Equazioni differenziali ordinarie

Risolutori di problemi di valore iniziale di equazioni differenziali ordinarie

I risolutori di equazioni differenziali ordinarie (ODE) di MATLAB^® risolvono problemi di valore iniziale con diverse proprietà. I solutori possono lavorare su problemi rigidi o non rigidi, problemi con una matrice di massa, equazioni algebriche differenziali (DAE) o problemi completamente impliciti. Per maggiori informazioni, vedere Scelta di un risolutore ODE.

Risolvere i problemi e impostare le opzioni utilizzando un'interfaccia visiva con l’attività Solve ODE (Risolvi Ode) in Live Editor.

Oggetti

`ode`	Ordinary differential equations (Da R2023b)
`odeMassMatrix`	ODE mass matrix (Da R2023b)
`odeJacobian`	ODE Jacobian matrix (Da R2023b)
`odeEvent`	ODE event definition (Da R2023b)
`odeSensitivity`	ODE sensitivity analysis (Da R2024a)
`odeDelay`	ODE delay definition (Da R2025a)
`ODEResults`	Results of ODE integration (Da R2023b)

Attività di Live Editor

Solve ODE

Solve system of ordinary differential equations in the Live Editor (Da R2024b)

Funzioni

espandi tutto

Risolutori non rigidi

`ode45`	Solve nonstiff differential equations — medium order method
`ode23`	Solve nonstiff differential equations — low order method
`ode78`	Solve nonstiff differential equations — high order method (Da R2021b)
`ode89`	Solve nonstiff differential equations — high order method (Da R2021b)
`ode113`	Solve nonstiff differential equations — variable order method

Risolutori rigidi

`ode15s`	Solve stiff differential equations and DAEs — variable order method
`ode23s`	Solve stiff differential equations — low order method
`ode23t`	Solve moderately stiff ODEs and DAEs — trapezoidal rule
`ode23tb`	Solve stiff differential equations — trapezoidal rule + backward differentiation formula

Risolutori completamente impliciti

`ode15i`	Solve fully implicit differential equations — variable order method
`decic`	Compute consistent initial conditions for `ode15i`

Opzioni ottenere/impostare

`odeget`	Extract ODE option values
`odeset`	Create or modify options structure for ODE and PDE solvers

Valutazione ed estensione della soluzione

`deval`	Evaluate differential equation solution structure
`odextend`	Extend solution to ODE

Argomenti

Risoluzione delle ODE

Scelta di un risolutore ODE
Informazioni generali sull'ODE, descrizioni dei risolutori, algoritmi e riepilogo degli esempi.
Riepilogo delle opzioni ODE
Utilizzo di odeset e della tabella che indica quali opzioni funzionano con ciascun risolutore ODE.
ODE Event Location
Detect events during solution of ODE.
Solve Nonstiff ODEs
This page contains two examples of solving nonstiff ordinary differential equations using ode45.
Solve Stiff ODEs
This page contains two examples of solving stiff ordinary differential equations using ode15s.
Solve Differential Algebraic Equations (DAEs)
Solve ODEs with a singular mass matrix.
Nonnegative ODE Solution
This topic shows how to constrain the solution of an ODE to be nonnegative.
Troubleshoot Common ODE Problems
FAQ containing common problems and solutions.

Differenziazione automatica nelle ODE

Resolve Error: Unable to Assign Expression During Automatic Differentiation
Troubleshoot expression initialization for automatic differentiation.
Supported Operations on Expressions
List of supported operations on expressions for automatic differentiation.
Automatic Differentiation Background
Learn the theoretical aspects of automatic differentiation and its forward and reverse modes.

Esempi in primo piano

Solve Predator-Prey Equations

Solve a differential equation representing a predator/prey model using variable step size Runge-Kutta integration methods. The ode23 method uses a 2nd and 3rd order pair of formulas for medium accuracy, and the ode45 method uses a 4th and 5th order pair for higher accuracy.

Apri live script

Solve Equations of Motion for Baton Thrown into Air

Solves a system of ordinary differential equations that model the dynamics of a baton thrown into the air [1]. The baton is modeled as two particles with masses $m_{1}$ and $m_{2}$ connected by a rod of length $L$ . The baton is thrown into the air and subsequently moves in the vertical xy-plane subject to the force due to gravity. The rod forms an angle $θ$ with the horizontal and the coordinates of the first mass are $(x, y)$ . With this formulation, the coordinates of the second mass are $(x + L \cos θ, y + L \sin θ)$ .

Apri live script

Sensitivity Analysis of Epidemic ODE Parameters

Perform sensitivity analysis on the parameters of an ordinary differential equation (ODE) system that models the spread of a disease in an epidemic.

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Solve Celestial Mechanics Problem with High-Order Solvers

Use ode78 and ode89 to solve a celestial mechanics problem that requires high accuracy in each step from the ODE solver for a successful integration. Both ode45 and ode113 are unable to solve the problem using the default error tolerances. Even with more stringent error thresholds, ode89 performs best on the problem due to the high accuracy of the Runge-Kutta formulas it uses in each step.

Apri live script

Solve Stiff Transistor Differential Algebraic Equation

Solve a stiff differential algebraic equation (DAE) that describes an electrical circuit [1]. The one-transistor amplifier problem coded in the example file amp1dae.m can be rewritten in semi-explicit form, but this example solves it in its original form $M u^{'} = ϕ (u)$ . The problem includes a constant, singular mass matrix $M$ .

Apri live script

Solve System of ODEs with Multiple Initial Conditions

Compares two techniques to solve a system of ordinary differential equations with multiple sets of initial conditions.

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Solve ODE with Strongly State-Dependent Mass Matrix

Solve Burgers' equation using a moving mesh technique [1]. The problem includes a mass matrix, and options are specified to account for the strong state dependence and sparsity of the mass matrix, making the solution process more efficient.

Apri live script