Local linearization method: Difference between revisions

Revision as of 15:10, 6 July 2017

Revision as of 15:30, 6 July 2017

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Consider the d-dimensional [[Ordinary differential equation|Ordinary Differential Equation]] (ODE).

<math display="inline">

<math>\frac{d\mathbf{x}\left( t\right) }{dt}=\mathbf{f}\left( t,\mathbf{x}\left(

t\right) \right) ,\qquad t\in \left[ t_{0},T\right], \qquad \qquad \qquad \qquad (1).</math>

</math>

with initial condition <math>\mathbf{x}(t_{0})=\mathbf{x}_{0}</math>, where <math>\mathbf{f}</math> is a differentiable function.

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Let <math>\left( t\right) _{h}=\{t_{n}:n=0,..,N\}</math> be a time discretization of the time interval <math>[t_{0},T]</math> with maximum stepsize '''h''' such that <math>t_{n}<t_{n+1} \quad and \quad h_{n}=t_{n+1}-t_{n}\leq h</math>. After the local linearization of the equation (1) at the time step <math>t_{n}</math> the [[Variation of parameters#First order equation|variation of constants formula]] yields

{{centrar|

<math>\mathbf{x}(t_{n}+h)=\mathbf{x}(t_{n})+\mathbf{\phi }(t_{n},\mathbf{x}

(t_{n});h)+\mathbf{r}(t_{n},\mathbf{x}(t_{n});h),

</math>

}}

where