Nonlinear Second-Order ODEs (Mathematica Tutorial)

The general form of a nonlinear second-order ODE is For simplicity, assume that the equation can be solved for the highest-order derivative y^ ′′(x) to give There are a few ...

Projection Method for NDSolve (Mathematica Tutorial)

When a differential system has a certain structure, it is advantageous if a numerical integration method preserves the structure. In certain situations it is useful to solve ...

Complex Numbers (Mathematica Guide)

Mathematica has fundamental support for both explicit complex numbers and symbolic complex variables. All applicable mathematical functions support arbitrary-precision ...

Function Approximations Package ... (Function Approximations Package Tutorial)

[1] Hairer, E. and G. Wanner. Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems. Springer, 1991. [2] Iserles, A. and S. P. Nørsett. Order ...

ButcherBetaBar (Numerical Differential Equation Analysis Package Symbol)

ButcherBetaBar[tree] gives the number of ways of labeling tree with ButcherOrder[tree] distinct labels such that every vertex is labeled.ButcherBetaBar[n, tree] gives the ...

ButcherBeta (Numerical Differential Equation Analysis Package Symbol)

ButcherBeta[tree] gives the number of ways of labeling tree with ButcherOrder[tree] - 1 distinct labels such that the root is not labeled, but every other vertex is labeled. ...

ButcherColumnConditions (Numerical Differential Equation Analysis Package Symbol)

ButcherColumnConditions[p, s] gives the column-simplifying conditions up to and including order p for s stages. ButcherColumnConditions[p] gives the column-simplifying ...

ButcherPlot (Numerical Differential Equation Analysis Package Symbol)

ButcherPlot[tree] gives a plot of tree.ButcherPlot[{tree_1, tree_2, ...}] gives an array of plots of tree_1, tree_2, ....

ButcherQuadratureConditions (Numerical Differential Equation Analysis Package Symbol)

ButcherQuadratureConditions[p, s] gives the quadrature conditions up to and including order p for s stages. ButcherQuadratureConditions[p] gives the quadrature conditions in ...

ButcherRowConditions (Numerical Differential Equation Analysis Package Symbol)

ButcherRowConditions[p, s] gives the row-simplifying conditions up to and including order p for s stages. ButcherRowConditions[p] gives the row-simplifying conditions in ...