odextend

Extend solution to ODE

Syntax

solext = odextend(sol,odefun,tfinal)

solext = odextend(sol,[],tfinal)

solext = odextend(sol,odefun,tfinal,y0)

solext = odextend(sol,odefun,tfinal,y0,options)

Description

solext = odextend(sol,odefun,tfinal) extends the solution sol by integrating odefun from sol.x(end) to tfinal, using the same ODE solver that created sol. The function odefun can be different than the original function used to compute sol. The lower bound for the independent variable in solext is the same as in sol, that is, sol.x(1). By default, odextend uses:

• The initial conditions y = sol.y(:,end) for the subsequent integration.

• The same integration properties and additional input arguments that the ODE solver originally used to compute sol. This information is stored in the solution structure sol and is later returned in solext. Unless you want to change these values, you do not need to pass them to odextend.

solext = odextend(sol,[],tfinal) extends the solution to the same ODE function that was solved to obtain sol.

solext = odextend(sol,odefun,tfinal,y0) specifies new initial conditions y0 for the extended integration instead of using sol.y(:,end).

For the ode15i solver: y0 must be an m-by-2 matrix containing column vectors of initial conditions for the solution components and their derivatives, y0 = [yinit ypinit].

solext = odextend(sol,odefun,tfinal,y0,options) uses the integration options defined by options, which is an argument created using the odeset function. The specified options override the options that the ODE solver originally used to compute sol. You can optionally specify y0 = [] to use default initial conditions.

Examples

Evaluate and Extend Solution Structure

The van der Pol equation is a second order ODE

$$y_1^{\prime \prime }-\mu (1-y_1^2)y_1^{\prime }+y_1=0.$$

Solve the van der Pol equation with $$\mu =1$$ using ode45. The function vdp1.m ships 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 ... ] (1x60 double)
          y: [2x60 double]
      stats: [1x1 struct]
      idata: [1x1 struct]

Use linspace to generate 250 points in the interval [0 20]. Evaluate the solution at these points using deval.

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

Plot the first component of the solution.

plot(x,y(1,:))

Extend the solution to $$t_f=35$$ using odextend and add the result to the original plot.

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

Input Arguments

sol — Solution structure

structure

Solution structure, specified as a structure returned by an ODE solver. When you call an ODE solver with a single output argument, it returns a solution structure.

Example: sol = ode45(myode,tspan,y0)

Data Types: struct

odefun — Function to solve

[] | function handle

Function to solve, specified as a function handle. Use this input to extend the solution using a new or modified ODE function. To continue using the original ODE function used to create the solution structure sol, specify odefun as an empty input [].

Data Types: function_handle

tfinal — Final integration time

scalar

Final integration time, specified as a scalar.

Data Types: single | double

y0 — Initial conditions

[] | scalar | vector | matrix

Initial conditions, specified as a scalar, vector, or matrix. By default odextend uses the initial conditions y = sol.y(:,end) to extend the integration. Use this input to specify new initial conditions for the extended integration.

For the ode15i solver: y0 must be an m-by-2 matrix containing column vectors of initial conditions for the solution components and their derivatives, y0 = [yinit ypinit].

Data Types: single | double

options — Options structure

structure

Options structure. By default, odextend uses the same options and additional inputs as the ODE solver originally used to compute sol. Use this input to specify a new options structure that overrides the options used to create sol.

Use the odeset function to create or modify an ODE options structure.

Data Types: struct

Output Arguments

solext — Extended solution

structure

Extended solution, returned as a structure. Use this structure with the deval function to evaluate the solution at any point in the interval [t0 tf]. The solext structure array always includes these fields:

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

Additionally, if you specify the Events option and events are detected, then solext also 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.

Version History

Introduced before R2006a