MATLAB Function Reference 
Solve boundary value problems (BVPs) for ordinary differential equations
Syntax
Arguments
odefun 
A function handle that evaluates the differential equations . It can have the form where x is a scalar corresponding to , and y is a column vector corresponding to . parameters is a vector of unknown parameters. The output dydx is a column vector.  
bcfun 
A function handle that computes the residual in the boundary conditions. For twopoint boundary value conditions of the form , bcfun can have the formwhere ya and yb are column vectors corresponding to and . parameters is a vector of unknown parameters. The output res is a column vector.See Multipoint Boundary Value Problems for a description of bcfun for multipoint boundary value problems.  
solinit 
A structure containing the initial guess for a solution. You create solinit using the function bvpinit . solinit has the following fields.  

x 
Ordered nodes of the initial mesh. Boundary conditions are imposed at = solinit.x(1) and = solinit.x(end) . 

y 
Initial guess for the solution such that solinit.y(:,i) is a guess for the solution at the node solinit.x(i) . 

parameters 
Optional. A vector that provides an initial guess for unknown parameters. 

The structure can have any name, but the fields must be named x , y , and parameters . You can form solinit with the helper function bvpinit . See bvpinit for details.  
options 
Optional integration argument. A structure you create using the bvpset function. See bvpset for details. 
Description
sol = bvp4c(odefun,bcfun,solinit)
integrates a system of ordinary differential equations of the form
on the interval [a,b] subject to twopoint boundary value conditions
odefun
and bcfun
are function handles. See Function Handles in the MATLAB Programming documentation for more information.
Parameterizing Functions Called by Function Functions, in the MATLAB mathematics documentation, explains how to provide additional parameters to the function odefun
, as well as the boundary condition function bcfun
, if necessary.
bvp4c
can also solve multipoint boundary value problems. See Multipoint Boundary Value Problems. You can use the function bvpinit
to specify the boundary points, which are stored in the input argument solinit
. See the reference page for bvpint
for more information.
The bvp4c
solver can also find unknown parameters for problems of the form
where corresponds to parameters
. You provide bvp4c
an initial guess for any unknown parameters in solinit.parameters
. The bvp4c
solver returns the final values of these unknown parameters in sol.parameters
.
bvp4c
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 bvp4c
to evaluate the solution at specific points xint
in the interval [a,b]
.
The structure sol
returned by bvp4c
has the following fields:
The structure sol
can have any name, and bvp4c
creates the fields x
, y
, yp
, parameters
, and solver
.
sol = bvp4c(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)
forms the initial guess solinit
with the vector params
of guesses for the unknown parameters.
Singular Boundary Value Problems
bvp4c
solves a class of singular boundary value problems, including problems with unknown parameters p
, of the form
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 and the initial guess should satisfy this condition.
Multipoint Boundary Value Problems
bvp4c
can solve multipoint boundary value problems where
are boundary points in the interval . The points represent interfaces that divide into regions. bvp4c
enumerates the regions from left to right (from a to b), with indices starting from 1. In region k, , bvp4c
evaluates the derivative as
In the boundary conditions function
yleft(:, k) is the solution at the left boundary of . Similarly, yright(:, k)
is the solution at the right boundary of region k. In particular,
For example, if there just one equation and the boundary points are 0 < 1 < 2, to specify the boundary conditions
yleft
and yright
have the following values.
The boundary condition function bcfun
has the form
function res = bc(yleft, yright) res = [ yleft(1)  4 yright(1)  4.5 yleft(2)  5 yright(2)  5.5];
When you create an initial guess with
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 bpv4c
, 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.
For an example of solving a threepoint boundary value problem, enter
Examples
Example 1. Boundary value problems can have multiple solutions and one purpose of the initial guess is to indicate which solution you want. The second order differential equation
has exactly two solutions that satisfy the boundary conditions
Prior to solving this problem with bvp4c
, you must write the differential equation as a system of two first order ODEs
Here and . This system has the required form
The function and the boundary conditions are coded in MATLAB as functions twoode
and twobc
.
function dydx = twoode(x,y) dydx = [ y(2) abs(y(1))]; function res = twobc(ya,yb) res = [ ya(1) yb(1) + 2];
Form a guess structure consisting of an initial mesh of five equally spaced points in [0,4] and a guess of constant values and with the command
Evaluate the numerical solution at 100 equally spaced points and plot with
You can obtain the other solution of this problem with the initial guess
Example 2. This boundary value problem involves an unknown parameter. The task is to compute the fourth () eigenvalue of Mathieu's equation
Because the unknown parameter is present, this second order differential equation is subject to three boundary conditions
It is convenient to use subfunctions to place all the functions required by bvp4c
in a single Mfile.
function mat4bvp lambda = 15; solinit = bvpinit(linspace(0,pi,10),@mat4init,lambda); sol = bvp4c(@mat4ode,@mat4bc,solinit); fprintf('The fourth eigenvalue is approximately %7.3f.\n',... sol.parameters) xint = linspace(0,pi); Sxint = deval(sol,xint); plot(xint,Sxint(1,:)) axis([0 pi 1 1.1]) title('Eigenfunction of Mathieu''s equation.') xlabel('x') ylabel('solution y') %  function dydx = mat4ode(x,y,lambda) q = 5; dydx = [ y(2) (lambda  2*q*cos(2*x))*y(1) ]; %  function res = mat4bc(ya,yb,lambda) res = [ ya(2) yb(2) ya(1)1 ]; %  function yinit = mat4init(x) yinit = [ cos(4*x) 4*sin(4*x) ];
The differential equation (converted to a first order system) and the boundary conditions are coded as subfunctions mat4ode
and mat4bc
, respectively. Because unknown parameters are present, these functions must accept three input arguments, even though some of the arguments are not used.
The guess structure solinit
is formed with bvpinit
. An initial guess for the solution is supplied in the form of a function mat4init
. We chose because it satisfies the boundary conditions and has the correct qualitative behavior (the correct number of sign changes). In the call to bvpinit
, the third argument (lambda = 15
) provides an initial guess for the unknown parameter .
After the problem is solved with bvp4c
, the field sol.parameters
returns the value , and the plot shows the eigenfunction associated with this eigenvalue.
Algorithms
bvp4c
is a finite difference code that implements the threestage Lobatto IIIa formula. This is a collocation formula and the collocation polynomial provides a C^{1}continuous solution that is fourth order accurate uniformly in [a,b]
. Mesh selection and error control are based on the residual of the continuous solution.
See Also
function_handle
(@
), bvpget
, bvpinit
, bvpset
, deval
References
[1] Shampine, L.F., M.W. Reichelt, and J. Kierzenka, "Solving Boundary Value
Problems for Ordinary Differential Equations in MATLAB with bvp4c,"
available at www.mathworks.com/bvp_tutorial
.
builtin  bvpget 
© 19942005 The MathWorks, Inc.