# ParametricNDSolveValue

ParametricNDSolveValue[eqns,expr,{x,xmin,xmax},pars]

gives the value of expr with functions determined by a numerical solution to the ordinary differential equations eqns with the independent variable x in the range xmin to xmax with parameters pars.

ParametricNDSolveValue[eqns,expr,{x,xmin,xmax},{y,ymin,ymax},pars]

solves the partial differential equations eqns over a rectangular region.

ParametricNDSolveValue[eqns,expr,{x,y}Ω,pars]

solves the partial differential equations eqns over the region Ω.

ParametricNDSolveValue[eqns,expr,{t,tmin,tmax},{x,y}Ω,pars]

solves the time-dependent partial differential equations eqns over the region Ω.

# Details and Options

• ParametricNDSolveValue 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
• Typically expr will depend on the parameters indirectly, through the solution of the differential equations, but may depend explicitly on the parameters.
• Derivatives of the resulting ParametricFunction object with respect to the parameters are computed using a combination of symbolic and numerical sensitivity methods when possible.
• ParametricNDSolveValue takes the same options and settings as NDSolve.
• NDSolve and ParametricNDSolveValue 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

# Examples

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## Basic Examples(3)

Get a parametric function of the parameter a for the value of y:

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

Get the value at :

Plot the solutions for several different values of the parameter:

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

This plots the value as a function of the parameter a:

Use the function with FindRoot to find a root:

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

The sensitivity with respect to a increases with t:

The sensitivity with respect to b does not increase with t:

## Scope(6)

### Parameter Dependence(4)

ParametricNDSolveValue returns a ParametricFunction object:

Get the solution for :

Plot the solution for :

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 :

Return a function of the solution :

:

:

:

:

:

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 a:

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(2)

Solve , for various values of WorkingPrecision and plot the error:

Consider finding a solution to a highly nonlinear problem that NDSolveValue cannot solve directly. Set up a boundary condition, a region and the equation that depends on a parameter :

NDSolveValue cannot find a solution:

Set up an initial seeding function:

Create a ParametricNDSolveValue function based on the parameter :

Solve the equation for :

Reset the seeding to use the solution from :

With the reset seeding, find the solution for :

Visualize the solution:

## Options(2)

### Method(2)

#### ParametricCaching(1)

Prevent caching of solutions to save memory:

With no caching, the only extra memory required is for the processed equations:

The default is to cache the most recently computed solution:

With caching, the memory requirement is much greater:

#### ParametricSensitivity(1)

Specify that no sensitivity should be computed:

The function evaluates quickly:

The derivative is not computed:

## Applications(14)

### Parameter Sweeps(7)

Simulate bouncing balls being dropped from various heights:

Find an initial value for which the solution of a differential equation will have :

Find a solution with :

Check the resulting solution:

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 the classical harmonic oscillator with . Start by exploring the possible parameter values:

Find the exact values for :

Plot them:

Find the first three eigenfunctions of the quantum harmonic oscillator , , . Start by plotting solutions of for :

Plot for :

The roots are the approximate eigenvalues. Find the first three:

Plot the approximate eigenfunctions together with solutions for nearby :

These only differ from the exact eigenfunctions, the Hermite functions, by scaling factors:

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:

Simulate the response of an RLC circuit to a step in the voltage v1 at time :

Show the step response when varying component values:

Dependence on resistance r:

The inductance l:

The capacitance c:

### Parameter Sensitivities(5)

Perturb a parameter in a differential equation and view several of the resulting perturbed solutions:

A plot of the solution with its sensitivity band is qualitatively similar:

Simulate an inverted pendulum stabilized by a base oscillating with frequency ω and amplitude a:

With , the pendulum is stabilized near an inverted position , but the sensitivity increases:

Find the sensitivity of the Lorenz equations to a parameter:

Parametric dependence of the heat equation , :

Find the sensitivities and :

Plot the corresponding sensitivity bands:

Indicate sensitivity to a and c by changing the color of the solution surface:

Parametric dependence of the wave equation , :

Find the sensitivities and :

Plot the corresponding sensitivity bands:

Indicate sensitivity to a and c by changing the color of the solution surface:

### 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:

Convert data to kelvins and find initial and final (ambient) temperatures:

Find solutions to Newton's law of cooling depending on parameters k1 and k2:

Fit the parameters in the differential equation to the given data:

## Properties & Relations(3)

Use NDSolveValue to solve differential equations with parameters:

For a given parameter value, each function call takes roughly the same amount of time:

ParametricNDSolve caches the solution and subsequently reuses the cached solution values:

DSolve can solve some differential equations with parameters in closed form:

Use ParametricNDSolveValue to solve the example numerically:

The sensitivity is the same from both paths:

Use SystemModelParametricSimulate to simulate larger hierarchical system models:

Simulate with two sets of resistor and spring damper parameters:

Compare the resulting angular velocities:

## Possible Issues(1)

Solution sampling with WhenEvent, Reap, and Sow only works on the first call for each parameter value:

Values are not sown if the solution has already been cached:

Wolfram Research (2012), ParametricNDSolveValue, Wolfram Language function, https://reference.wolfram.com/language/ref/ParametricNDSolveValue.html (updated 2014).

#### Text

Wolfram Research (2012), ParametricNDSolveValue, Wolfram Language function, https://reference.wolfram.com/language/ref/ParametricNDSolveValue.html (updated 2014).

#### CMS

Wolfram Language. 2012. "ParametricNDSolveValue." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2014. https://reference.wolfram.com/language/ref/ParametricNDSolveValue.html.

#### APA

Wolfram Language. (2012). ParametricNDSolveValue. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ParametricNDSolveValue.html

#### BibTeX

@misc{reference.wolfram_2024_parametricndsolvevalue, author="Wolfram Research", title="{ParametricNDSolveValue}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/ParametricNDSolveValue.html}", note=[Accessed: 16-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_parametricndsolvevalue, organization={Wolfram Research}, title={ParametricNDSolveValue}, year={2014}, url={https://reference.wolfram.com/language/ref/ParametricNDSolveValue.html}, note=[Accessed: 16-September-2024 ]}