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bvpset

Create or alter options structure of boundary value problem

Syntax

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

Description

options = bvpset('name1',value1,'name2',value2,...) creates a structure options that you can supply to the boundary value problem solver bvp4c, in which the named properties have the specified values. Any unspecified properties retain their default values. For all properties, it is sufficient to type only the leading characters that uniquely identify the property. bvpset ignores case for property names.

options = bvpset(oldopts,'name1',value1,...) alters an existing options structure oldopts. This overwrites any values in oldopts that are specified using name/value pairs and returns the modified structure as the output argument.

options = bvpset(oldopts,newopts) combines an existing options structure oldopts with a new options structure newopts. Any values set in newopts overwrite the corresponding values in oldopts.

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

You can use the function bvpget to query the options structure for the value of a specific property.

BVP Properties

bvpset enables you to specify properties for the boundary value problem solver bvp4c. There are several categories of properties that you can set:

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 S(x) is the exact solution of a perturbed problem S′(x) = f(x,S(x)) + res(x). 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.

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 f(x,y). The default, 1e-3, corresponds to 0.1% accuracy.

The computed solution S(x) is the exact solution of S′(x) = F(x,S(x)) + res(x). On each subinterval of the mesh, the residual res(x) satisfies

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.

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) ...]. That is, your ODE function can pass to the solver a whole array of column vectors at once. This enables 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 ∂f/∂y of the differential equations, and analytical partial derivatives, ∂bc/∂ya and ∂bc/∂yb, of the boundary conditions. If the problem involves unknown parameters, you must also provide partial derivatives, ∂f/∂p and ∂bc/∂p, with respect to the parameters.

The following table describes the analytical partial derivatives properties.

BVP Analytical Partial Derivative Properties

Property

Value

Description

FJacobian

Function handle

A function handle that computes the analytical partial derivatives of f(x,y). When solving y′ = f(x,y), set this property to @fjac if dfdy = fjac(x,y) evaluates the Jacobian ∂f/∂y. If the problem involves unknown parameters p, [dfdy,dfdp] = fjac(x,y,p) must also return the partial derivative ∂f/∂p. For problems with constant partial derivatives, set this property to the value of dfdy or to a cell array {dfdy,dfdp}.

BCJacobian

Function handle

A function handle that computes the analytical partial derivatives of bc(ya,yb). For boundary conditions bc(ya,yb), set this property to @bcjac if [dbcdya,dbcdyb] = bcjac(ya,yb) evaluates the partial derivatives ∂bc/∂ya, and ∂bc/∂yb. If the problem involves unknown parameters p, [dbcdya,dbcdyb,dbcdp] = bcjac(ya,yb,p) must also return the partial derivative ∂bc/∂p. 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 [0,b] where b > 0. For such problems, specify the constant matrix S as the value of SingularTerm. For equations of this form, odefun evaluates only the f(x,y,p) term, where p represents unknown parameters, if any.

Singular BVP Property

Property

Value

Description

SingularTerm

Constant matrix

Singular term of singular BVPs. Set to the constant matrix S for equations of the form

posed on the interval [0,b] where b > 0.

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.

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.

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 f(x,y)

• The number of times it called the boundary condition function bcfun to evaluate bc(y(a),y(b))

Examples

To create an options structure that changes the relative error tolerance of bvp4c from the default value of 1e-3 to 1e-4, enter

`options = bvpset('RelTol',1e-4);`

To recover the value of 'RelTol' from options, enter

```bvpget(options,'RelTol')

ans =

1.0000e-004```