naginterfaces.library.ode.bvp_​coll_​sys

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

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://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/d02/d02jbf.html

Parameters

nint

$n$, the order of the system of differential equations.

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

$cf$ defines the system of differential equations (see Notes).

It must return the value of a coefficient function $a_{i,j}\left(x\right)$, of $A$, at a given point $x$, or of a right-hand side function $r_i\left(x\right)$ if $j=0$.

Parameters

iint

Indicate the function to be evaluated, namely $a_{i,j}\left(x\right)$ if $1\le j\le n$, or $r_i\left(x\right)$ if $j=0$.

$1\le i\le n$, $0\le j\le n$.

jint

Indicate the function to be evaluated, namely $a_{i,j}\left(x\right)$ if $1\le j\le n$, or $r_i\left(x\right)$ if $j=0$.

$1\le i\le n$, $0\le j\le n$.

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 $a_{i,j}\left(x\right)$, of $A$, at a given point $x$, or of a right-hand side function $r_i\left(x\right)$ if $j=0$.

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

$bc$ defines the $n$ boundary conditions, which have the form $y_k\left(x_0\right)=s$ or $y_k\left(x_1\right)=s$.

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 $-k$ if the $i$th boundary condition is $y_k\left(x_0\right)=s$, or to $+k$ if it is $y_k\left(x_1\right)=s$.

$j$ must not be set to the same value $k$ for two different values of $i$.

rhsfloat

The value $s$.

x0float

The left- and right-hand boundaries, $x_0$ and $x_1$, respectively.

x1float

The left- and right-hand boundaries, $x_0$ and $x_1$, 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 $k1-1$).

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 $(k1,n)$

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

$$y_k={\sum ^{\prime }}_{i=1}^{k1}c[i-1,k-1]T_{i-1}\left(x\right)\text{,}1\le k\le n$$

where $T_i\left(x\right)$ is the $i$th Chebyshev polynomial of the first kind, and ${\sum }^{\prime }$ denotes that the first coefficient, $c[0,k-1]$, is halved.

Raises

NagValueError

(errno $1$)

On entry, $kp=⟨value⟩$ and $k1=⟨value⟩$.

Constraint: $kp+1\ge k1$.

(errno $1$)

On entry, $k1=⟨value⟩$.

Constraint: $k1\ge 2$.

(errno $1$)

On entry, $x1=⟨value⟩$ and $x0=⟨value⟩$.

Constraint: $x1>x0$.

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 1$.

(errno $3$)

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 $4$)

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 $n$th-order system of first-order ordinary differential equations as a Chebyshev series in the interval $(x_0,x_1)$. The differential equation

$$y^{\prime }=A\left(x\right)y+r\left(x\right)$$

is defined by $cf$, and the boundary conditions at the points $x_0$ and $x_1$ are defined by $bc$.

You specify the degree of Chebyshev series required, $k1-1$, and the number of collocation points, $kp$. The function sets up a system of linear equations for the Chebyshev coefficients, $n$ 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 $(-1,1)$.

fit.dim1_cheb_eval2 can be used to evaluate the components of the solution at any point on the interval $(x_0,x_1)$. 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