MATLAB Function Reference

bvp4c

Solve boundary value problems for ordinary differential equations

Syntax

sol = bvp4c(odefun,bcfun,solinit) sol = bvp4c(odefun,bcfun,solinit,options) solinit = bvpinit(x, yinit, params)

Arguments

`odefun`	A function handle that evaluates the differential equations . It can have the form dydx = odefun(x,y) dydx = odefun(x,y,parameters) 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 two-point boundary value conditions of the form , `bcfun` can have the form res = bcfun(ya,yb) res = bcfun(ya,yb,parameters) where `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 two-point boundary value conditions

odefun and bcfun are function handles. See Function Handles in the MATLAB^® Programming documentation for more information.

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 bvpinit 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].

sxint = deval(sol,xint)

The structure sol returned by bvp4c has the following fields:

`sol.x`	Mesh selected by `bvp4c`
`sol.y`	Approximation to at the mesh points of `sol.x`
`sol.yp`	Approximation to at the mesh points of `sol.x`
`sol.parameters`	Values returned by `bvp4c` for the unknown parameters, if any
`sol.solver`	'`bvp4c`'

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

yp = odefun(x, y, k)

In the boundary conditions function

bcfun(yleft, yright)

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,

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 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 three-point boundary value problem, type threebvp at the MATLAB command prompt to run a demonstration.

Note The bvp5c function is used exactly like bvp4c, with the exception of the meaning of error tolerances between the two solvers. If S(x) approximates the solution y(x), bvp4c controls the residual |S'(x) - f(x,S(x))|. This controls indirectly the true error |y(x) - S(x)|. bvp5c controls the true error directly. bvp5c is more efficient than bvp4c for small error tolerances.

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 software 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

solinit = bvpinit(linspace(0,4,5),[1 0]);

Now solve the problem with

sol = bvp4c(@twoode,@twobc,solinit);

Evaluate the numerical solution at 100 equally spaced points and plot with

x = linspace(0,4);
y = deval(sol,x);
plot(x,y(1,:));

You can obtain the other solution of this problem with the initial guess

solinit = bvpinit(linspace(0,4,5),[-1 0]);

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 M-file.

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 three-stage Lobatto IIIa formula. This is a collocation formula and the collocation polynomial provides a C¹-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.

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 http://www.mathworks.com/bvp_tutorial