ode45

Solve nonstiff differential equations — medium order method

collapse all in page

Syntax

[t,y] = ode45(odefun,tspan,y0)

[t,y] = ode45(odefun,tspan,y0,options)

[t,y,te,ye,ie] = ode45(odefun,tspan,y0,options)

sol = ode45(___)

Description

example

[t,y] = ode45(odefun,tspan,y0), wheretspan = [t0 tf], integrates the system of differential equations $y' = f (t, y)$ fromt0totfwith initial conditionsy0. Each row in the solution arrayycorresponds to a value returned in column vectort.

All MATLAB^®ODE solvers can solve systems of equations of the form $y' = f (t, y)$ , or problems that involve a mass matrix, $M (t, y) y' = f (t, y)$ . The solvers all use similar syntaxes. Theode23ssolver only can solve problems with a mass matrix if the mass matrix is constant.ode15sandode23tcan solve problems with a mass matrix that is singular, known as differential-algebraic equations (DAEs). Specify the mass matrix using theMassoption ofodeset.

ode45is a versatile ODE solver and is the first solver you should try for most problems. However, if the problem is stiff or requires high accuracy, then there are other ODE solvers that might be better suited to the problem. SeeChoose an ODE Solverfor more information.

example

[t,y] = ode45(odefun,tspan,y0,options)also uses the integration settings defined byoptions, which is an argument created using theodeset有趣的ction. For example, use theAbsTolandRelToloptions to specify absolute and relative error tolerances, or theMassoption to provide a mass matrix.

[t,y,te,ye,ie] = ode45(odefun,tspan,y0,options)additionally finds where functions of(t,y), called event functions, are zero. In the output,teis the time of the event,yeis the solution at the time of the event, andieis the index of the triggered event.

For each event function, specify whether the integration is to terminate at a zero and whether the direction of the zero crossing matters. Do this by setting the'Events'property to a function, such asmyEventFcnor@myEventFcn, and creating a corresponding function: [value,isterminal,direction] =myEventFcn(t,y). For more information, seeODE Event Location.

example

sol= ode45(___)returns a structure that you can use withdevalto evaluate the solution at any point on the interval[t0 tf]. You can use any of the input argument combinations in previous syntaxes.

Examples

collapse all

ODE with Single Solution Component

Open Live Script

Simple ODEs that have a single solution component can be specified as an anonymous function in the call to the solver. The anonymous function must accept two inputs(t,y), even if one of the inputs is not used in the function.

Solve the ODE

$y^{'} = 2 t .$

Specify a time interval of[0 5]and the initial conditiony0 = 0.

tspan = [0 5]; y0 = 0; [t,y] = ode45(@(t,y) 2*t, tspan, y0);

Plot the solution.

plot(t,y,'-o')

图包含一个坐标轴对象。坐标轴对象有限公司ntains an object of type line.

Solve Nonstiff Equation

Open Live Script

The van der Pol equation is a second-order ODE

$y_{1}^{''} - μ (1 - y_{1}^{2}) y_{1}^{'} + y_{1} = 0,$

where $μ > 0$ is a scalar parameter. Rewrite this equation as a system of first-order ODEs by making the substitution $y_{1}^{'} = y_{2}$ . The resulting system of first-order ODEs is

$\begin{array}{cl} y_{1}^{'} & = y_{2} \\ y_{2}^{'} & = μ (1 - y_{1}^{2}) y_{2} - y_{1} . \end{array}$

The function filevdp1.mrepresents the van der Pol equation using $μ = 1$ . The variables $y_{1}$ and $y_{2}$ are the entriesy(1)andy(2)of a two-element vectordydt.

有趣的ctiondydt = vdp1(t,y)%VDP1 Evaluate the van der Pol ODEs for mu = 1%% See also ODE113, ODE23, ODE45.% Jacek Kierzenka and Lawrence F. Shampine% Copyright 1984-2014 The MathWorks, Inc.dydt = [y(2); (1-y(1)^2)*y(2)-y(1)];

Solve the ODE using theode45有趣的ction on the time interval[0 20]with initial values[2 0]. The resulting output is a column vector of time pointstand a solution arrayy. Each row inycorresponds to a time returned in the corresponding row oft. The first column ofycorresponds to $y_{1}$ , and the second column corresponds to $y_{2}$ .

[t,y] = ode45(@vdp1,[0 20],[2; 0]);

Plot the solutions for $y_{1}$ and $y_{2}$ againstt.

情节(t y (: 1),'-o',t,y(:,2),'-o') title(van der Po的解决方案l Equation (\mu = 1) with ODE45'); xlabel('Time t'); ylabel('Solution y'); legend('y_1','y_2')

图包含一个坐标轴对象。The axes object with title Solution of van der Pol Equation ( mu blank = blank 1 ) with ODE45, xlabel Time t, ylabel Solution y contains 2 objects of type line. These objects represent y_1, y_2.

Pass Extra Parameters to ODE Function

Open Live Script

ode45works only with functions that use two input arguments,tandy. However, you can pass extra parameters by defining them outside the function and passing them in when you specify the function handle.

Solve the ODE

$y^{''} = \frac{A}{B} t y .$

Rewriting the equation as a first-order system yields

$\begin{array}{cl} y_{1}^{'} & = y_{2} \\ y_{2}^{'} & = \frac{A}{B} t y_{1} . \end{array}$

odefcn, a local function included at the end of this example, represents this system of equations as a function that accepts four input arguments:t,y,A, andB.

有趣的ctiondydt = odefcn(t,y,A,B) dydt = zeros(2,1); dydt(1) = y(2); dydt(2) = (A/B)*t.*y(1);end

Solve the ODE usingode45. Specify the function handle so that it passes the predefined values forAandBtoodefcn.

A = 1; B = 2; tspan = [0 5]; y0 = [0 0.01]; [t,y] = ode45(@(t,y) odefcn(t,y,A,B), tspan, y0);

Plot the results.

情节(t y (: 1),'-o',t,y(:,2),'-.')

图包含一个坐标轴对象。坐标轴对象有限公司ntains 2 objects of type line.

有趣的ctiondydt = odefcn(t,y,A,B) dydt = zeros(2,1); dydt(1) = y(2); dydt(2) = (A/B)*t.*y(1);end

Solve ODE with Multiple Initial Conditions

Open Live Script

For simple ODE systems with one equation, you can specifyy0as a vector containing multiple initial conditions. This technique creates a system of independent equations through scalar expansion, one for each initial value, andode45solves the system to produce results for each initial value.

Create an anonymous function to represent the equation $f (t, y) = - 2 y + 2 \cos (t) \sin (2 t)$ . The function must accept two inputs fortandy.

yprime = @(t,y) -2*y + 2*cos(t).*sin(2*t);

Create a vector of different initial conditions in the range $[- 5, 5]$ .

y0 = -5:5;

解方程为每一个初始条件the time interval $[0, 3]$ usingode45.

tspan = [0 3]; [t,y] = ode45(yprime,tspan,y0);

Plot the results.

plot(t,y) gridonxlabel('t') ylabel('y') title('Solutions of y'' = -2y + 2 cos(t) sin(2t), y(0) = -5,-4,...,4,5','interpreter','latex')

图包含一个坐标轴对象。The axes object with title Solutions of y' = -2y + 2 cos(t) sin(2t), y(0) = -5,-4,...,4,5, xlabel t, ylabel y contains 11 objects of type line.

This technique is useful for solving simple ODEs with several initial conditions. However, the technique also has some tradeoffs:

You cannot solve systems of equations with multiple initial conditions. The technique only works when solving one equation with multiple initial conditions.
The time step chosen by the solver at each step is based on the equation in the system that needs to take the smallest step. This means the solver can take small steps to satisfy the equation for one initial condition, but the other equations, if solved on their own, would use different step sizes. Despite this, solving for multiple initial conditions at the same time is generally faster than solving the equations separately using afor-loop.

ODE with Time-Dependent Terms

Open Script

Consider the following ODE with time-dependent parameters

$$y'(t) + f(t)y(t) = g(t).$$

The initial condition is $y_0 = 1$ . The functionf(t)is defined by the n-by-1 vectorfevaluated at timesft. The functiong(t)is defined by the m-by-1 vectorgevaluated at timesgt.

Create the vectorsfandg.

ft = linspace(0,5,25); f = ft.^2 - ft - 3; gt = linspace(1,6,25); g = 3*sin(gt-0.25);

Write a function namedmyodethat interpolatesfandgto obtain the value of the time-dependent terms at the specified time. Save the function in your current folder to run the rest of the example.

Themyode有趣的ction accepts extra input arguments to evaluate the ODE at each time step, butode45only uses the first two input argumentstandy.

有趣的ctiondydt = myode(t,y,ft,f,gt,g) f = interp1(ft,f,t);% Interpolate the data set (ft,f) at time tg = interp1(gt,g,t);% Interpolate the data set (gt,g) at time tdydt = -f.*y + g;% Evaluate ODE at time t

Solve the equation over the time interval[1 5]usingode45. Specify the function using a function handle so thatode45uses only the first two input arguments ofmyode. Also, loosen the error thresholds usingodeset.

tspan = [1 5]; ic = 1; opts = odeset('RelTol',1e-2,'AbsTol',1e-4); [t,y] = ode45(@(t,y) myode(t,y,ft,f,gt,g), tspan, ic, opts);

Plot the solution,y, as a function of the time points,t.

plot(t,y)

Evaluate and Extend Solution Structure

Open Live Script

The van der Pol equation is a second order ODE

$y_{1}^{''} - μ (1 - y_{1}^{2}) y_{1}^{'} + y_{1} = 0 .$

Solve the van der Pol equation with $μ = 1$ usingode45. The functionvdp1.mships with MATLAB® and encodes the equations. Specify a single output to return a structure containing information about the solution, such as the solver and evaluation points.

tspan = [0 20]; y0 = [2 0]; sol = ode45(@vdp1,tspan,y0)

sol =struct with fields:solver: 'ode45' extdata: [1x1 struct] x: [0 1.0048e-04 6.0285e-04 0.0031 0.0157 0.0785 0.2844 0.5407 0.8788 1.4032 1.8905 2.3778 2.7795 3.1285 3.4093 3.6657 3.9275 4.2944 4.9013 5.3506 5.7998 6.2075 6.5387 6.7519 6.9652 7.2247 7.5719 8.1226 8.6122 9.1017 9.5054 9.8402 10.1157 ... ] y: [2x60 double] stats: [1x1 struct] idata: [1x1 struct]

Uselinspaceto generate 250 points in the interval[0 20]. Evaluate the solution at these points usingdeval.

x = linspace(0,20,250); y = deval(sol,x);

Plot the first component of the solution.

plot(x,y(1,:))

图包含一个坐标轴对象。坐标轴对象有限公司ntains an object of type line.

Extend the solution to $t_{f} = 35$ usingodextendand add the result to the original plot.

sol_new = odextend(sol,@vdp1,35); x = linspace(20,35,350); y = deval(sol_new,x); holdonplot(x,y(1,:),'r')

图包含一个坐标轴对象。坐标轴对象有限公司ntains 2 objects of type line.

Input Arguments

collapse all

`odefun`—Functions to solve
有趣的ction handle

Functions to solve, specified as a function handle that defines the functions to be integrated.

The functiondydt = odefun(t,y), for a scalartand a column vectory, must return a column vectordydtof data typesingleordoublethat corresponds to $f (t, y)$ .odefunmust accept both input argumentstandy, even if one of the arguments is not used in the function.

For example, to solve $y' = 5 y - 3$ , use the function:

有趣的ctiondydt = odefun(t,y) dydt = 5*y-3;end

For a system of equations, the output ofodefunis a vector. Each element in the vector is the solution to one equation. For example, to solve

$\begin{array}{l} y'_{1} = y_{1} + 2 y_{2} \\ y'_{2} = 3 y_{1} + 2 y_{2} \end{array}$

use the function:

有趣的ctiondydt = odefun(t,y) dydt = zeros(2,1); dydt(1) = y(1)+2*y(2); dydt(2) = 3*y(1)+2*y(2);end

For information on how to provide additional parameters to the functionodefun, seeParameterizing Functions.

Example:@myFcn

Data Types:有趣的ction_handle

`tspan`—Interval of integration
vector

Interval of integration, specified as a vector. At a minimum,tspanmust be a two-element vector[t0 tf]specifying the initial and final times. To obtain solutions at specific times betweent0andtf, use a longer vector of the form[t0,t1,t2,...,tf]. The elements intspanmust be all increasing or all decreasing.

The solver imposes the initial conditions given byy0at the initial timetspan(1), and then integrates fromtspan(1)totspan(end):

Iftspanhas two elements[t0 tf], then the solver returns the solution evaluated at each internal integration step within the interval.
Iftspanhas more than two elements[t0,t1,t2,...,tf], then the solver returns the solution evaluated at the given points. However, the solver does not step precisely to each point specified intspan. Instead, the solver uses its own internal steps to compute the solution, and then evaluates the solution at the requested points intspan. The solutions produced at the specified points are of the same order of accuracy as the solutions computed at each internal step.
Specifying several intermediate points has little effect on the efficiency of computation, but can affect memory management for large systems.

The values oftspanare used by the solver to calculate suitable values forInitialStepandMaxStep:

Iftspancontains several intermediate points[t0,t1,t2,...,tf], then the specified points give an indication of the scale for the problem, which can affect the value ofInitialStepused by the solver. Therefore, the solution obtained by the solver might be different depending on whether you specifytspanas a two-element vector or as a vector with intermediate points.
The initial and final values intspanare used to calculate the maximum step sizeMaxStep. Therefore, changing the initial or final values intspancan cause the solver to use a different step sequence, which might change the solution.

Example:[1 10]

Example:[1 3 5 7 9 10]

Data Types:single|double

`y0`—Initial conditions
vector

Initial conditions, specified as a vector.y0must be the same length as the vector output ofodefun, so thaty0contains an initial condition for each equation defined inodefun.

Data Types:single|double

`options`—Option structure
structure array

Option structure, specified as a structure array. Use theodeset有趣的ction to create or modify theoptionsstructure. SeeSummary of ODE Optionsfor a list of the options compatible with each solver.

Example:options = odeset('RelTol',1e-5,'Stats','on','OutputFcn',@odeplot)specifies a relative error tolerance of1e-5, turns on the display of solver statistics, and specifies the output function@odeplotto plot the solution as it is computed.

Data Types:struct

Output Arguments

collapse all

`t`— Evaluation points
column vector

Evaluation points, returned as a column vector.

Iftspancontains two elements[t0 tf], thentcontains the internal evaluation points used to perform the integration.
Iftspancontains more than two elements, thentis the same astspan.

`y`— Solutions
array

Solutions, returned as an array. Each row inycorresponds to the solution at the value returned in the corresponding row oft.

`te`— Time of events
column vector

Time of events, returned as a column vector. The event times intecorrespond to the solutions returned inye, andiespecifies which event occurred.

`ye`— Solution at time of events
array

Solution at time of events, returned as an array. The event times intecorrespond to the solutions returned inye, andiespecifies which event occurred.

`ie`— Index of triggered event function
column vector

Index of triggered event function, returned as a column vector. The event times intecorrespond to the solutions returned inye, andiespecifies which event occurred.

`sol`——结构评价
structure array

Structure for evaluation, returned as a structure array. Use this structure with thedeval有趣的ction to evaluate the solution at any point in the interval[t0 tf]. Thesolstructure array always includes these fields:

Structure Field	Description
`sol.x`	Row vector of the steps chosen by the solver.
`sol.y`	Solutions. Each column`sol.y(:,i)`contains the solution at time`sol.x(i)`.
`sol.solver`	Solver name.

Additionally, if you specify theEventsoption ofodesetand events are detected, thensolalso includes these fields:

Structure Field	Description
`sol.xe`	Points when events occurred.`sol.xe(end)`contains the exact point of a terminal event, if any.
`sol.ye`	Solutions that correspond to events in`sol.xe`.
`sol.ie`	Indices into the vector returned by the function specified in the`Events`option. The values indicate which event the solver detected.

Algorithms

ode45is based on an explicit Runge-Kutta (4,5) formula, the Dormand-Prince pair. It is a single-step solver – in computingy(t_n), it needs only the solution at the immediately preceding time point,y(t_n-1)[1],[2].

References

[1] Dormand, J. R. and P. J. Prince, “A family of embedded Runge-Kutta formulae,”J. Comp. Appl. Math., Vol. 6, 1980, pp. 19–26.

[2] Shampine, L. F. and M. W. Reichelt, “The MATLAB ODE Suite,”SIAM Journal on Scientific Computing, Vol. 18, 1997, pp. 1–22.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

Allodesetoption arguments must be constant.
Code generation does not support a constant mass matrix in the options structure. Provide a mass matrix as a function.
You must provide at least the two output argumentsTandY.
Input types must be homogeneous—all double or all single.
Variable-sizing support must be enabled. Code generation requires dynamic memory allocation whentspanhas two elements or you use event functions.

Version History

Introduced before R2006a

ode45

Syntax

Description

Examples

ODE with Single Solution Component

Solve Nonstiff Equation

Pass Extra Parameters to ODE Function

Solve ODE with Multiple Initial Conditions

ODE with Time-Dependent Terms

Evaluate and Extend Solution Structure

Input Arguments

`odefun`—Functions to solve
有趣的ction handle

`tspan`—Interval of integration
vector

`y0`—Initial conditions
vector

`options`—Option structure
structure array

Output Arguments

`t`— Evaluation points
column vector

`y`— Solutions
array

`te`— Time of events
column vector

`ye`— Solution at time of events
array

`ie`— Index of triggered event function
column vector

`sol`——结构评价
structure array

Algorithms

References

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

Version History

See Also

Topics

ode45

Syntax

Description

Examples

ODE with Single Solution Component

Solve Nonstiff Equation

Pass Extra Parameters to ODE Function

Solve ODE with Multiple Initial Conditions

ODE with Time-Dependent Terms

Evaluate and Extend Solution Structure

Input Arguments

odefun—Functions to solve有趣的ction handle

tspan—Interval of integrationvector

y0—Initial conditionsvector

options—Option structurestructure array

Output Arguments

t— Evaluation pointscolumn vector

y— Solutionsarray

te— Time of eventscolumn vector

ye— Solution at time of eventsarray

ie— Index of triggered event functioncolumn vector

sol——结构评价structure array

Algorithms

References

Extended Capabilities

C/C++ Code GenerationGenerate C and C++ code using MATLAB® Coder™.

Version History

See Also

Topics

`odefun`—Functions to solve
有趣的ction handle

`tspan`—Interval of integration
vector

`y0`—Initial conditions
vector

`options`—Option structure
structure array

`t`— Evaluation points
column vector

`y`— Solutions
array

`te`— Time of events
column vector

`ye`— Solution at time of events
array

`ie`— Index of triggered event function
column vector

`sol`——结构评价
structure array

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.