naginterfaces.library.ode.bvp_​coll_​sys

naginterfaces.library.ode.bvp_coll_sys(n, cf, bc, x0, x1, k1, kp, comm, data=None)[source]

bvp_coll_sys solves a regular linear two-point boundary value problem for a system of ordinary differential equations by Chebyshev series using collocation and least squares.

For full information please refer to the NAG Library document for d02jb

https://www.nag.com/numeric/nl/nagdoc_28.6/flhtml/d02/d02jbf.html

Parameters
nint

, the order of the system of differential equations.

cfcallable retval = cf(i, j, x, data=None)

defines the system of differential equations (see Notes).

It must return the value of a coefficient function , of , at a given point , or of a right-hand side function if .

Parameters
iint

Indicate the function to be evaluated, namely if , or if .

, .

jint

Indicate the function to be evaluated, namely if , or if .

, .

xfloat

The point at which the function is to be evaluated.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns
retvalfloat

The value of a coefficient function , of , at a given point , or of a right-hand side function if .

bccallable (j, rhs) = bc(i, data=None)

defines the boundary conditions, which have the form or .

The boundary conditions may be specified in any order.

Parameters
iint

The index of the boundary condition to be defined.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns
jint

Must be set to if the th boundary condition is , or to if it is .

must not be set to the same value for two different values of .

rhsfloat

The value .

x0float

The left- and right-hand boundaries, and , respectively.

x1float

The left- and right-hand boundaries, and , respectively.

k1int

The number of coefficients to be returned in the Chebyshev series representation of the components of the solution (hence the degree of the polynomial approximation is ).

kpint

The number of collocation points to be used.

commdict, communication object, modified in place

Communication structure.

On initial entry: need not be set.

dataarbitrary, optional

User-communication data for callback functions.

Returns
cfloat, ndarray, shape

The computed Chebyshev coefficients of the th component of the solution, ; that is, the computed solution is:

where is the th Chebyshev polynomial of the first kind, and denotes that the first coefficient, , is halved.

Raises
NagValueError
(errno )

On entry, and .

Constraint: .

(errno )

On entry, .

Constraint: .

(errno )

On entry, and .

Constraint: .

(errno )

On entry, .

Constraint: .

(errno )

Either the boundary conditions are not linearly independent, or the coefficient matrix is rank deficient. Increasing the number of collocation points may overcome this latter problem.

(errno )

Iterative refinement in the least squares solution has failed to converge. The coefficient matrix is too ill-conditioned.

Notes

No equivalent traditional C interface for this routine exists in the NAG Library.

bvp_coll_sys calculates the solution of a regular two-point boundary value problem for a regular linear th-order system of first-order ordinary differential equations as a Chebyshev series in the interval . The differential equation

is defined by , and the boundary conditions at the points and are defined by .

You specify the degree of Chebyshev series required, , and the number of collocation points, . The function sets up a system of linear equations for the Chebyshev coefficients, equations for each collocation point and one for each boundary condition. The boundary conditions are solved exactly, and the remaining equations are then solved by a least squares method. The result produced is a set of coefficients for a Chebyshev series solution for each component of the solution of the system of differential equations on an interval normalized to .

fit.dim1_cheb_eval2 can be used to evaluate the components of the solution at any point on the interval . fit.dim1_cheb_deriv followed by fit.dim1_cheb_eval2 can be used to evaluate their derivatives.

References

Picken, S M, 1970, Algorithms for the solution of differential equations in Chebyshev-series by the selected points method, Report Math. 94, National Physical Laboratory