BUILT-IN MATHEMATICA SYMBOL

# ParametricNDSolve

ParametricNDSolve[eqns, u, {t, tmin, tmax}, pars]
finds a numerical solution to the ordinary differential equations eqns for the function u with the independent variable t in the range to with parameters pars.

ParametricNDSolve[eqns, u, {t, tmin, tmax}, {x, xmin, xmax}, pars]
finds a numerical solution to the partial differential equations eqns.

ParametricNDSolve[eqns, {u1, u2, ...}, {t, tmin, tmax}, pars]
finds numerical solutions for the functions .

## Details and OptionsDetails and Options

• ParametricNDSolve gives results in terms of ParametricFunction objects.
• A specification for the parameters pars of can be used to specify ranges.
• Possible forms for are:
•  p p has range Reals or Complexes Element[p,Reals] p has range Reals Element[p,Complexes] p has range Complexes Element[p,{v1,...}] p has discrete range {p,pmin,pmax} p has range
• In ParametricNDSolve[eqns, {u1, u2, ...}, ...], can be any expression. Typically, will depend on the parameters indirectly through the solution of the differential equations but may depend explicitly on the parameters. A ParametricFunction object that will return a list can be obtained using ParametricNDSolve[eqns, {{u1, u2, ...}}, ...] or by using ParametricNDSolveValue[eqns, {u1, u2, ...}, ...].
• Derivatives of the resulting ParametricFunction objects with respect to the parameters are computed using a combination of symbolic and numerical sensitivity methods when possible.
• ParametricNDSolve takes the same options and settings as NDSolve.
• NDSolve and ParametricNDSolve typically solve differential equations by going through several different stages, depending on the type of equations. With Method->{s1->m1, s2->m2, ...}, stage is handled by method . The actual stages used and their order are determined by NDSolve, based on the problem to be solved.
• Possible solution stages are the same as for NDSolve, with the addition of:
•  "ParametricCaching" caching of computed solutions "ParametricSensitivity" computation of derivatives with respect to parameters

## ExamplesExamplesopen allclose all

### Basic Examples (3)Basic Examples (3)

Get a parametric solution for :

 Out[1]=

Evaluating with a numerical value of gives an approximate function solution for :

 Out[2]=

Evaluate at a time :

 Out[3]=

Plot the solutions for several different values of the parameter:

 Out[4]=

Get a function of the parameter a that gives the function f at :

 Out[1]=

This plots the value of f[10] as a function of the parameter a:

 Out[2]=

Find a value of a for which :

 Out[3]=

Show the sensitivity of the solution of a differential equation to parameters:

 Out[1]=

The sensitivity with respect to a increases with t:

 Out[2]=

The sensitivity with respect to does not increase with t:

 Out[3]=