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Additional resources for Differentialgleichungen Lösungsmethoden und Lösungen

Example text

Q' O· ro n° I" n erfüllen, läßt sich die Lösung vdx), ... , V n (x) mit den Anfangswerten Yp(O) = cp (p = 1, ... , n) nach Potenzen von (1, Cl' . . J qo+· . +gn;;; I m (x) "qOcql ••. c qn • Tp,qo, ... ,qn ~ 1 n ' und diese Reihen sind im Bereich I, I Il 'I e c 1 + '"i+l_n_ < I) vv__ e-,'{nttA+l) 00 ~_ +_~_ To rn 1'1 1 v! v=l absolut konvergent. 6·4. Beziehung zu partiellen DiHereiltialgleichungen 2 ). 3) -;- dx h + I}lv(x, YI' ... , Yn);=·0. cy. n v~I Sind nämlich die Iv in dem Gebiet ~ (x, YI' ...

28 nebst der Lösung auf S. 307. Hunge-König, ~umerisches Rechnen, S. 238-262, 300-311. 6 A. § 1. Differentialgleichungen erster Ordnung. Geometrisch bedeutet das Iterationsverfahren folgendes (Fig. 3): Kennt man die Näherungskurve Y = IPn (x), so werden ihre Punkte durch die Richtungskoeffi. zienten f (x, IPn (x)) zu Linienelementen ergänzt. 3. 4. baren Kurve Y = IPn+1(x) zusammenschließen. Wie L. Vietoris ' ) gezeigt hat, kann die Verschiebung auch längs anderer Kurven erfolgen, z. B. senkrecht zur Richtung des einzelnen Linienelementes; die Konvergenz kann dabei besser werden, aber man bekommt die neue Näherungsfunktion in einern veränderten Intervall (Fig.

K 7/fSfr. Ritzungsberiehte Preuß. Akad. 3, S. 171- 174. M. Müller, Math. 8) 3,l9-3fJD. 3) 111. Fukuha((l, Proccedings Acad. Tokyo G (1930) 360- 362. E. Kamke, Acta Math. ) 71f. E. l, Math. Zeitschrift 39 (1934) ID7~-IGO. ßd. XVlll. Kamkc. Ditlerentialglelchungen I. 3 34 I. A. § 2. Systeme von allgemeinen expliziten Differentialgleichungen. Li pschi tz- Bedingungen Yl' ... , Yn) - I. (x, Yl> ... , Yn) I ~ M ( 2 ) l f . (x, n 17 IYk - Yk I k~l erfüllen i). 5·3. Existenzsatz von Caratheodory. Die Funktionen Iv seien in dem BereIch <»: a < x < b; -- 00 < Yl' ...

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