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bvp5c

Solve boundary value problems for ordinary differential equations

Syntax

sol = bvp5c(odefun,bcfun,solinit)
sol = bvp5c(odefun,bcfun,solinit,options)
solinit = bvpinit(x, yinit, params)

Arguments

odefun

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 x and a column vector y, odefun(x,y) must return a column vector, dydx, representing f(x,y). parameters is a vector of unknown parameters.

bcfun

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)), bcfun can have the form

res = bcfun(ya,yb)
res = bcfun(ya,yb,parameters)

where ya and yb are column vectors corresponding to y(a) and y(b). parameters is a vector of unknown parameters. The output res is a column vector.

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 a = solinit.x(1) and b = 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 = 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 the function_handle reference page 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:

sol.x

Mesh selected by bvp5c

sol.y

Approximation to y(x) at the mesh points of sol.x

sol.parameters

Values returned by bvp5c for the unknown parameters, if any

sol.solver

'bvp5c'

sol.stats

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) forms the initial guess solinit with the vector params of guesses for the unknown parameters.

Singular Boundary Value Problems

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.

Multipoint Boundary Value Problems

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.

More About

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Algorithms

bvp5c is a finite difference code that implements the four-stage Lobatto IIIa formula. This is a collocation formula and the collocation polynomial provides a C1-continuous solution that is fifth-order accurate uniformly in [a,b]. The formula is implemented as an implicit Runge-Kutta formula. bvp5c solves the algebraic equations directly; bvp4c uses analytical condensation. bvp4c handles unknown parameters directly; while bvp5c augments the system with trivial differential equations for unknown parameters.

References

[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 that this tutorial uses the bvp4c function, however in most cases the solvers can be used interchangeably.

See Also

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