| Title:
|
Die allgemeine Lösung einer zylindrischen Differentialgleichung vierter Ordnung nullten Parameterwertes (German) |
| Title:
|
The general solution of a cylindrical differential equation of fourth order with parameter value zero (English) |
| Author:
|
Panc, Vladimír |
| Language:
|
German |
| Journal:
|
Aplikace matematiky |
| ISSN:
|
0373-6725 |
| Volume:
|
16 |
| Issue:
|
3 |
| Year:
|
1971 |
| Pages:
|
203-214 |
| Summary lang:
|
English |
| Summary lang:
|
Czech |
| . |
| Category:
|
math |
| . |
| Summary:
|
The paper gives a comprehensive review of general solutions of the ordinary linear differential equation (1) $\Delta^2w+2\epsilon\Delta w+w=0, \ \Delta=d^2/d\rho^2+(1/\rho)(d/d\rho)$, the particular solution of which is represented by the Bessel function $w=Z_0(\rho\sqrt{\lambda})$of zero index with a real, imaginary or complex argument, respectively.
In the case $\epsilon = \pm1$ the corresponding characteristic equation $\lambda^2 - 2\epsilon\lambda + 1=0$ evidently zields one double root $\lambda=\pm 1$; then another independent particular solution of Eq. (1) is represented by the function $w=\rho Z_1(\rho\sqrt{\pm 1})$.
Generally it is proved that the general solution of a double Bessel equation of the $v$-th index (2) $[\Delta + \lambda - (v/\rho)^2]^2w=0$ can be written in the form $w=A_1J_v(\rho sqrt{\lambda})+A_2\rho J_{v+1}(\rho\sqrt{\lambda})+A_3Y_v(\rho\sqrt{\lambda})+A_4\rho Y_{v+1}(\rho\sqrt{\lambda})$ where $A_1$ to $A_4$ denote the constants of integration and $J_v(\rho \sqrt{\lambda}),\ Y_v(\rho \sqrt{\lambda})$ are the Bessel functions of the $v$-th index of the first and second kinds, respectively. () |
| MSC:
|
30D05 |
| MSC:
|
34M99 |
| MSC:
|
39B22 |
| MSC:
|
39B32 |
| MSC:
|
74Bxx |
| idZBL:
|
Zbl 0224.34008 |
| DOI:
|
10.21136/AM.1971.103346 |
| . |
| Date available:
|
2008-05-20T17:50:38Z |
| Last updated:
|
2020-07-28 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/103346 |
| . |
| Reference:
|
[1] B. G. Korenjew: Einige in Besselschen Funktionen lösbare Aufgaben der Elastizitatstheorie und Wärmeleitung.(russisch), Moskau 1960. |
| Reference:
|
[2] V. Pane: Theorie der schubweichen Kreisplatte auf elastischer Unterlage.Acta mechanica, Vol. 1/3, 1965, 294-317. 10.1007/BF01387240 |
| Reference:
|
[3] F. Schleicher: Kreisplatten auf elastischer Unterlage.Berlin 1926. |
| Reference:
|
[4] : Table of the Bessel Functions $J_0 (z)$ and $J_1 (z)$ for Complex Arguments.New York 1943, Moskau 1963. Zbl 0061.30206 |
| Reference:
|
[5] : Table of the Bessel Functions $Y_0 (z)$ and $Y_1 (z)$ for Complex Arguments.New York 1950, Moskau 1963. Zbl 0041.24504 |
| Reference:
|
[6] Jahnke-Emde-Lösch: Tafeln höherer Funktionen.Stuttgart 1960. |
| Reference:
|
[7] E. Kamke: Differentialgleichungen, Lösungsmethoden und Lösungen.B. 1, Gewöhnliche Differentialgleichungen, Leipzig 1951. |
| Reference:
|
[8] E. T. Whittaker G. N. Watson: A Course of Modern Analysis.Cambridge 1927. |
| . |
