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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 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
Examplesopen all
Basic Examples (3)
Get a parametric solution for 
:
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Evaluating with a numerical value of 
gives an approximate function solution for 
:
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Plot the solutions for several different values of the parameter:
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Get a function of the parameter a that gives the function f at 
:
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This plots the value of f[10] as a function of the parameter a:
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Show the sensitivity of the solution of a differential equation to parameters:
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The sensitivity with respect to a increases with t:
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The sensitivity with respect to 
does not increase with t:
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