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linear_first-order_ode_system [2014/03/06 22:34] nikolaj |
linear_first-order_ode_system [2017/01/17 01:06] nikolaj |
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===== Linear first-order ODE system ===== | ===== Linear first-order ODE system ===== | ||
==== Set ==== | ==== Set ==== | ||
- | | @#88DDEE: $ A:\mathbb R\to\mathrm{Matrix}(n,\mathbb R) $ | | + | | @#55CCEE: context | @#55CCEE: $ A:\mathbb R\to\mathrm{Matrix}(n,\mathbb R) $ | |
- | | @#88DDEE: $ b:\mathbb R\to\mathbb R^n $ | | + | | @#55CCEE: context | @#55CCEE: $ b:\mathbb R\to\mathbb R^n $ | |
+ | | @#FFBB00: definiendum | @#FFBB00: $ y \in \mathrm{it} $ | | ||
+ | | @#55EE55: postulate | @#55EE55: $ y:C^k(\mathbb R,\mathbb R^n) $ | | ||
+ | | @#55EE55: postulate | @#55EE55: $ y'(t)=A(t)\ y(t)+b(t) $ | | ||
- | | @#FFBB00: $ y \in \mathrm{it} $ | | + | ----- |
- | + | ||
- | | @#55EE55: $ y:C^k(\mathbb R,\mathbb R^n) $ | | + | |
- | + | ||
- | | @#55EE55: $ y'(t)=A(t)\ y(t)+b(t) $ | | + | |
- | + | ||
- | ==== Discussion ==== | + | |
=== Theorems === | === Theorems === | ||
There exists a matrix $S(t,s)$ such that the solution of the equation above is of the form | There exists a matrix $S(t,s)$ such that the solution of the equation above is of the form | ||
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* The equation $y'(t)=A(t)y(t)$ is solved by $y(t)=\mathrm{e}^{\int A(t)\,\mathrm dt}y(0)$. | * The equation $y'(t)=A(t)y(t)$ is solved by $y(t)=\mathrm{e}^{\int A(t)\,\mathrm dt}y(0)$. | ||
- | We can in fact sketch how to deal with this equation in cases where $A(t)$ is a more general operator. Dyson series: Say we at least know how to apply $A(t)$. The iterative solution technique for the equation is $y_{n+1}(t)=y(0)+\int_{0}^t A(t)\,y_n(t)\,\mathrm dt$. Note that "$f(x):=y(0)+\mathrm{int}x $" iterated with initial condition $y(0)$ gives $\left(\sum_{n=0}^\infty\mathrm{int}^n\right)y(0)$. Factors $\frac{1}{n!}$ are introduces when time-ordering the integrand and the resulting series is hence mnemonically written as $y(t)=\mathcal T\exp(int_{t_0}^tA(t))y(0)$ | + | We can in fact sketch how to deal with this equation in cases where $A(t)$ is a more general operator. Dyson series: Say we at least know how to apply $A(t)$. The iterative solution technique for the equation is $y_{n+1}(t)=y(0)+\int_{0}^t A(t)\,y_n(t)\,\mathrm dt$. Note that "$f(x):=y(0)+\mathrm{int}x $" iterated with initial condition $y(0)$ gives $\left(\sum_{n=0}^\infty\mathrm{int}^n\right)y(0)$. Factors $\frac{1}{n!}$ are introduces when time-ordering the integrand and the resulting series is hence mnemonically written as $y(t)=\mathcal T\exp(\mathrm{int}_{t_0}^tA(t))y(0)$ |
* For $A(t),b(t)$ one-dimensional one has | * For $A(t),b(t)$ one-dimensional one has | ||
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=== Reference === | === Reference === | ||
Wikipedia: [[http://en.wikipedia.org/wiki/Dyson_series|Dyson series]] | Wikipedia: [[http://en.wikipedia.org/wiki/Dyson_series|Dyson series]] | ||
- | ==== Parents ==== | + | |
+ | ----- | ||
=== Context === | === Context === | ||
[[Square matrix]] | [[Square matrix]] | ||
=== Subset of === | === Subset of === | ||
[[ODE system]] | [[ODE system]] |