ParametricNDSolve
ParametricNDSolve[eqns,u,{x,xmin,xmax},pars]
finds a numerical solution to the ordinary differential equations eqns for the function u with the independent variable x in the range xmin to xmax with parameters pars.
ParametricNDSolve[eqns,u,{x,xmin,xmax},{y,ymin,ymax},pars]
solves the partial differential equations eqns over a rectangular region.
ParametricNDSolve[eqns,u,{x,y}∈Ω,pars]
solves the partial differential equations eqns over the region Ω.
ParametricNDSolve[eqns,u,{t,tmin,tmax},{x,y}∈Ω,pars]
solves the time-dependent partial differential equations eqns over the region Ω.
ParametricNDSolve[eqns,{u1,u2,…},…]
solves for the functions ui.
Details and Options
- ParametricNDSolve gives results in terms of ParametricFunction objects.
- A specification for the parameters pars of {pspec1,pspec2,…} can be used to specify ranges.
- Possible forms for pspeci 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 {v1,…} {p,pmin,pmax} p has range 
- In ParametricNDSolve[eqns,{u1,u2,…},…], ui can be any expression. Typically, ui 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 si is handled by method mi. 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 -
AccuracyGoal Automatic digits of absolute accuracy sought Compiled Automatic whether expressions should be compiled automatically DependentVariables Automatic the list of all dependent variables EvaluationMonitor None expression to evaluate whenever the function is evaluated InitialSeeding {} seeding equations for some algorithms InterpolationOrder Automatic the continuity degree of the final output MaxStepFraction 1/10 maximum fraction of range to cover in each step MaxSteps Automatic maximum number of steps to take MaxStepSize Automatic maximum size of each step Method Automatic method to use NormFunction Automatic the norm to use for error estimation PrecisionGoal Automatic digits of precision sought StartingStepSize Automatic initial step size used StepMonitor None expression to evaluate when a step is taken WorkingPrecision MachinePrecision precision to use in internal computations
List of all options
Examples
open allclose allBasic Examples (3)
Get a parametric solution for y:
Evaluating with a numerical value of a gives an approximate function solution for y:
Plot the solutions for several different values of the parameter:
Get a function of the parameter a that gives the function f at 
:
This plots the value of f[10] as a function of the parameter a:
Find a value of a for which y[10]=0:
Show the sensitivity of the solution of a differential equation to parameters:
Scope (5)
Parameter Dependence (3)
ParametricNDSolve returns a substitution to a ParametricFunction object:
Plot solutions for values of 
ranging from 
to 
:
Initial conditions can be specified as parameters:
Plot solutions with 
with 
for values of 
ranging from 
to 
:
Plot solutions with 
with 
for values of 
ranging from 
to 
:
Differential equation coefficients and boundary conditions can be specified as parameters:
Plot solutions with 
for values of 
ranging from 
to 
and with 
and 
:
Parameter Sensitivity (2)
Solve the classical harmonic oscillator with parametric amplitude 
:
Plot the solution for 
and several nearby values of 
:
The sensitivity of 
with respect to 
is by definition 
. Plot the sensitivity at 
:
Plot the sensitivity 
as a band around the solution 
for 
:
Sensitivity analysis of a differential equation with multiple parameters:
Plot the sensitivity with respect to the initial condition 
at 
, 
:
Plot the sensitivity with respect to the initial condition 
at 
, 
:
Generalizations & Extensions (1)
Solve 
, 
for various values of WorkingPrecision and plot the error:
Options (2)
Method (2)
ParametricCaching (1)
Applications (10)
Parameter Sweeps (4)
Find an initial value 
for which the solution 
of a differential equation will have 
:
Compare to nearby values of the parameter s:
Find several solutions to the boundary value problem 
, 
, 
. First consider several possible values for 
:
Run a parameter sweep to determine nontrivial solution values of 
:
Using approximate initial values from the graph above:
Plot the solutions that were found:
Find all eigenvalues 
and eigenfunctions 
for 
with 
. Start by exploring the possible parameter values:
Find the value of 
for which the solution of 
, 
has minimal arc length from 
to 
. Begin by plotting the solutions for values of 
ranging from 0 to 1:
Plot 
versus the arc length of the solution:
The minimum arc length solution for 
seems to occur at 
:
Find the local minimum, which appears near 
:
Plot the corresponding solution of (locally) minimal arc length together with some nearby solutions:
Parameter Sensitivities (4)
Perturb a parameter in a differential equation and view several of the resulting perturbed solutions:
Plotting the sensitivity solutions gives qualitatively the same result:
Simulate an inverted pendulum stabilized by an oscillating base:
Sensitivity of 
with respect to the amplitude a increases with time:
Parametric dependence of the heat equation 
, 
:
Plot the corresponding sensitivity bands:
Indicate sensitivity to a and c by changing the color of the solution surface:
Find the sensitivity of the Lorenz equations to a parameter:
Parameter Fitting (2)
Sample the solution of a differential equation and add noise:
Fit a trigonometric model to the noisy data:
A quadratic model is a better fit:
Find the parameters that make the solution of a differential equation the best fit to data:
The data comes from a cooling body, so use Newton's law of cooling:
Properties & Relations (1)
Use SystemModelParametricSimulate to simulate larger hierarchical system models:
Simulate with two sets of resistor and spring damper parameters:
Text
Wolfram Research (2012), ParametricNDSolve, Wolfram Language function, https://reference.wolfram.com/language/ref/ParametricNDSolve.html (updated 2014).
CMS
Wolfram Language. 2012. "ParametricNDSolve." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2014. https://reference.wolfram.com/language/ref/ParametricNDSolve.html.
APA
Wolfram Language. (2012). ParametricNDSolve. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ParametricNDSolve.html