secod kid, J ad Y, ad odified Bessel fuctios of the first ad the secod kid, I ad K. Each kid cotais orders fro zero to three, i.e., Jo. J, J, J3, Yo, Y, Y, Y3, Io, I, I, I3, ad Ko, K, K, K3. The results are i the for of a uerical tables, with a icreet of. i the idepedet variable R. The tables are saved i C-drive, with file aes as TABLEBRG.FOR for the Bessel fuctios, ad TABLEMBS.FOR for the odified Bessel fuctios. Usig Microsoft Excel, these tables ca be plotted i graphical fors. To calculate Bessel fuctios ad odified Bessel fuctios for a give value of R, key i the value of R ad press ENTER (R is betwee. ad 3. for the Bessel fuctios, ad betwee. ad 5. for the odified Bessel fuctios. To teriate the calculatio, key i -. ad press ENTER. The results fro each of these calculatios are stored i a file i the C-drive, aely BesselSG.FOR for the Bessel fuctios, ad BesselMSG.FOR for the odified Bessel fuctios. Figure. The Coputer File BesselPF.FOR B. For the output data file BesselRG.FOR, the uerical table cosists of two pages. A portio of the first page is show i Figure 3. The icreet of the idepedet variable r (or R) is., ad the rage is fro. to 3.. The order of the Bessel fuctio is fro zero to three, which is ore tha ay of the tables i the existig published books. Through the use of Microsoft Excel, this table ca be represeted i a graphical for, as show i Figure 4 ad Figure 5. The Bessel fuctios of the first kid possess a fiite uerical value at R=, which is either. or.. While the Bessel fuctios of the secod kid all approach egative ifiite as R approaches zero. As the value of R icreases, all Bessel fuctios display the characteristic of oscillatig waves ad at the sae tie decreasig values, that is, a dapig effect. THE ROOTS OF THE ZERO-ORDER BESSEL FUNCTION OF THE FIRST KIND, Jo(R): R J J J J3 Y Y Y Y

5 Figure 3. The Output Data File BesselRG.FOR C. The file BesselSG.FOR is show i Figure 6. For the iteractive coputatio, the iput value for r (or R) ca be i ay arbitrary sequece, as log as it is withi the rage of. to 3.. For each value of r, eight values of the Bessel fuctio are evaluated. Bessel Fuctios of The First Kid J(R)..9.8 Jo.7.6 J.5 J.4 J R Figure 4. Graphical Presetatio of the Bessel Fuctio of the First Kid

6 Bessel Fuctios of The Secod Kid.8.6 Yo Y Y.4 Y Y(R) R Figure 5. Graphical Presetatio of the Bessel Fuctio of the Secod Kid R=. Jo=.9975 J=.4994 J=.5 J3=. Yo= Y= Y= Y3= R=. Jo=.765 J=.445 J=.49 J3=.956 Yo=.886 Y= -.78 Y= Y3= R=. Jo=.389 J=.5767 J=.3583 J3=.894 Yo=.538 Y= -.73 Y= Y3= R= 3. Jo= -.65 J=.3396 J=.4869 J3=.396 Yo= Y=.3467 Y= -.64 Y3= R= 4. Jo= J= J=.3643 J3=.437 Yo= Y= Y=.59 Y3= -.8 R= 5. Jo= J= J=.4657 J3= Yo= Y=.4786 Y= Y3=.467 R= 6. Jo=.565 J= J= J3=.477 Yo= Y= -.75 Y=.986 Y3=.385 R= 7. Jo=.38 J= J= -.34 J3= Yo= Y= Y= Y3=.688 Figure 6. The cotet of the File BesselSG.FOR

7 D. The data file BesselMBS.FOR cotais the uerical table for the odified Bessel fuctios, which exteds over two pages. A portio of the first page is show i Figure 7. The Microsoft Excel graphical display is show i Figure 8. At R=, the uerical value of the odified Bessel fuctio of the first kid of zero order, that is, I, is., while the values of all higher order fuctios, such as I, I, ad I 3, are.. The values of these fuctios icrease as the value of R icreases. O the other had, for the odified Bessel fuctios of the secod kid, such as K, K, K, ad K 3, the uerical values approach positive ifiite as the value of R approaches zero. These fuctios all becoe very sall i values as the value of R becoes large. R I I I I3 K K K K Figure 7. The output Data File BesselMBS.FOR

8 Modified Bessel Fuctios of the first kid (I) ad secod kid (K) K K3 I(R) or K(R) 5 Io I K I 5 Ko I R Figure 8. Graphical Presetatio of the Modified Bessel Fuctios of the First (I) ad Secod (K) Kids E. For the iteractive coputatio of the odified Bessel fuctios, the output file BesselMSG.FOR is show i Figure 9. The file icludes all iput values of R ad the correspodig values of the odified Bessel fuctios. To start iteractive coputatio of odified Bessel fuctios, key i the value of R ad press ENTER. To stop the calculatio, key i -. ad press ENTER. The results are stored i a file i the C-drive, aed BesselMSG.FOR R=.9 Io=.99 I=.4973 I=.86 I3=.597 Ko= K=.7653 K=.793 K3= R= 6. Io= I= I= I3= Ko=.5 K=.34 K=.69 K3=.48 R= 3.5 Io= I= I= 3.83 I3=.8639 Ko=.96 K=.4 K=.33 K3=.596 R= 7. Io= I= I= 4.3 I3= Ko=.47 K=.4 K=.57 K3=.77 R=.55 Io=.6976 I=.34 I=.3654 I3=.8995 Ko=.4 K=.586 K= K3= R=.85 Io=.8 I=.454 I=.9 I3=.

9 Ko=.5875 K=.6336 K= K3= 34.9 R=. Io= I= I= 8.59 I3= Ko=. K=. K=. K3=. Figure 9. The Cotet of the File BesselMSG.FOR SUMMARY For the evaluatio of Bessel fuctios ad odified Bessel fuctios, a executable coputer progra has bee developed i the preset study. The progra is aed BESSEL.EXE. A free copy of this progra ca be obtaied fro the author by cotactig hi through his e-ail address. No specific coputer prograig laguage copiler is required i the coputer syste itself, as log as the achie is IBM copatible. The progra creates uerical tables for the Bessel ad odified Bessel fuctios, respectively, for both the first kid ad the secod kid, for order fro zero to third. The rage of the idepedet variable ca be fro to 3., ad the icreet is.. To obtai iediate feedback fro the progra, it ca be ru iteractively, by eterig a uerical value of the idepedet variable betwee. ad 3.. All coputed fuctio values are stored i a file i the C: drive, ad also displayed o the coputer display scree. The iteractive process ca be repeated as ay ties as eeded. REFERENCES [] Broshtei, I. N., Seedyayev, K. A., Had Book of Matheatics, Va Nostrad Reihold Copay, New. 985, pages 3-4, 4-4. [] Carslaw, H. S., ad Jaeger, J. C. Coductio of Heat i Solids. d ed Oxford Uiversity Press. Pages [3] Morse,Philip M., ad Feshbach, Hera. Methods of Theoretical Physics. McGraw-Hill Book Copay, Ic Page 94, 3-3. [4] Scheider, P. J. Coductio Heat Trasfer, Addiso-Wesley Publishig Copay, Readig, Massachusetts. 955, pages 46-5, 56, [5] Yeh, P. S. Teperature Distributio i a Cylidrical Core ad Coil Assebly with Heat Geeratio. ASEE Southeast Sectio Coferece Proceedigs. April 4-6, 4. Aubur Uiversity, Aubur, Alabaa. [6] Yeh, P. S. Two-diesioal Teperature Distributio i a Trasforer Coil Assebly with Heat Geeratio. ASEE Southeast Sectio Coferece Proceedigs. April 3-5, 5. The Uiversity of Teessee at Chattaooga, Chattaooga, Teessee. [7] Duffy, Dea G. Advaced Egieerig Matheatics, CRC Press, New York Pages [8] Hilderbrad, Fraces B. Advaced Calculus for Applicatios. Pretice Hall, Ic. 96. Pages P. S. Yeh P. S. Yeh received his B.S. degree fro the Natioal Taiwa Uiversity i Taipei, Taiwa, the M.S. degree fro the Uiversity of Illiois i Urbaa-Chapaig, Illiois, ad the Ph.D. degree fro Rutgers Uiversity i New Bruswick, New Jersey, all i Mechaical Egieerig. He is a Professor Eeritus i Egieerig at Jacksoville State Uiversity, i Jacksoville, Alabaa. His ajor areas of teachig ad research are i fluid echaics, therodyaics, heat trasfer, coputer-aided desig, coputer prograig, acoustics, ad eviroetal egieerig.