In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time (typically denoted t = 0). For a system of order k (the number of time lags in discrete time, or the order of the largest derivative in continuous time) and dimension n (that is, with n different evolving variables, which together can be denoted by an n-dimensional coordinate vector), generally nk initial conditions are needed in order to trace the system's variables forward through time.
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| - Condition initiale (fr)
- Initial condition (en)
- Condizioni iniziali (it)
- 初始條件 (zh)
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| - En physique ou en mathématique, on définit comme conditions initiales les éléments nécessaires à la détermination de la solution complète et si possible unique d'un problème, éléments qui décrivent l'état du système à l'instant initial, c'est-à-dire l'état de départ. Plus formellement, on appelle « condition initiale » l'espace d'état d'un système étudié à l'instant initial.C'est ce qui permet de déterminer les coefficients des solutions des équations différentielles, par exemple les équations de mouvement des corps. Les conditions initiales sont à différencier des conditions aux limites. (fr)
- In un sistema fisico descritto da un certo numero di , le condizioni iniziali sono rappresentate dall'insieme dei valori assunti da tali variabili in un certo istante t0 di riferimento detto istante iniziale. Tali condizioni permettono così di definire lo stato in cui si trova il sistema in quel dato istante ed a partire dallo stato iniziale è possibile prevedere l'evoluzione fisica degli stati successivi del sistema a mezzo di equazioni differenziali che descrivono il sistema stesso. (it)
- 在数学以及动力系统中，初始條件（initial condition），有時也稱為種子值（seed value），是系統未知變數在初始時間（一般表示為t = 0）下的值。考慮以下的初值問題，其中的和即為初值條件。 針對k階微分方程系統（若在離散時間系統下，是時間延遲的次數，若是連續時間系統，則是微分的總次數），其维数為n（表示有n個變數，也可以組成n維的向量），一般會需要nk個初始條件，才能完整的追蹤系統的變數。 在連續時間下的微分方程或是離散時間下的遞迴關係式中，初始條件都會影響後續時間的變數值。若是連續時間系統，針對一動力系統以及其初始條件，要求得其狀態變數相對時間函數的解析解，稱為初值問題。離散系統中也有對應的問題。若無法求得解析解，可能會用迭代的方式，逐步計算各變數在不同時間下的值，不過因為誤差的關係，在長時間後，數值偏差可能會越來越大。 (zh)
- In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time (typically denoted t = 0). For a system of order k (the number of time lags in discrete time, or the order of the largest derivative in continuous time) and dimension n (that is, with n different evolving variables, which together can be denoted by an n-dimensional coordinate vector), generally nk initial conditions are needed in order to trace the system's variables forward through time. (en)
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| - A nonsmooth initial condition for a vibrating string, and the evolution thereof (en)
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| - Another initial condition (en)
- Evolution from the initial condition (en)
- The initial condition of a vibrating string (en)
- Evolution of this initial condition for an example PDE (en)
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| - LF-Initial.png (en)
- LF-Solution.png (en)
- Vibrating string oscillation for nonsmooth initial condition.gif (en)
- Nonsmooth initial condition for vibrating string.svg (en)
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| - In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time (typically denoted t = 0). For a system of order k (the number of time lags in discrete time, or the order of the largest derivative in continuous time) and dimension n (that is, with n different evolving variables, which together can be denoted by an n-dimensional coordinate vector), generally nk initial conditions are needed in order to trace the system's variables forward through time. In both differential equations in continuous time and difference equations in discrete time, initial conditions affect the value of the dynamic variables (state variables) at any future time. In continuous time, the problem of finding a closed form solution for the state variables as a function of time and of the initial conditions is called the initial value problem. A corresponding problem exists for discrete time situations. While a closed form solution is not always possible to obtain, future values of a discrete time system can be found by iterating forward one time period per iteration, though rounding error may make this impractical over long horizons. (en)
- En physique ou en mathématique, on définit comme conditions initiales les éléments nécessaires à la détermination de la solution complète et si possible unique d'un problème, éléments qui décrivent l'état du système à l'instant initial, c'est-à-dire l'état de départ. Plus formellement, on appelle « condition initiale » l'espace d'état d'un système étudié à l'instant initial.C'est ce qui permet de déterminer les coefficients des solutions des équations différentielles, par exemple les équations de mouvement des corps. Les conditions initiales sont à différencier des conditions aux limites. (fr)
- In un sistema fisico descritto da un certo numero di , le condizioni iniziali sono rappresentate dall'insieme dei valori assunti da tali variabili in un certo istante t0 di riferimento detto istante iniziale. Tali condizioni permettono così di definire lo stato in cui si trova il sistema in quel dato istante ed a partire dallo stato iniziale è possibile prevedere l'evoluzione fisica degli stati successivi del sistema a mezzo di equazioni differenziali che descrivono il sistema stesso. (it)
- 在数学以及动力系统中，初始條件（initial condition），有時也稱為種子值（seed value），是系統未知變數在初始時間（一般表示為t = 0）下的值。考慮以下的初值問題，其中的和即為初值條件。 針對k階微分方程系統（若在離散時間系統下，是時間延遲的次數，若是連續時間系統，則是微分的總次數），其维数為n（表示有n個變數，也可以組成n維的向量），一般會需要nk個初始條件，才能完整的追蹤系統的變數。 在連續時間下的微分方程或是離散時間下的遞迴關係式中，初始條件都會影響後續時間的變數值。若是連續時間系統，針對一動力系統以及其初始條件，要求得其狀態變數相對時間函數的解析解，稱為初值問題。離散系統中也有對應的問題。若無法求得解析解，可能會用迭代的方式，逐步計算各變數在不同時間下的值，不過因為誤差的關係，在長時間後，數值偏差可能會越來越大。 (zh)
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