NDSolve

• NDSolve gives results in terms of InterpolatingFunction objects.

• NDSolve[eqns, y[x], {x, x_min, x_max}] gives solutions for y[x] rather than for the function y itself.

• Differential equations must be stated in terms of derivatives such as y'[x], obtained with D, not total derivatives obtained with Dt.

• NDSolve solves a wide range of ordinary differential equations as well as many partial differential equations.

• In ordinary differential equations the functions y_i must depend only on the single variable x. In partial differential equations they may depend on more than one variable.

• The differential equations must contain enough initial or boundary conditions to determine the solutions for the y_i completely.

• Initial and boundary conditions are typically stated in the form y[x₀]c₀, y'[x₀]dc₀, etc., but may consist of more complicated equations.

• The c₀, dc₀, etc. can be lists, specifying that y[x] is a function with vector or general list values.

• Periodic boundary conditions can be specified using y[x₀]y[x₁].

• The point x₀ that appears in the initial or boundary conditions need not lie in the range x_min to x_max over which the solution is sought.

• The differential equations in NDSolve can involve complex numbers.

• NDSolve can solve many differential-algebraic equations, in which some of the eqns are purely algebraic, or some of the variables are implicitly algebraic.

• The y_i can be functions of the dependent variables, and need not include all such variables.

• The following options can be given:

• NDSolve adapts its step size so that the estimated error in the solution is just within the tolerances specified by PrecisionGoal and AccuracyGoal.

• The option NormFunction->f specifies that the estimated errors for each of the y_i should be combined using f[{e₁, e₂, ...}].

• AccuracyGoal effectively specifies the absolute local error allowed at each step in finding a solution, while PrecisionGoal specifies the relative local error.

• If solutions must be followed accurately when their values are close to zero, AccuracyGoal should be set larger, or to Infinity.

• The default setting of Automatic for AccuracyGoal and PrecisionGoal is equivalent to WorkingPrecision/2.

• The setting for MaxStepFraction specifies the maximum step to be taken by NDSolve as a fraction of the range of values for each independent variable.

• With DependentVariables->Automatic, NDSolve attempts to determine the dependent variables by analyzing the equations given.

• Possible explicit settings for the Method option include:

• With Method->{"controller", Method->"submethod"} or Method->{"controller", Method->{m₁, m₂, ...}} possible controller methods include:

• Methods used mainly as submethods include:

• The setting InterpolationOrder->All specifies that NDSolve should generate solutions that use interpolation of the same order as the underlying method used. »