Local linearization method: Difference between revisions

Revision as of 15:23, 7 July 2017

Revision as of 15:26, 7 July 2017

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<div style="text-align: center;">

<math>\mathbf{y}_{n+1}=\mathbf{y}_{n}+\mathbf{u}_{4}+\frac{h_{n}}{6}(2\mathbf{k}%_{2}+2\mathbf{k}_{3}+\mathbf{k}_{4}), \qquad \qquad (8)</math>

_{2}+2\mathbf{k}_{3}+\mathbf{k}_{4}), \qquad \qquad (8)</math>

</div>

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<div style="text-align: center;">

<math>\mathbf{u}_{j}=\mathbf{L}(\mathbf{P}_{p,q}(2^{-\kappa _{j}}\mathbf{M}_{n}c_{j}h_{n}))^{2^{\kappa _{j}}}\mathbf{r} </math>

_{n}c_{j}h_{n}))^{2^{\kappa _{j}}}\mathbf{r} </math>

</div>

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<div style="text-align: center;">

<math>\mathbf{k}_{j}=\mathbf{f}\left( t_{n}+c_{j}h_{n},\mathbf{y}_{n}+\mathbf{u}%

_{j}+c_{j}h_{n}\mathbf{k}_{j-1}\right) -\mathbf{f}\left( t_{n},\mathbf{y}%

_{n}\right) -\mathbf{f}_{\mathbf{x}}\left( t_{n},\mathbf{y}_{n}\right)

\mathbf{u}_{j}\ -\mathbf{f}_{t}\left( t_{n},\mathbf{y}_{n}\right)

Line 300 ⟶ 298:

0 & \frac{1}{2} & \frac{1}{2} & 1

\end{array}

\right] , </math> and p + q > 3. For large systems of ODEs, the vector <math>\mathbf{u}_{j} </math> in the above scheme is replaced by <math>\mathbf{u}_{j}=\mathbf{L\mathbf{k}

scheme is replaced by <math>\mathbf{u}_{j}=\mathbf{L\mathbf{k}

}_{m_{j},k_{j}}^{p,q}(c_{j}h_{n},\mathbf{M}_{n},\mathbf{r}).</math>