The backward differentiation formula (BDF) is a family of implicit methods for the numerical integration of ordinary differential equations. They are linear multistep methods that, for a given function and time, approximate the derivative of that function using information from already computed time points, thereby increasing the accuracy of the approximation. These methods are especially used for the solution of stiff differential equations. The methods were first introduced by Charles F. Curtiss and Joseph O. Hirschfelder in 1952. In 1967 the field was formalized by C. William Gear in a seminal paper based on his earlier unpublished work.
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| - BDF-Verfahren (de)
- Backward differentiation formula (en)
- Formula di differenziazione all'indietro (it)
- 後退微分法 (ja)
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| - Die BDF-Verfahren (englisch Backward Differentiation Formulas) sind lineare Mehrschrittverfahren zur numerischen Lösung von Anfangswertproblemen gewöhnlicher Differentialgleichungen: . Dabei wird für eine Näherungslösung an den Zwischenstellen berechnet: . Die Verfahren wurden 1952 von Charles Francis Curtiss und Joseph Oakland Hirschfelder eingeführt und sind seit dem Erscheinen der Arbeiten von C. William Gear 1971 als Löser für steife Anfangswertprobleme weit verbreitet. (de)
- The backward differentiation formula (BDF) is a family of implicit methods for the numerical integration of ordinary differential equations. They are linear multistep methods that, for a given function and time, approximate the derivative of that function using information from already computed time points, thereby increasing the accuracy of the approximation. These methods are especially used for the solution of stiff differential equations. The methods were first introduced by Charles F. Curtiss and Joseph O. Hirschfelder in 1952. In 1967 the field was formalized by C. William Gear in a seminal paper based on his earlier unpublished work. (en)
- 後退微分法（こうたいびぶんほう、英: backward differentiation formula; BDF）は常微分方程式の数値解法の一つである。線型多段法の一種で、過去の複数の値を用いて現在値を計算する方法である。特にの解を計算するときに使われている。 (ja)
- La formula di differenziazione all'indietro (BDF, dall'inglese backward differentiation formula) è una famiglia di metodi impliciti per l'integrazione numerica di equazioni differenziali ordinarie. Sono metodi lineari multipasso che, dati una funzione e un istante temporale, forniscono un valore approssimato della derivata della funzione utilizzando i risultati dei calcoli per istanti temporali precedenti e aumentando così l'accuratezza dell'approssimazione. Questi metodi sono usati specialmente per la soluzione di equazioni differenziali rigide. (it)
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| - Die BDF-Verfahren (englisch Backward Differentiation Formulas) sind lineare Mehrschrittverfahren zur numerischen Lösung von Anfangswertproblemen gewöhnlicher Differentialgleichungen: . Dabei wird für eine Näherungslösung an den Zwischenstellen berechnet: . Die Verfahren wurden 1952 von Charles Francis Curtiss und Joseph Oakland Hirschfelder eingeführt und sind seit dem Erscheinen der Arbeiten von C. William Gear 1971 als Löser für steife Anfangswertprobleme weit verbreitet. (de)
- The backward differentiation formula (BDF) is a family of implicit methods for the numerical integration of ordinary differential equations. They are linear multistep methods that, for a given function and time, approximate the derivative of that function using information from already computed time points, thereby increasing the accuracy of the approximation. These methods are especially used for the solution of stiff differential equations. The methods were first introduced by Charles F. Curtiss and Joseph O. Hirschfelder in 1952. In 1967 the field was formalized by C. William Gear in a seminal paper based on his earlier unpublished work. (en)
- 後退微分法（こうたいびぶんほう、英: backward differentiation formula; BDF）は常微分方程式の数値解法の一つである。線型多段法の一種で、過去の複数の値を用いて現在値を計算する方法である。特にの解を計算するときに使われている。 (ja)
- La formula di differenziazione all'indietro (BDF, dall'inglese backward differentiation formula) è una famiglia di metodi impliciti per l'integrazione numerica di equazioni differenziali ordinarie. Sono metodi lineari multipasso che, dati una funzione e un istante temporale, forniscono un valore approssimato della derivata della funzione utilizzando i risultati dei calcoli per istanti temporali precedenti e aumentando così l'accuratezza dell'approssimazione. Questi metodi sono usati specialmente per la soluzione di equazioni differenziali rigide. (it)
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