WOLFRAM

Wolfram Language & System Documentation Center
Wolfram Language Home Page »

ParametricNDSolveValue
✖
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:
  • pp 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

    • List of all options

Examples

open allclose all

Basic Examples  (3)Summary of the most common use cases

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

In[1]:=1
✖
https://wolfram.com/xid/0yv78vsgx7ga-9ofqzx
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/0yv78vsgx7ga-xubub
Out[2]=2

Get the value at :

In[3]:=3
✖
https://wolfram.com/xid/0yv78vsgx7ga-mf30av
Out[3]=3

Plot the solutions for several different values of the parameter:

In[4]:=4
✖
https://wolfram.com/xid/0yv78vsgx7ga-e0upg4
Out[4]=4

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

In[1]:=1
✖
https://wolfram.com/xid/0yv78vsgx7ga-4egbx
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/0yv78vsgx7ga-fl17ka
Out[2]=2

Use the function with FindRoot to find a root:

In[3]:=3
✖
https://wolfram.com/xid/0yv78vsgx7ga-blgdzz
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/0yv78vsgx7ga-1cvjb

The sensitivity with respect to a increases with t:

In[2]:=2
✖
https://wolfram.com/xid/0yv78vsgx7ga-ie4uv8
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0yv78vsgx7ga-b3liub
Out[3]=3

Scope  (6)Survey of the scope of standard use cases

Parameter Dependence  (4)

ParametricNDSolveValue returns a ParametricFunction object:

In[1]:=1
✖
https://wolfram.com/xid/0yv78vsgx7ga-zfh55g
Out[1]=1

Get the solution for :

In[2]:=2
✖
https://wolfram.com/xid/0yv78vsgx7ga-djp5ax
Out[2]=2

Plot the solution for :

In[3]:=3
✖
https://wolfram.com/xid/0yv78vsgx7ga-bejqm
Out[3]=3

Plot solutions for values of ranging from to :

In[4]:=4
✖
https://wolfram.com/xid/0yv78vsgx7ga-gpsnsc
Out[4]=4

Initial conditions can be specified as parameters:

In[1]:=1
✖
https://wolfram.com/xid/0yv78vsgx7ga-7eiz8x
Out[1]=1

Plot solutions with with for values of ranging from to :

In[2]:=2
✖
https://wolfram.com/xid/0yv78vsgx7ga-lzmhb1
Out[2]=2

Plot solutions with with for values of ranging from to :

In[3]:=3
✖
https://wolfram.com/xid/0yv78vsgx7ga-nozxts
Out[3]=3

Return a function of the solution :

In[1]:=1
✖
https://wolfram.com/xid/0yv78vsgx7ga-c4ykdz

:

In[2]:=2
✖
https://wolfram.com/xid/0yv78vsgx7ga-u1nvpj
In[3]:=3
✖
https://wolfram.com/xid/0yv78vsgx7ga-d5pejo
Out[3]=3

:

In[4]:=4
✖
https://wolfram.com/xid/0yv78vsgx7ga-50cdym
In[5]:=5
✖
https://wolfram.com/xid/0yv78vsgx7ga-yx8whm
Out[5]=5

:

In[6]:=6
✖
https://wolfram.com/xid/0yv78vsgx7ga-cq7zvm
In[7]:=7
✖
https://wolfram.com/xid/0yv78vsgx7ga-5092j
Out[7]=7

:

In[8]:=8
✖
https://wolfram.com/xid/0yv78vsgx7ga-bslq9n
In[9]:=9
✖
https://wolfram.com/xid/0yv78vsgx7ga-0cy5h
Out[9]=9

:

In[10]:=10
✖
https://wolfram.com/xid/0yv78vsgx7ga-4mw5fm
In[11]:=11
✖
https://wolfram.com/xid/0yv78vsgx7ga-osel5
Out[11]=11

Differential equation coefficients and boundary conditions can be specified as parameters:

In[1]:=1
✖
https://wolfram.com/xid/0yv78vsgx7ga-lah9g5
Out[1]=1

Plot solutions with for values of ranging from to and with and :

In[2]:=2
✖
https://wolfram.com/xid/0yv78vsgx7ga-r3uqlm
Out[2]=2

Parameter Sensitivity  (2)

Solve the classical harmonic oscillator with parametric amplitude a:

In[1]:=1
✖
https://wolfram.com/xid/0yv78vsgx7ga-mre2ud

Plot the solution for and several nearby values of :

In[2]:=2
✖
https://wolfram.com/xid/0yv78vsgx7ga-u3oe4m
Out[2]=2

The sensitivity of with respect to is by definition . Plot the sensitivity at :

In[3]:=3
✖
https://wolfram.com/xid/0yv78vsgx7ga-tmuiqz
Out[3]=3

Plot the sensitivity as a band around the solution for :

In[4]:=4
✖
https://wolfram.com/xid/0yv78vsgx7ga-2j0nt1
Out[4]=4

Sensitivity analysis of a differential equation with multiple parameters:

In[1]:=1
✖
https://wolfram.com/xid/0yv78vsgx7ga-b1364h
In[2]:=2
✖
https://wolfram.com/xid/0yv78vsgx7ga-ukeauf
Out[2]=2

Plot the sensitivity with respect to the initial condition at , :

In[3]:=3
✖
https://wolfram.com/xid/0yv78vsgx7ga-gm8vih
In[4]:=4
✖
https://wolfram.com/xid/0yv78vsgx7ga-06v5e8
Out[4]=4

Plot the sensitivity with respect to the initial condition at , :

In[5]:=5
✖
https://wolfram.com/xid/0yv78vsgx7ga-i8bjzq
In[6]:=6
✖
https://wolfram.com/xid/0yv78vsgx7ga-i34ytz
Out[6]=6

Generalizations & Extensions  (2)Generalized and extended use cases

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

In[1]:=1
✖
https://wolfram.com/xid/0yv78vsgx7ga-c5ob98
In[2]:=2
✖
https://wolfram.com/xid/0yv78vsgx7ga-ks1ou9
Out[2]=2

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 :

In[1]:=1
✖
https://wolfram.com/xid/0yv78vsgx7ga-rsbzzl

NDSolveValue cannot find a solution:

In[2]:=2
✖
https://wolfram.com/xid/0yv78vsgx7ga-ek5hdg
Out[2]=2

Set up an initial seeding function:

In[3]:=3
✖
https://wolfram.com/xid/0yv78vsgx7ga-b1r3fc

Create a ParametricNDSolveValue function based on the parameter :

In[4]:=4
✖
https://wolfram.com/xid/0yv78vsgx7ga-bxj24s
Out[4]=4

Solve the equation for :

In[5]:=5
✖
https://wolfram.com/xid/0yv78vsgx7ga-xbduu1
Out[5]=5

Reset the seeding to use the solution from :

In[6]:=6
✖
https://wolfram.com/xid/0yv78vsgx7ga-dgs70g

With the reset seeding, find the solution for :

In[7]:=7
✖
https://wolfram.com/xid/0yv78vsgx7ga-kux87b
Out[7]=7

Visualize the solution:

In[8]:=8
✖
https://wolfram.com/xid/0yv78vsgx7ga-njcmvh
Out[8]=8

Options  (2)Common values & functionality for each option

Method  (2)

ParametricCaching  (1)

Prevent caching of solutions to save memory:

In[1]:=1
✖
https://wolfram.com/xid/0yv78vsgx7ga-nx48a1

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

In[2]:=2
✖
https://wolfram.com/xid/0yv78vsgx7ga-id8402

The default is to cache the most recently computed solution:

In[3]:=3
✖
https://wolfram.com/xid/0yv78vsgx7ga-i6i594

With caching, the memory requirement is much greater:

In[4]:=4
✖
https://wolfram.com/xid/0yv78vsgx7ga-k9lpy

ParametricSensitivity  (1)

Specify that no sensitivity should be computed:

In[1]:=1
✖
https://wolfram.com/xid/0yv78vsgx7ga-dmawrp

The function evaluates quickly:

In[2]:=2
✖
https://wolfram.com/xid/0yv78vsgx7ga-bpx145
Out[2]=2

The derivative is not computed:

In[3]:=3
✖
https://wolfram.com/xid/0yv78vsgx7ga-hnh6tu
Out[4]=4

Applications  (14)Sample problems that can be solved with this function

Parameter Sweeps  (7)

Simulate bouncing balls being dropped from various heights:

In[1]:=1
✖
https://wolfram.com/xid/0yv78vsgx7ga-fgr8l1
In[2]:=2
✖
https://wolfram.com/xid/0yv78vsgx7ga-uwjq1y
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0yv78vsgx7ga-czre8w

Find a solution with :

In[2]:=2
✖
https://wolfram.com/xid/0yv78vsgx7ga-ciwnwr
Out[2]=2

Check the resulting solution:

In[3]:=3
✖
https://wolfram.com/xid/0yv78vsgx7ga-jb7t4u
In[4]:=4
✖
https://wolfram.com/xid/0yv78vsgx7ga-hlktr
Out[4]=4

Compare to nearby values of the parameter s:

In[5]:=5
✖
https://wolfram.com/xid/0yv78vsgx7ga-c6mlcd
In[6]:=6
✖
https://wolfram.com/xid/0yv78vsgx7ga-fibgs1
Out[6]=6

Find several solutions to the boundary value problem , , . First consider several possible values for :

In[1]:=1
✖
https://wolfram.com/xid/0yv78vsgx7ga-bbbwh5
In[2]:=2
✖
https://wolfram.com/xid/0yv78vsgx7ga-e0ez5y
Out[2]=2

Run a parameter sweep to determine nontrivial solution values of :

In[3]:=3
✖
https://wolfram.com/xid/0yv78vsgx7ga-jvi3g8
In[4]:=4
✖
https://wolfram.com/xid/0yv78vsgx7ga-mh6wwi
Out[4]=4

Using approximate initial values from the graph above:

In[5]:=5
✖
https://wolfram.com/xid/0yv78vsgx7ga-b1hi4j
Out[5]=5

Plot the solutions that were found:

In[6]:=6
✖
https://wolfram.com/xid/0yv78vsgx7ga-d4p60
Out[6]=6

Find all eigenvalues and eigenfunctions for the classical harmonic oscillator with . Start by exploring the possible parameter values:

In[1]:=1
✖
https://wolfram.com/xid/0yv78vsgx7ga-bi1fzy
In[2]:=2
✖
https://wolfram.com/xid/0yv78vsgx7ga-m3k6gv
Out[2]=2

Find the exact values for :

In[3]:=3
✖
https://wolfram.com/xid/0yv78vsgx7ga-2tgzz
In[4]:=4
✖
https://wolfram.com/xid/0yv78vsgx7ga-lwug5
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0yv78vsgx7ga-fyxm0
Out[5]=5

Plot them:

In[6]:=6
✖
https://wolfram.com/xid/0yv78vsgx7ga-h7n1dm
Out[6]=6

Find the first three eigenfunctions of the quantum harmonic oscillator , , lim_(TemplateBox[{x}, Abs]->infty) y(x)=0. Start by plotting solutions of for :

In[1]:=1
✖
https://wolfram.com/xid/0yv78vsgx7ga-zxvqc2
In[2]:=2
✖
https://wolfram.com/xid/0yv78vsgx7ga-51ndvq
Out[2]=2

Plot for :

In[3]:=3
✖
https://wolfram.com/xid/0yv78vsgx7ga-cwum7u
Out[3]=3

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

In[4]:=4
✖
https://wolfram.com/xid/0yv78vsgx7ga-il5lvu
Out[4]=4

Plot the approximate eigenfunctions together with solutions for nearby :

In[5]:=5
✖
https://wolfram.com/xid/0yv78vsgx7ga-j1he8f
Out[5]=5

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

In[6]:=6
✖
https://wolfram.com/xid/0yv78vsgx7ga-zbbk8x
In[7]:=7
✖
https://wolfram.com/xid/0yv78vsgx7ga-n1ae63
Out[7]=7

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:

In[1]:=1
✖
https://wolfram.com/xid/0yv78vsgx7ga-6dq08b
In[2]:=2
✖
https://wolfram.com/xid/0yv78vsgx7ga-l85qal
Out[2]=2

Plot versus the arc length of the solution:

In[3]:=3
✖
https://wolfram.com/xid/0yv78vsgx7ga-tc7xy3
Out[3]=3

The minimum arc length solution for seems to occur at :

In[4]:=4
✖
https://wolfram.com/xid/0yv78vsgx7ga-hpj15o
Out[4]=4

Find the local minimum, which appears near :

In[5]:=5
✖
https://wolfram.com/xid/0yv78vsgx7ga-c4gpl9
Out[5]=5

Plot the corresponding solution of (locally) minimal arc length together with some nearby solutions:

In[6]:=6
✖
https://wolfram.com/xid/0yv78vsgx7ga-zylubx
Out[6]=6

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

In[1]:=1
✖
https://wolfram.com/xid/0yv78vsgx7ga-sacejk
In[2]:=2
✖
https://wolfram.com/xid/0yv78vsgx7ga-64ijga
In[3]:=3
✖
https://wolfram.com/xid/0yv78vsgx7ga-00vuam

Show the step response when varying component values:

In[4]:=4
✖
https://wolfram.com/xid/0yv78vsgx7ga-glqebs
Out[4]=4

Dependence on resistance r:

In[5]:=5
✖
https://wolfram.com/xid/0yv78vsgx7ga-l8e9ek
Out[5]=5

The inductance l:

In[6]:=6
✖
https://wolfram.com/xid/0yv78vsgx7ga-bmr4z3
Out[6]=6

The capacitance c:

In[7]:=7
✖
https://wolfram.com/xid/0yv78vsgx7ga-gqslli
Out[7]=7

Parameter Sensitivities  (5)

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

In[1]:=1
✖
https://wolfram.com/xid/0yv78vsgx7ga-hvg8a6
In[2]:=2
✖
https://wolfram.com/xid/0yv78vsgx7ga-vg4bhd
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0yv78vsgx7ga-yn7wmx
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/0yv78vsgx7ga-dq1ieq

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

In[2]:=2
✖
https://wolfram.com/xid/0yv78vsgx7ga-qca8es
In[3]:=3
✖
https://wolfram.com/xid/0yv78vsgx7ga-ebi95k
Out[3]=3

Find the sensitivity of the Lorenz equations to a parameter:

In[1]:=1
✖
https://wolfram.com/xid/0yv78vsgx7ga-dxt8gt
In[2]:=2
✖
https://wolfram.com/xid/0yv78vsgx7ga-9iy0r
Out[2]=2

Parametric dependence of the heat equation , :

In[1]:=1
✖
https://wolfram.com/xid/0yv78vsgx7ga-gd4twp

Find the sensitivities and :

In[2]:=2
✖
https://wolfram.com/xid/0yv78vsgx7ga-uydjl4

Plot the corresponding sensitivity bands:

In[3]:=3
✖
https://wolfram.com/xid/0yv78vsgx7ga-j2ighn
Out[3]=3

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

In[4]:=4
✖
https://wolfram.com/xid/0yv78vsgx7ga-bob1jo
In[5]:=5
✖
https://wolfram.com/xid/0yv78vsgx7ga-cms6oo
Out[5]=5

Parametric dependence of the wave equation , :

In[1]:=1
✖
https://wolfram.com/xid/0yv78vsgx7ga-ibt8ar

Find the sensitivities and :

In[2]:=2
✖
https://wolfram.com/xid/0yv78vsgx7ga-dlav1e

Plot the corresponding sensitivity bands:

In[3]:=3
✖
https://wolfram.com/xid/0yv78vsgx7ga-gh6skk
Out[3]=3

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

In[4]:=4
✖
https://wolfram.com/xid/0yv78vsgx7ga-550brj
In[5]:=5
✖
https://wolfram.com/xid/0yv78vsgx7ga-f1o9jg
Out[5]=5

Parameter Fitting  (2)

Sample the solution of a differential equation and add noise:

In[1]:=1
✖
https://wolfram.com/xid/0yv78vsgx7ga-0k356a
In[2]:=2
✖
https://wolfram.com/xid/0yv78vsgx7ga-m4vr3k
Out[2]=2

Fit a trigonometric model to the noisy data:

In[3]:=3
✖
https://wolfram.com/xid/0yv78vsgx7ga-dx6g6w
In[4]:=4
✖
https://wolfram.com/xid/0yv78vsgx7ga-2aauvl
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0yv78vsgx7ga-zepq1o
Out[5]=5

A quadratic model is a better fit:

In[6]:=6
✖
https://wolfram.com/xid/0yv78vsgx7ga-hg2rw1
In[7]:=7
✖
https://wolfram.com/xid/0yv78vsgx7ga-zftehu
Out[7]=7
In[8]:=8
✖
https://wolfram.com/xid/0yv78vsgx7ga-jorips
Out[8]=8

Find the parameters that make the solution of a differential equation the best fit to data:

In[1]:=1
✖
https://wolfram.com/xid/0yv78vsgx7ga-v47bs

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

In[2]:=2
✖
https://wolfram.com/xid/0yv78vsgx7ga-nbzm45

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

In[3]:=3
✖
https://wolfram.com/xid/0yv78vsgx7ga-gwdpy
In[4]:=4
✖
https://wolfram.com/xid/0yv78vsgx7ga-g8at25

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

In[5]:=5
✖
https://wolfram.com/xid/0yv78vsgx7ga-dctar9
In[6]:=6
✖
https://wolfram.com/xid/0yv78vsgx7ga-eeyps5
Out[6]=6
In[7]:=7
✖
https://wolfram.com/xid/0yv78vsgx7ga-dh6yyh
Out[7]=7

Properties & Relations  (3)Properties of the function, and connections to other functions

Use NDSolveValue to solve differential equations with parameters:

In[1]:=1
✖
https://wolfram.com/xid/0yv78vsgx7ga-dcqxze

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

In[2]:=2
✖
https://wolfram.com/xid/0yv78vsgx7ga-ce29gz
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0yv78vsgx7ga-ps3mt4
In[4]:=4
✖
https://wolfram.com/xid/0yv78vsgx7ga-b6uk
Out[4]=4

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

In[1]:=1
✖
https://wolfram.com/xid/0yv78vsgx7ga-zib9g0
Out[1]=1

Use ParametricNDSolveValue to solve the example numerically:

In[2]:=2
✖
https://wolfram.com/xid/0yv78vsgx7ga-fh2l53

The sensitivity is the same from both paths:

In[3]:=3
✖
https://wolfram.com/xid/0yv78vsgx7ga-briqet
Out[3]=3

Use SystemModelParametricSimulate to simulate larger hierarchical system models:

In[1]:=1
✖
https://wolfram.com/xid/0yv78vsgx7ga-zi06e7
Out[1]=1

Simulate with two sets of resistor and spring damper parameters:

In[2]:=2
✖
https://wolfram.com/xid/0yv78vsgx7ga-b6usds

Compare the resulting angular velocities:

In[3]:=3
✖
https://wolfram.com/xid/0yv78vsgx7ga-5nu2jm
Out[3]=3

Possible Issues  (1)Common pitfalls and unexpected behavior

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

In[1]:=1
✖
https://wolfram.com/xid/0yv78vsgx7ga-teyrdw
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0yv78vsgx7ga-eo642a
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0yv78vsgx7ga-sho2oi
Out[3]=3

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

In[4]:=4
✖
https://wolfram.com/xid/0yv78vsgx7ga-0xl0wc
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0yv78vsgx7ga-yrle11
Out[5]=5
Wolfram Research (2012), ParametricNDSolveValue, Wolfram Language function, https://reference.wolfram.com/language/ref/ParametricNDSolveValue.html (updated 2014).
✖
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).

✖
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.

✖
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

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

BibTeX

@misc{reference.wolfram_2025_parametricndsolvevalue, author="Wolfram Research", title="{ParametricNDSolveValue}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/ParametricNDSolveValue.html}", note=[Accessed: 20-May-2025 ]}

✖
@misc{reference.wolfram_2025_parametricndsolvevalue, author="Wolfram Research", title="{ParametricNDSolveValue}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/ParametricNDSolveValue.html}", note=[Accessed: 20-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_parametricndsolvevalue, organization={Wolfram Research}, title={ParametricNDSolveValue}, year={2014}, url={https://reference.wolfram.com/language/ref/ParametricNDSolveValue.html}, note=[Accessed: 20-May-2025 ]}

✖
@online{reference.wolfram_2025_parametricndsolvevalue, organization={Wolfram Research}, title={ParametricNDSolveValue}, year={2014}, url={https://reference.wolfram.com/language/ref/ParametricNDSolveValue.html}, note=[Accessed: 20-May-2025 ]}