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Solve fully implicit differential equations, variable order method
[T,Y] = ode15i(odefun,tspan,y0,yp0)
[T,Y] = ode15i(odefun,tspan,y0,yp0,options)
[T,Y,TE,YE,IE] = ode15i(odefun,tspan,y0,yp0,options...)
sol = ode15i(odefun,[t0 tfinal],y0,yp0,...)
The following table lists the input arguments for ode15i.
A function handle that evaluates the left side of the differential equations, which are of the form f(t,y,y′) = 0. | |
A vector specifying the interval of integration, [t0,tf]. To obtain solutions at specific times (all increasing or all decreasing), use tspan = [t0,t1,...,tf]. | |
Vectors of initial conditions for y and y′ respectively. | |
Optional integration argument created using the odeset function. See odeset for details. |
The following table lists the output arguments for ode15i.
T | Column vector of time points |
Y | Solution array. Each row in y corresponds to the solution at a time returned in the corresponding row of t. |
[T,Y] = ode15i(odefun,tspan,y0,yp0) with tspan = [t0 tf] integrates the system of differential equations f(t,y,y′) = 0 from time t0 to tf with initial conditions y0 and yp0. odefun is a function handle. Function ode15i solves ODEs and DAEs of index 1. The initial conditions must be consistent, meaning that f(t0,y0,yp0) = 0. You can use the function decic to compute consistent initial conditions close to guessed values. Function odefun(t,y,yp), for a scalar t and column vectors y and yp, must return a column vector corresponding to f(t,y,y′). Each row in the solution array Y corresponds to a time returned in the column vector T. To obtain solutions at specific times t0,t1,...,tf (all increasing or all decreasing), use tspan = [t0,t1,...,tf].
Parameterizing Functions explains how to provide additional parameters to the function odefun, if necessary.
[T,Y] = ode15i(odefun,tspan,y0,yp0,options) solves as above with default integration parameters replaced by property values specified in options, an argument created with the odeset function. Commonly used options include a scalar relative error tolerance RelTol (1e-3 by default) and a vector of absolute error tolerances AbsTol (all components 1e-6 by default). See odeset for details.
[T,Y,TE,YE,IE] = ode15i(odefun,tspan,y0,yp0,options...) with the 'Events' property in options set to a function events, solves as above while also finding where functions of (t,y,y′), called event functions, are zero. The function events is of the form [value,isterminal,direction] = events(t,y,yp) and includes the necessary event functions. Code the function events so that the ith element of each output vector corresponds to the ith event. For the ith event function in events:
value(i) is the value of the function.
isterminal(i) = 1 if the integration is to terminate at a zero of this event function and 0 otherwise.
direction(i) = 0 if all zeros are to be computed (the default), +1 if only the zeros where the event function increases, and -1 if only the zeros where the event function decreases.
Output TE is a column vector of times at which events occur. Rows of YE are the corresponding solutions, and indices in vector IE specify which event occurred. See Integrator Options in the MATLAB^{®} Mathematics documentation for more information.
sol = ode15i(odefun,[t0 tfinal],y0,yp0,...) returns a structure that can be used with deval to evaluate the solution at any point between t0 and tfinal. The structure sol always includes these fields:
Steps chosen by the solver. If you specify the Events option and a terminal event is detected, sol.x(end) contains the end of the step at which the event occurred. | |
Each column sol.y(:,i) contains the solution at sol.x(i). |
If you specify the Events option and events are detected, sol also includes these fields:
Points at which events, if any, occurred. sol.xe(end) contains the exact point of a terminal event, if any. | |
Solutions that correspond to events in sol.xe. | |
Indices into the vector returned by the function specified in the Events option. The values indicate which event the solver detected. |
ode15i accepts the following parameters in options. For more information, see odeset and Changing ODE Integration Properties in the MATLAB Mathematics documentation.
Error control | RelTol, AbsTol, NormControl |
Solver output | OutputFcn, OutputSel, Refine, Stats |
Event location | Events |
Step size | MaxStep, InitialStep |
Jacobian matrix | Jacobian, JPattern, Vectorized |
If you specify an output function as the value of the OutputFcn property, the solver calls it with the computed solution after each time step. Four output functions are provided: odeplot, odephas2, odephas3, odeprint. When you call the solver with no output arguments, it calls the default odeplot to plot the solution as it is computed. odephas2 and odephas3 produce two- and three-dimensional phase plane plots, respectively. odeprint displays the solution components on the screen. By default, the ODE solver passes all components of the solution to the output function. You can pass only specific components by providing a vector of indices as the value of the OutputSel property. For example, if you call the solver with no output arguments and set the value of OutputSel to [1,3], the solver plots solution components 1 and 3 as they are computed.
The Jacobian matrices ∂f/∂y and ∂f/∂y′ are critical to reliability and efficiency. You can provide these matrices as one of the following:
Function of the form [dfdy,dfdyp] = FJAC(t,y,yp) that computes the Jacobian matrices. If FJAC returns an empty matrix [] for either dfdy or dfdyp, then ode15i approximates that matrix by finite differences.
Cell array of two constant matrices {dfdy,dfdyp}, either of which could be empty.
Use odeset to set the Jacobian option to the function or cell array. If you do not set the Jacobian option, ode15i approximates both Jacobian matrices by finite differences.
For ode15i, Vectorized is a two-element cell array. Set the first element to 'on' if odefun(t,[y1,y2,...],yp) returns [odefun(t,y1,yp),odefun(t,y2,yp),...]. Set the second element to 'on' if odefun(t,y,[yp1,yp2,...]) returns [odefun(t,y,yp1),odefun(t,y,yp2),...]. The default value of Vectorized is {'off','off'}.
For ode15i, JPattern is also a two-element sparse matrix cell array. If ∂f/∂y or ∂f/∂y′ is a sparse matrix, set JPattern to the sparsity patterns, {SPDY,SPDYP}. A sparsity pattern of ∂f/∂y is a sparse matrix SPDY with SPDY(i,j) = 1 if component i of f(t,y,yp) depends on component j of y, and 0 otherwise. Use SPDY = [] to indicate that ∂f/∂y is a full matrix. Similarly for ∂f/∂y′ and SPDYP. The default value of JPattern is {[],[]}.
This example uses a helper function, decic, to hold fixed the initial value for y(t0) and compute a consistent initial value for y'(t0) for the Weissinger implicit ODE. The Weissinger function evaluates the residual of the implicit ODE.
t0 = 1; y0 = sqrt(3/2); yp0 = 0; [y0,yp0] = decic(@weissinger,t0,y0,1,yp0,0);
Use ode15i to solve the ODE, and then plot the numerical solution, y, against the analytical solution, ytrue.
[t,y] = ode15i(@weissinger,[1 10],y0,yp0);
ytrue = sqrt(t.^2 + 0.5);
plot(t,y,t,ytrue,'o')
The files, ihb1dae.m and iburgersode.m, are examples of implicit ODEs.