Mathematics

Changing BVP Integration Properties

The default integration properties in the BVP solver bvp4c are selected to handle common problems. In some cases, you can improve solver performance by changing these defaults. To do this, supply bvp4c with one or more property values in an options structure.

• sol = bvp4c(odefun,bcfun,solinit,options)
  

This section:

• Explains how to create, modify, and query an options structure
• Describes the properties that you can use in an options structure

In this and subsequent property tables, the most commonly used property categories are listed first, followed by more advanced categories.

BVP Property Categories  

Properties Category	Property Names
Error control	RelTol, AbsTol
Vectorization	Vectorized
Analytical partial derivatives	FJacobian, BCJacobian
Singular BVPs	SingularTerm
Mesh size	NMax
Output displayed	Stats

Note    For other ways to improve solver efficiency, check Using Continuation to Make a Good Initial Guess and the tutorial, "Solving Boundary Value Problems for Ordinary Differential Equations in MATLAB with bvp4c," available at ftp://ftp.mathworks.com/pub/doc/papers/bvp/.

Creating and Maintaining a BVP Options Structure

The bvpset function creates an options structure that you can supply to bvp4c. You can use bvpget to query the options structure for the value of a specific property.

Creating an Options Structure.   The bvpset function accepts property name/property value pairs using the syntax

• options = bvpset('name1',value1,'name2',value2,...)
  

This creates a structure options in which the named properties have the specified values. Unspecified properties retain their default values. For all properties, it is sufficient to type only the leading characters that uniquely identify the property name. bvpset ignores case for property names.

With no arguments, bvpset displays all property names and their possible values, indicating defaults with braces {}.

Modifying an Existing Options Structure.   To modify an existing options argument, use

• options = bvpset(oldopts,'name1',value1,...)
  

This overwrites any values in oldopts that are specified using name/value pairs. The modified structure is returned as the output argument. In the same way, the command

• options = bvpset(oldopts,newopts)
  

combines the structures oldopts and newopts. In options, any values set in newopts overwrite those in oldopts.

Querying an Options Structure.   The bvpget function extracts a property value from an options structure created with bvpset.

• o = bvpget(options,'name')
  

This returns the value of the specified property, or an empty matrix [] if the property value is unspecified in the options structure.

As with bvpset, it is sufficient to type only the leading characters that uniquely identify the property name; case is ignored for property names.

Error Tolerance Properties

Because bvp4c uses a collocation formula, the numerical solution is based on a mesh of points at which the collocation equations are satisfied. Mesh selection and error control are based on the residual of this solution, such that the computed solution is the exact solution of a perturbed problem

. On each subinterval of the mesh, a norm of the residual in the ith component of the solution, res(i), is estimated and is required to be less than or equal to a tolerance. This tolerance is a function of the relative and absolute tolerances, RelTol and AbsTol, defined by the user.

The following table describes the error tolerance properties. Use bvpset to set these properties.

BVP Error Tolerance Properties  

Property	Value	Description
RelTol	Positive scalar {1e-3}	A relative error tolerance that applies to all components of the residual vector. It is a measure of the residual relative to the size of . The default, 1e-3, corresponds to 0.1% accuracy.
AbsTol	Positive scalar or vector {1e-6}	Absolute error tolerances that apply to the corresponding components of the residual vector. AbsTol(i) is a threshold below which the values of the corresponding components are unimportant. If a scalar value is specified, it applies to all components.

Vectorization

The following table describes the BVP vectorization property. Vectorization of the ODE function used by bvp4c differs from the vectorization used by the ODE solvers:

• For bvp4c, the ODE function must be vectorized with respect to the first argument as well as the second one, so that F([x1 x2 ...],[y1 y2 ...]) returns [F(x1,y1) F(x2,y2) ...].
• bvp4c benefits from vectorization even when analytical Jacobians are provided. For stiff ODE solvers, vectorization is ignored when analytical Jacobians are used.

Use bvpset to set this property.

Vectorization Properties

Property	Value	Description
Vectorized	on | {off}	Set on to inform bvp4c that you have coded the ODE function F so that F([x1 x2 ...],[y1 y2 ...]) returns [F(x1,y1) F(x2,y2) ...]. This allows the solver to reduce the number of function evaluations, and may significantly reduce solution time. With the MATLAB array notation, it is typically an easy matter to vectorize an ODE function. In the shockbvp example shown previously, the shockODE function has been vectorized using colon notation into the subscripts and by using the array multiplication (.*) operator. function dydx = shockODE(x,y,e) pix = pi*x; dydx = [ y(2,:) -x/e.*y(2,:)-pi^2*cos(pix)-pix/e.*sin(pix) ];

Analytical Partial Derivatives

By default, the bvp4c solver approximates all partial derivatives with finite differences. bvp4c can be more efficient if you provide analytical partial derivatives of the differential equations, and analytical partial derivatives, and , of the boundary conditions. If the problem involves unknown parameters, you must also provide partial derivatives, and , with respect to the parameters.

The following table describes the analytical partial derivatives properties. Use bvpset to set these properties.

BVP Analytical Partial Derivative Properties  

Property	Value	Description
FJacobian	Function	The function computes the analytical partial derivatives of . When solving , set this property to @fjac if dfdy = fjac(x,y) evaluates the Jacobian . If the problem involves unknown parameters , [dfdy,dfdp] = fjac(x,y,p) must also return the partial derivative . For problems with constant partial derivatives, set this property to the value of dfdy or to a cell array {dfdy,dfdp}.
BCJacobian	Function	The function computes the analytical partial derivatives of . For boundary conditions , set this property to @bcjac if [dbcdya,dbcdyb] = bcjac(ya,yb) evaluates the partial derivatives , and . If the problem involves unknown parameters , [dbcdya,dbcdyb,dbcdp] = bcjac(ya,yb,p) must also return the partial derivative . For problems with constant partial derivatives, set this property to a cell array {dbcdya,dbcdyb} or {dbcdya,dbcdyb,dbcdp}.

Singular BVPs

bvp4c can solve singular problems of the form

posed on the interval where . For such problems, specify the constant matrix as the value of SingularTerm. For equations of this form, odefun evaluates only the term, where represents unknown parameters, if any.

Singular BVP Property  

Property	Value	Description
SingularTerm	Constant matrix	Singular term of singular BVPs. Set to the constant matrix for equations of the form posed on the interval where .

Mesh Size Property

bvp4c solves a system of algebraic equations to determine the numerical solution to a BVP at each of the mesh points. The size of the algebraic system depends on the number of differential equations (n) and the number of mesh points in the current mesh (N). When the allowed number of mesh points is exhausted, the computation stops, bvp4c displays a warning message and returns the solution it found so far. This solution does not satisfy the error tolerance, but it may provide an excellent initial guess for computations restarted with relaxed error tolerances or an increased value of NMax.

The following table describes the mesh size property. Use bvpset to set this property.

BVP Mesh Size Property  

Property	Value	Description
NMax	positive integer {floor(1000/n)}	Maximum number of mesh points allowed when solving the BVP, where n is the number of differential equations in the problem. The default value of NMax limits the size of the algebraic system to about 1000 equations. For systems of a few differential equations, the default value of NMax should be sufficient to obtain an accurate solution.

Solution Statistic Property

The Stats property lets you view solution statistics.

The following table describes the solution statistics property. Use bvpset to set this property.

BVP Solution Statistic Property  

Property	Value	Description
Stats	on | {off}	Specifies whether statistics about the computations are displayed. If the stats property is on, after solving the problem, bvp4c displays: The number of points in the mesh The maximum residual of the solution The number of times it called the differential equation function odefun to evaluate The number of times it called the boundary condition function bcfun to evaluate

Solving Singular BVPs

Partial Differential Equations