Solve boundary value problems for ordinary differential equations
sol = bvp5c(odefun,bcfun,solinit)
sol = bvp5c(odefun,bcfun,solinit,options)solinit = bvpinit(x, yinit,
params)
| A function handle that evaluates the differential equations f(x,y). It can have the form dydx = odefun(x,y) dydx = odefun(x,y,parameters) For a scalar | |
| A function handle that computes
the residual in the boundary conditions. For two-point boundary value
conditions of the form bc(y(a),y(b)), res = bcfun(ya,yb) res = bcfun(ya,yb,parameters) where | |
| A structure containing the
initial guess for a solution. You create | |
| Ordered nodes of the initial mesh. Boundary conditions
are imposed at a = | |
| Initial guess for the solution such that | |
| Optional. | |
The structure can have any
name, but the fields must be named | ||
| Optional integration argument.
A structure you create using the | |
sol = bvp5c(odefun,bcfun,solinit) integrates
a system of ordinary differential equations of the form
y′ = f(x,y)
on the interval [a,b] subject to two-point boundary value conditions
bc(y(a),y(b)) = 0
odefun and bcfun are function
handles. See Create Function Handle for more information.
Parameterizing Functions explains how to provide additional
parameters to the function odefun, as well as the
boundary condition function bcfun, if necessary.
You can use the function bvpinit to
specify the boundary points, which are stored in the input argument solinit.
The bvp5c solver can also find unknown
parameters p for problems of the form
y′ = f(x,y,p)
0 = bc(y(a),y(b),p)
where p corresponds to parameters.
You provide bvp5c an initial guess for any unknown
parameters in solinit.parameters. The bvp5c solver
returns the final values of these unknown parameters in sol.parameters.
bvp5c produces a solution that is continuous
on [a,b] and has a continuous first derivative
there. Use the function deval and
the output sol of bvp5c to
evaluate the solution at specific points xint in
the interval [a,b].
sxint = deval(sol,xint)
The structure sol returned by bvp5c has
the following fields:
| Mesh selected by |
| Approximation to y(x)
at the mesh points of |
| Values returned by |
|
|
| Computational cost statistics (also displayed when the stats option
is set with bvpset). |
The structure sol can have any name, and bvp5c creates
the fields x, y, parameters,
and solver.
sol = bvp5c(odefun,bcfun,solinit,options) solves
as above with default integration properties replaced by the values
in options, a structure created with the bvpset function.
See bvpset for details.
solinit = bvpinit(x, yinit,
params)solinit with
the vector params of guesses for the unknown parameters.
bvp5c solves a class of singular boundary
value problems, including problems with unknown parameters p,
of the form
y′ = S· y/x + f(x,y,p)
0 = bc(y(0),y(b),p)
The interval is required to be [0, b] with
b > 0. Often such problems arise when computing a smooth solution
of ODEs that result from partial differential equations (PDEs) due
to cylindrical or spherical symmetry. For singular problems, you specify
the (constant) matrix S as the value of the 'SingularTerm' option
of bvpset, and odefun evaluates
only f(x,y,p).
The boundary conditions must be consistent with the necessary condition S· y(0)
= 0 and the initial guess should satisfy this condition.
bvp5c can solve multipoint boundary value
problems where a = a0 < a1 < a2 <
... < an = b are
boundary points in the interval [a,b].
The points a1,a2,
... ,an–1 represent
interfaces that divide [a,b]
into regions. bvp5c enumerates the regions from
left to right (from a to b),
with indices starting from 1. In region k, [ak–1,ak], bvp5c evaluates
the derivative as
yp = odefun(x,y,k)
In the boundary conditions function
bcfun(yleft,yright)
yleft(:, k) is the solution at the left boundary of [ak–1,ak].
Similarly, yright(:,k) is the solution at the right
boundary of region k. In particular,
yleft(:,1) = y(a)
and
yright(:,end) = y(b)
When you create an initial guess with
solinit = bvpinit(xinit,yinit),
use double entries in xinit for each interface
point. See the reference page for bvpinit for
more information.
If yinit is a function, bvpinit calls y
= yinit(x, k) to get an initial guess for the solution at x in
region k. In the solution structure sol returned
by bvp5c, sol.x has double entries
for each interface point. The corresponding columns of sol.y contain
the left and right solution at the interface, respectively.
To see an example of that solves a three-point boundary value
problem, type threebvp at the MATLAB® command
prompt.
Note:
The |
[1] Shampine, L.F., M.W. Reichelt, and J.
Kierzenka "Solving Boundary Value Problems for Ordinary Differential
Equations in MATLAB with bvp4c" http://www.mathworks.com/bvp_tutorial.
Note:
This tutorial uses the |