NDSolveValue
NDSolveValue[eqns,expr,{x,x_{min},x_{max}}]
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 x_{min} to x_{max}.
NDSolveValue[eqns,expr,{x,x_{min},x_{max}},{y,y_{min},y_{max}}]
solves the partial differential equations eqns over a rectangular region.
NDSolveValue[eqns,expr,{x,y}∈Ω]
solves the partial differential equations eqns over the region Ω.
NDSolveValue[eqns,u,{t,t_{min},t_{max}},{x,y}∈Ω]
solves the timedependent partial differential equations eqns over the region Ω.
Details and Options
 NDSolveValue[eqns,y[x],{x,x_{min},x_{max}}] gives solutions for y[x] rather than for the function y itself.
 Differential equations must be stated in terms of derivatives such as y'[x], obtained with D, not total derivatives obtained with Dt.
 Partial differential equations may also be specified using the differential operators Grad (∇), Div (∇.), Laplacian (∇^{2}), and Curl (∇). Typically these operators are used as in Inactive[op] to keep the operator form from evaluating.
 NDSolveValue solves a wide range of ordinary differential equations as well as many partial differential equations.
 NDSolveValue can also solve many delay differential equations.
 In ordinary differential equations, the functions y_{i} must depend only on the single variable x. In partial differential equations, they may depend on more than one variable.
 The differential equations must contain enough initial or boundary conditions to determine the solutions for the y_{i} completely.
 Initial and boundary conditions are typically stated in the form y[x_{0}]==c_{0}, y'[x_{0}]==dc_{0}, etc., but may consist of more complicated equations.
 The c_{0}, dc_{0}, etc. can be lists, specifying that y[x] is a function with vector or general list values.
 Periodic boundary conditions can be specified using y[x_{0}]==y[x_{1}].
 The point x_{0} that appears in the initial or boundary conditions need not lie in the range x_{min} to x_{max} over which the solution is sought.
 In delay differential equations, initial history functions are given in the form y[x/;x<x_{0}]==c_{0}, where c_{0} is in general a function of x.
 WhenEvent[event,action] may be included in the equations eqns to specify an action that occurs when event becomes True.
 Boundary values may also be specified using DirichletCondition and NeumannValue.
 The differential equations in NDSolveValue can involve complex numbers.
 NDSolveValue can solve many differential‐algebraic equations, in which some of the eqns are purely algebraic, or some of the variables are implicitly algebraic.
 The y_{i} can be functions of the dependent variables, and need not include all such variables.
 The following options can be given:

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  NDSolveValue adapts its step size so that the estimated error in the solution is just within the tolerances specified by PrecisionGoal and AccuracyGoal.
 The option NormFunction>f specifies that the estimated errors for each of the y_{i} should be combined using f[{e_{1},e_{2},…}].
 AccuracyGoal effectively specifies the absolute local error allowed at each step in finding a solution, while PrecisionGoal specifies the relative local error.
 If solutions must be followed accurately when their values are close to zero, AccuracyGoal should be set larger, or to Infinity.
 The default setting of Automatic for AccuracyGoal and PrecisionGoal is equivalent to WorkingPrecision/2.
 The default setting of Automatic for MaxSteps estimates the maximum number of steps to be taken by NDSolve, depending on start and stop time and an estimate of the step size. Should this not be possible, a fixed number of steps is taken.
 The setting for MaxStepFraction specifies the maximum step to be taken by NDSolve as a fraction of the range of values for each independent variable.
 With DependentVariables>Automatic, NDSolve attempts to determine the dependent variables by analyzing the equations given.
 NDSolveValue typically solves differential equations by going through several different stages depending on the type of equations. With Method>{s_{1}>m_{1},s_{2}>m_{2},…}, stage s_{i} is handled by method m_{i}. The actual stages used and their order are determined by NDSolve, based on the problem to solve.
 Possible solution stages are:

"TimeIntegration" time integration for systems of differential equations "BoundaryValues" ordinary differential equation boundary value solutions "DiscontinuityProcessing" symbolic processing for handling of discontinuous differential equations "EquationSimplification" simplification of equation form for numerical evaluation "IndexReduction" symbolic index reduction for differential algebraic equations "DAEInitialization" consistent initialization for differential algebraic equations "PDEDiscretization" discretization for partial differential equations  With Method>m_{1} or Method>{m_{1},s_{2}>m_{2},…}, the method m_{1} is assumed to be for time integration, so Method>m_{1} is equivalent to Method>{"TimeIntegration">m_{1}}.
 Possible explicit time integration settings for the Method option include:

"Adams" predictor‐corrector Adams method with orders 1 through 12 "BDF" implicit backward differentiation formulas with orders 1 through 5 "ExplicitRungeKutta" adaptive embedded pairs of 2(1) through 9(8) Runge–Kutta methods "ImplicitRungeKutta" families of arbitrary‐order implicit Runge–Kutta methods "SymplecticPartitionedRungeKutta" interleaved Runge–Kutta methods for separable Hamiltonian systems  With Method>{"controller",Method>"submethod"} or Method>{"controller",Method>{m_{1},m_{2},…}}, possible controller methods include:

"Composition" compose a list of submethods "DoubleStep" adapt step size by the double‐step method "EventLocator" respond to specified events "Extrapolation" adapt order and step size using polynomial extrapolation "FixedStep" use a constant step size "OrthogonalProjection" project solutions to fulfill orthogonal constraints "Projection" project solutions to fulfill general constraints "Splitting" split equations and use different submethods "StiffnessSwitching" switch from explicit to implicit methods if stiffness is detected  Methods used mainly as submethods include:

"ExplicitEuler" forward Euler method "ExplicitMidpoint" midpoint rule method "ExplicitModifiedMidpoint" midpoint rule method with Gragg smoothing "LinearlyImplicitEuler" linearly implicit Euler method "LinearlyImplicitMidpoint" linearly implicit midpoint rule method "LinearlyImplicitModifiedMidpoint" linearly implicit Bader‐smoothed midpoint rule method "LocallyExact" numerical approximation to locally exact symbolic solution  The setting InterpolationOrder>All specifies that NDSolve should generate solutions that use interpolation of the same order as the underlying method used. »
Examples
open allclose allBasic Examples (7)
Solve a firstorder ordinary differential equation:
Use the function and its derivative in a plot:
NDSolveValue can also directly return the above values:
Secondorder nonlinear ordinary differential equation:
Plot the function and its first two derivatives:
Alternatively, substitute the functions to plot directly into NDSolveValue:
System of ordinary differential equations:
This solves the heat equation in one dimension:
Solve the Poisson equation over a Disk:
Find a minimal surface over a Disk with a sinusoidal boundary condition:
Solve a coupled nonlinear sineGordon equation over a region:
Scope (25)
Ordinary Differential Equations (7)
Specify any order equation; reduction to normal form is done automatically:
Directly differentiate the solution to make a phase plot:
Directly specify a system of equations:
Solve for a vectorvalued function:
Plot the four components of the solution:
Different equivalent ways of specifying a harmonic oscillator as a secondorder equation:
As a system of firstorder equations:
Using a vector variable with the dimension deduced from the initial condition:
Use matrixvalued variables to compute the fundamental matrix solution:
Compare to the exact solution:
Define a Van der Pol equation:
The solution's "stiff" behavior that the default solver automatically handles:
Other methods may not be able to solve this system:
The solution ysol[x] is continuous, as it integrates the piecewise function once:
The solution ysol[x] is differentiable, whereas ysol'[x] is continuous only:
Partial Differential Equations (5)
Nonlinear advectiondiffusion equation in one dimension:
Define a system of PDEs of mixed parabolichyperbolic type:
Nonlinear sineGordon equation in two spatial dimensions with periodic boundary conditions:
Plot the solution at the final time:
Plot the time evolution of a radial cross section of the solution:
Solve a wave equation over a region with a slit:
Solve a Poisson equation with periodic boundary conditions on curved boundaries:
Boundary Value Problems (5)
A nonlinear multipoint boundary value problem:
Solve a nonlinear diffusion equation with Dirichlet and Neumann boundary conditions starting from an initial seed of :
Solve a nonlinear equation with Dirichlet boundary conditions starting from an initial seed of :
Solve a complexvalued nonlinear reaction equation with Dirichlet boundary conditions;
Delay Differential Equations (2)
Hybrid and Discontinuous Systems (5)
Options (28)
Use defaults to solve a celestial mechanics equation with sensitive dependence on initial conditions:
Higher accuracy and precision goals give a different result:
Increasing the goals extends the correct solution further:
DependentVariables (1)
EvaluationMonitor (2)
InitialSeedings (2)
InterpolationOrder (1)
Use InterpolationOrder>All to get interpolation the same order as the method:
This is more time consuming than the default interpolation order used:
MaxStepFraction (1)
Features with small relative size in the integration interval can be missed:
Use MaxStepFraction to ensure features are not missed, independent of interval size:
MaxSteps (1)
MaxStepSize (2)
The default step control may miss a suddenly varying feature:
A smaller MaxStepSize setting ensures that NDSolveValue catches the feature:
Attempting to compute the number of positive integers less than misses several events:
Setting a small enough MaxStepSize ensures that none of the events are missed:
Method (11)
TimeIntegration (5)
BoundaryValues (1)
DiscontinuityProcessing (1)
NDSolveValue automatically does processing for discontinuous functions like Sign:
If the processing is turned off, NDSolveValue may fail at the discontinuity point:
With some time integration methods, the solution may be very inaccurate:
An equivalent way to find the solution is to use "DiscontinuitySignature":
EquationSimplification (1)
IndexReduction (1)
An index 3 formulation of a constrained pendulum using index reduction:
The default method can only solve index 1 problems:
The problem resulting from symbolic index reduction can be solved:
Solve using reduction to index 0 and a projection method to maintain the constraints:
Plot implicit energy constraint for the two solutions at the time steps:
DAEInitialization (1)
Use forward collocation for initialization to avoid problems with the Abs term at 0:
NormFunction (1)
StartingStepSize (1)
StepMonitor (3)
Applications (12)
Lotka–Volterra Equations (1)
The Lotka–Volterra predatorprey equations [more info]:
Lorenz Equations (1)
The Lorenz equations [more info]:
Wolfram's Nonlinear Wave Equation (2)
Wolfram's nonlinear wave equation [more info]:
nBody Problems (1)
Reduced 3body problem [more info]:
A formulation suitable for a number of different initial conditions:
Properties & Relations (9)
NDSolveValue gives the solution function, while NDSolve returns the result as a rule:
The following two are equivalent:
NDSolveValue can directly return the result of the solution function at a specified point:
With NDSolve:
NDSolveValue can directly return the solution evaluated at several points:
With NDSolve:
NDSolveValue can directly return the result of any expression involving the solution:
With NDSolve:
If an equation has multiple solutions, NDSolveValue will pick one:
NDSolve will get both solution branches:
Numerically compute values of an integral at different points in an interval:
For functions of the independent variable, NDSolve effectively gives an indefinite integral:
Finding an event is related to finding a root of a function of the solution:
Event location finds the root accurately and efficiently:
This gives as a function of for a differential equation:
Solve the equivalent boundary value problem:
Use NDSolve as a solver for a SystemModel:
Plot variables from the simulation result:
Use SystemModel to model larger hierarchical models:
Possible Issues (10)
Multiple Solutions (1)
For equations with multiple solutions, NDSolveValue will return only one solution:
Use NDSolve to get all the solutions:
Numerical Error (3)
The error tends to grow as one goes further from the initial conditions:
Find the difference between numerical and exact solutions:
Error for a nonlinear equation:
For highorder methods, the default interpolation may have large errors between steps:
Interpolation with the order corresponding to the method reduces the error between steps:
Differential Algebraic Equations (2)
Here is a system of differentialalgebraic equations:
NDSolveValue may change the specified initial conditions if it cannot find the solution with :
NDSolveValue is limited to index 1, but the solution with has index 2:
The default method may not be able to converge to the default tolerances:
With lower AccuracyGoal and PrecisionGoal settings, a solution is found:
The "StateSpace" time integration method can solve this with default tolerances:
Partial Differential Equations (3)
The spatial discretization is based on the initial value, which varies less than the final value:
By increasing the minimal number of spatial grid points, you can accurately compute the final value:
The plot demonstrates the onset of a spatially more complex solution:
Define a heat equation with an initial value that is a step function:
Discontinuities in the initial value may result in too many spatial grid points:
Setting the number of spatial grid points smaller results in essentially as good a solution:
Define a Laplace equation with initial values:
The solver only works for equations well posed as initial value (Cauchy) problems: