d02tyc and its associated functions (d02tlc, d02tvc, d02txc and d02tzc) solve the two-point boundary value problem for a nonlinear mixed order system of ordinary differential equations

\begin{array}{rcl} y_{1}^{(m_{1})} (x) & = & f_{1} (x, y_{1}, y_{1}^{(1)}, \dots, y_{1}^{(m_{1} - 1)}, y_{2}, \dots, y_{n}^{(m_{n} - 1)}) \\ y_{2}^{(m_{2})} (x) & = & f_{2} (x, y_{1}, y_{1}^{(1)}, \dots, y_{1}^{(m_{1} - 1)}, y_{2}, \dots, y_{n}^{(m_{n} - 1)}) \\ ⋮ \\ y_{n}^{(m_{n})} (x) & = & f_{n} (x, y_{1}, y_{1}^{(1)}, \dots, y_{1}^{(m_{1} - 1)}, y_{2}, \dots, y_{n}^{(m_{n} - 1)}) \end{array}

over an interval

[a, b]

subject to

p

(

> 0

) nonlinear boundary conditions at

a

and

q

(

> 0

) nonlinear boundary conditions at

b

, where

p + q = \sum_{i = 1}^{n} m_{i}

. Note that

y_{i}^{(m)} (x)

is the

m

th derivative of the

i

th solution component. Hence

y_{i}^{(0)} (x) = y_{i} (x)

. The left boundary conditions at

a

are defined as

g_{i} (z (y (a))) = 0, i = 1, 2, \dots, p,

and the right boundary conditions at

b

{\bar{g}}_{j} (z (y (b))) = 0, j = 1, 2, \dots, q,

where

y = (y_{1}, y_{2}, \dots, y_{n})

and

z (y (x)) = (y_{1} (x), y_{1}^{(1)} (x), \dots, y_{1}^{(m_{1} - 1)} (x), y_{2} (x), \dots, y_{n}^{(m_{n} - 1)} (x)) .

First, d02tvc must be called to specify the initial mesh, error requirements and other details. Then, d02tlc can be used to solve the boundary value problem. After successful computation, d02tzc can be used to ascertain details about the final mesh and other details of the solution procedure, and d02tyc can be used to compute the approximate solution anywhere on the interval

[a, b]

using interpolation.

The functions are based on modified versions of the codes COLSYS and COLNEW (see Ascher et al. (1979) and Ascher and Bader (1987)). A comprehensive treatment of the numerical solution of boundary value problems can be found in Ascher et al. (1988) and Keller (1992).

4 References

Ascher U M and Bader G (1987) A new basis implementation for a mixed order boundary value ODE solver SIAM J. Sci. Stat. Comput. 8 483–500

Ascher U M, Christiansen J and Russell R D (1979) A collocation solver for mixed order systems of boundary value problems Math. Comput. 33 659–679

Ascher U M, Mattheij R M M and Russell R D (1988) Numerical Solution of Boundary Value Problems for Ordinary Differential Equations Prentice–Hall

Grossman C (1992) Enclosures of the solution of the Thomas–Fermi equation by monotone discretization J. Comput. Phys. 98 26–32

Keller H B (1992) Numerical Methods for Two-point Boundary-value Problems Dover, New York

5 Arguments

1: $x$ – double Input: On entry: $x$ , the independent variable.

Constraint: $a \leq x \leq b$ , i.e., not outside the range of the original mesh specified in the initialization call to d02tvc.
2: $y [neq \times mmax]$ – double Output: Note: where $Y (i, j)$ appears in this document, it refers to the array element $y [j \times neq + i - 1]$ .

On exit: $Y (i, j)$ contains an approximation to $y_{i}^{(j)} (x)$ , for $i = 1, 2, \dots, neq$ and $j = 0, 1, \dots, m_{i} - 1$ . The remaining elements of y (where $m_{i} < mmax$ ) are initialized to $0.0$ .
3: $neq$ – Integer Input: On entry: the number of differential equations.

Constraint: $neq$ must be the same value as supplied to d02tvc.
4: $mmax$ – Integer Input: On entry: the maximal order of the differential equations, $\max (m_{i})$ , for $i = 1, 2, \dots, neq$ .

Constraint: $mmax$ must contain the maximum value of the components of the argument m as supplied to d02tvc.
5: $rcomm [\dim]$ – double Communication Array: Note: the dimension, $\dim$ , of this array is dictated by the requirements of associated functions that must have been previously called. This array MUST be the same array passed as argument rcomm in the previous call to d02tlc.

On entry: this must be the same array as supplied to d02tlc and must remain unchanged between calls.

On exit: contains information about the solution for use on subsequent calls to associated functions.
6: $icomm [\dim]$ – const Integer Communication Array: Note: the dimension, $\dim$ , of this array is dictated by the requirements of associated functions that must have been previously called. This array MUST be the same array passed as argument icomm in the previous call to d02tlc.

On entry: this must be the same array as supplied to d02tlc and must remain unchanged between calls.

On exit: contains information about the solution for use on subsequent calls to associated functions.
7: $fail$ – NagError * Input/Output: The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_CONVERGENCE_SOL: The solver function did not produce any results suitable for interpolation.
NE_INT_2: On entry, $mmax = 〈value〉$ and $\max (m [i]) = 〈value〉$ in d02tvc.
Constraint: $mmax = \max (m [i])$ in d02tvc.
NE_INT_CHANGED: On entry, $neq = 〈value〉$ and $neq = 〈value〉$ in d02tvc.
Constraint: $neq = neq$ in d02tvc.
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_MISSING_CALL: The solver function does not appear to have been called.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_REAL: On entry, $x = 〈value〉$ .
Constraint: $x \leq 〈value〉$ .

On entry, $x = 〈value〉$ .
Constraint: $x \geq 〈value〉$ .
NW_NOT_CONVERGED: The solver function did not converge to a suitable solution.
A converged intermediate solution has been used.
Interpolated values should be treated with caution.
NW_TOO_MUCH_ACC_REQUESTED: The solver function did not satisfy the error requirements.
Interpolated values should be treated with caution.

7 Accuracy

If d02tyc returns the value

fail . code =

NE_NOERROR, the computed values of the solution components

y_{i}

should be of similar accuracy to that specified by the argument tols of d02tvc. Note that during the solution process the error in the derivatives

y_{i}^{(j)}

, for

j = 1, 2, \dots, m_{i} - 1

, has not been controlled and that the derivative values returned by d02tyc are computed via differentiation of the piecewise polynomial approximation to

y_{i}

. See also Section 9.

8 Parallelism and Performance

d02tyc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

If d02tyc returns the value fail.code

= NW_NOT_CONVERGED

, then the accuracy of the interpolated values may be proportional to the quantity ermx as returned by d02tzc.

If d02tlc returned a value for fail.code other than

fail . code =

NE_NOERROR, then nothing can be said regarding either the quality or accuracy of the values computed by d02tyc.

10 Example

The following example is used to illustrate that a system with singular coefficients can be treated without modification of the system definition. See also d02tlc, d02tvc, d02txc and d02tzc, for the illustration of other facilities.

Consider the Thomas–Fermi equation used in the investigation of potentials and charge densities of ionized atoms. See Grossman (1992), for example, and the references therein. The equation is

y^{''} = x^{- 1 / 2} y^{3 / 2}

with boundary conditions

y (0) = 1, y (a) = 0, a > 0 .

The coefficient

x^{- 1 / 2}

implies a singularity at the left-hand boundary

x = 0

We use the initial approximation

y (x) = 1 - x / a

, which satisfies the boundary conditions, on a uniform mesh of six points. For illustration we choose

a = 1

, as in Grossman (1992). Note that in ffun and fjac (see d02tlc) we have taken the precaution of setting the function value and Jacobian value to

0.0

in case a value of

y

becomes negative, although starting from our initial solution profile this proves unnecessary during the solution phase. Of course the true solution

y (x)

is positive for all

x < a

d02ty: FL CL CPP AD

NAG CL Interfaced02tyc (bvp_​coll_​nlin_​interp)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG CL Interface
d02tyc (bvp_coll_nlin_interp)