Lines Matching refs:dot

54 F(t,u,\dot{u}) = G(t,u), \quad u(t_0) = u_0.
66 - Function $F(t,u,\dot{u})$
74   time, input state $u$, input time derivative $\dot{u}$,
76   $F(t,u,\dot{u}) = \dot{u}$ then one need not call this
89   $\sigma F_{\dot{u}}(t^n,u^n,\dot{u}^n) + F_u(t^n,u^n,\dot{u}^n)$
105   $\dot{u}$, input shift $\sigma$, matrix $A$,
113 …\frac{d F}{d u^n} &  = &  \frac{\partial F}{\partial \dot{u}}|_{u^n} \frac{\partial \dot{u}}{\part…
116   For any ODE integration method the approximation of $\dot{u}$
118   $\frac{\partial \dot{u}}{\partial u}|_{u^n} = \sigma$, where
124 …\frac{d F}{d u^n} &  = &    \sigma F_{\dot{u}}(t^n,u^n,\dot{u}^n) + F_u(t^n,u^n,\dot{u}^n).\end{al…
132   F(t^n,u^n,\dot{u}^n) = F(t^n,u^n,w+\sigma*u^n)
139   \frac{d F}{d u^n} = \sigma F_{\dot{u}}(t^n,u^n,\dot{u}^n) + F_u(t^n,u^n,\dot{u}^n)
145   $F(t, u, \dot{u}) = \dot{u} - f(t, u)$ with
146   $\dot{u} = (u^n - u^{n-1})/\delta t$ and
147   $\frac{\partial \dot{u}}{\partial u}|_{u^n} = 1/\delta t$
152       \frac{d F}{d u^n} & = &   (1/\delta t)F_{\dot{u}} + F_u(t^n,u^n,\dot{u}^n).\end{aligned}
155   But $F_{\dot{u}} = 1$, in this special case, resulting in the
296 F(t, u, \dot{u}) = 0
307   \dot{u} &= f(t, u, z) \\
320   \dot{u} &= f(t, u, z) \\
329         0 &= \dot{h}(t, u) \\
330           &= \frac{dh}{du} \dot{u} + \frac{\partial h}{\partial t} \\
348 - “Stiff” part $F(t,u,\dot u)$ using `TSSetIFunction()`.
349 - Jacobian $F_u + \sigma F_{\dot u}$ using `TSSetIJacobian()`.
353 $F(t,u,\dot u) = M \dot u - f(t,u)$, where $M$ is not the
365 …dot{u})=G(t,u)$, where $F()$ is meant to be integrated implicitly and $G()$ explicitly. An IMEX fo…
372    * - :math:`\dot{u} = g(t,u)`
374      - :math:`\begin{aligned}F(t,u,\dot{u}) &= \dot{u} \\ G(t,u) &= g(t,u)\end{aligned}`
375    * - :math:`M \dot{u} = g(t,u)`
377      - :math:`\begin{aligned}F(t,u,\dot{u}) &= \dot{u} \\ G(t,u) &= M^{-1} g(t,u)\end{aligned}`
378    * - :math:`\dot{u} = f(t,u)`
380      - :math:`\begin{aligned}F(t,u,\dot{u}) &= \dot{u} - f(t,u) \\ G(t,u) &= 0\end{aligned}`
381    * - :math:`M \dot{u} = f(t,u)`
383      - :math:`\begin{aligned}F(t,u,\dot{u}) &= M \dot{u} - f(t,u) \\ G(t,u) &= 0\end{aligned}`
384    * - :math:`\dot{u} = f(t,u) + g(t,u)`
386      - :math:`\begin{aligned}F(t,u,\dot{u}) &= \dot{u} - f(t,u) \\ G(t,u) &= g(t,u)\end{aligned}`
387    * - :math:`M \dot{u} = f(t,u) + g(t,u)`
389 …- :math:`\begin{aligned}F(t,u,\dot{u}) &= M\dot{u} - f(t,u) \\ G(t,u) &= M^{-1} g(t,u)\end{aligned…
390    * - :math:`\begin{aligned}\dot{u} &= f(t,u,z) + g(t,u,z)\\0 &= h(t,y,z)\end{aligned}`
392 …- :math:`\begin{aligned}F(t,u,\dot{u}) &= \begin{pmatrix}\dot{u} - f(t,u,z)\\h(t, u, z)\end{pmatri…
393    * - :math:`f(t,u,\dot{u})=0`
395 …- :math:`\begin{aligned}F(t,u,\dot{u}) &= f(t,u,\dot{u})\\G(t,u) &= 0\end{aligned}`; the user need…
685 \dot{u}^{slow} & = f^{slow}(t, u^{slow},u^{fast}) \\
686 M \dot{u}^{fast} & = g^{fast}(t, u^{slow},u^{fast}) + f^{fast}(t, u^{slow},u^{fast})
881 \dot{u} = G(u,t).
886 `TSSetIFunction()` is equivalent to $\dot{u} + \tilde{F}(t,u)$)
911 - $\dot{u} = A u.$ First compute the matrix $A$ then call
927 - $\dot{u} = A(t) u.$ Use
1098 Discretized finite element problems often have the form $M \dot u = G(t, u)$ where $M$ is the mass …
1100 …n ODE, not a DAE), explicit integrators can be applied to $\dot u = M^{-1} G(t, u)$ or $\dot u = \…
1122 F(t,y,\dot{y},p) = 0, \quad y(t_0)=y_0(p) \quad t_0 \le t \le t_F
1279 system $F(t,y,\dot{y},p)$ (set by `TSSetIFunction()` ) and its
1280 corresponding Jacobian $F_y + \sigma F_{\dot y}$ (set by
1451 [^id5]: If the matrix $F_{\dot{u}}(t) = \partial F
1452     / \partial \dot{u}$ is nonsingular then it is an ODE and can be
1506 F(u,\dot{u}) = 0.