nag_ode_bvp_ps_lin_solve (d02ue) finds the solution of a linear constant coefficient boundary value problem by using the Chebyshev integration formulation on a Chebyshev Gauss–Lobatto grid.

Syntax

[bmat, f, uc, resid, ifail] = d02ue(n, a, b, c, bmat, y, bvec, f, 'm', m)

[bmat, f, uc, resid, ifail] = nag_ode_bvp_ps_lin_solve(n, a, b, c, bmat, y, bvec, f, 'm', m)

Description

nag_ode_bvp_ps_lin_solve (d02ue) solves the constant linear coefficient ordinary differential problem

\sum_{j = 0}^{m} f_{j + 1} \frac{d^{j} u}{d x^{j}} = f (x), x \in [a, b]

subject to a set of

m

linear constraints at points

y_{i} \in [a, b]

, for

i = 1, 2, \dots, m

\sum_{j = 0}^{m} B_{i, j + 1} {(\frac{d^{j} u}{{d x}^{j}})}_{(x = y_{i})} = β_{i},

where

1 \leq m \leq 4

B

is an

m \times (m + 1)

matrix of constant coefficients and

β_{i}

are constants. The points

y_{i}

are usually either

a

b

The function

f (x)

is supplied as an array of Chebyshev coefficients

c_{j}

j = 0, 1, \dots, n

for the function discretized on

n + 1

Chebyshev Gauss–Lobatto points (as returned by nag_ode_bvp_ps_lin_cgl_grid (d02uc)); the coefficients are normally obtained by a previous call to nag_ode_bvp_ps_lin_coeffs (d02ua). The solution and its derivatives (up to order

m

) are returned, in the form of their Chebyshev series representation, as arrays of Chebyshev coefficients; subsequent calls to nag_ode_bvp_ps_lin_cgl_vals (d02ub) will return the corresponding function and derivative values at the Chebyshev Gauss–Lobatto discretization points on

[a, b]

. Function and derivative values can be obtained on any uniform grid over the same range

[a, b]

by calling the interpolation function nag_ode_bvp_ps_lin_grid_vals (d02uw).

References

Clenshaw C W (1957) The numerical solution of linear differential equations in Chebyshev series Proc. Camb. Phil. Soc. 53 134–149

Coutsias E A, Hagstrom T and Torres D (1996) An efficient spectral method for ordinary differential equations with rational function coefficients Mathematics of Computation 65(214) 611–635

Greengard L (1991) Spectral integration and two-point boundary value problems SIAM J. Numer. Anal. 28(4) 1071–80

Lundbladh A, Hennigson D S and Johannson A V (1992) An efficient spectral integration method for the solution of the Navier–Stokes equations Technical report FFA–TN 1992–28 Aeronautical Research Institute of Sweden

Muite B K (2010) A numerical comparison of Chebyshev methods for solving fourth-order semilinear initial boundary value problems Journal of Computational and Applied Mathematics 234(2) 317–342

Parameters

Compulsory Input Parameters

1: $n$ – int64int32nag_int scalar: $n$ , where the number of grid points is $n + 1$ .

Constraint: $n \geq 8$ and n is even.
2: $a$ – double scalar: $a$ , the lower bound of domain $[a, b]$ .

Constraint: $a < b$ .
3: $b$ – double scalar: $b$ , the upper bound of domain $[a, b]$ .

Constraint: $b > a$ .
4: $c (n + 1)$ – double array: The Chebyshev coefficients $c_{j}$ , $j = 0, 1, \dots, n$ , for the right hand side of the boundary value problem. Usually these are obtained by a previous call of nag_ode_bvp_ps_lin_coeffs (d02ua).
5: $bmat (m, m + 1)$ – double array: $bmat (i, j + 1)$ must contain the coefficients $B_{i, j + 1}$ , for $i = 1, 2, \dots, m$ and $j = 0, 1, \dots, m$ , in the problem formulation of Description.
6: $y (m)$ – double array: The points, $y_{i}$ , for $i = 1, 2, \dots, m$ , where the boundary conditions are discretized.
7: $bvec (m)$ – double array: The values, $β_{i}$ , for $i = 1, 2, \dots, m$ , in the formulation of the boundary conditions given in Description.
8: $f (m + 1)$ – double array: The coefficients, $f_{j}$ , for $j = 1, 2, \dots, m + 1$ , in the formulation of the linear boundary value problem given in Description. The highest order term, $f (m + 1)$ , needs to be nonzero to have a well posed problem.

Optional Input Parameters

1: $m$ – int64int32nag_int scalar: Default: the first dimension of the array bmat and the dimension of the arrays y, bvec. (An error is raised if these dimensions are not equal.)
The order, $m$ , of the boundary value problem to be solved.

Constraint: $1 \leq m \leq 4$ .

Output Parameters

1: $bmat (m, m + 1)$ – double array

The coefficients have been scaled to form an equivalent problem defined on the domain

[- 1, 1]

2: $f (m + 1)$ – double array

The coefficients have been scaled to form an equivalent problem defined on the domain

[- 1, 1]

3: $uc (n + 1, m + 1)$ – double array

The Chebyshev coefficients in the Chebyshev series representations of the solution and derivatives of the solution to the boundary value problem. The

n + 1

elements

uc (1 : n + 1, 1)

contain the coefficients representing the solution

U (x_{i})

, for

i = 0, 1, \dots, n

uc (1 : n + 1, j + 1)

contains the coefficients representing the

j

th derivative of

U

, for

j = 1, 2, \dots, m

4: $resid$ – double scalar

The maximum residual resulting from substituting the solution vectors returned in uc into both linear equations of Description representing the linear boundary value problem and associated boundary conditions. That is

\max \{\max_{i = 1, m} (|\sum_{j = 0}^{m} B_{i, j + 1} {(\frac{d^{j} u}{d x^{j}})}_{(x = y_{i})} - β_{i}|), \max_{i = 1, n + 1} (|\sum_{j = 0}^{m} f_{j + 1} {(\frac{d^{j} u}{{d x}^{j}})}_{(x = x_{i})} - f (x)|)\} .

5: $ifail$ – int64int32nag_int scalar

ifail = 0

unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

$ifail = 1$: Constraint: n is even.

Constraint: $n \geq 8$ .

$ifail = 2$: Constraint: $a < b$ .

$ifail = 3$: On entry, $f (m + 1) = 0.0$ .

$ifail = 6$: Constraint: $1 \leq m \leq 4$ .

$ifail = 7$: Internal error while unpacking matrix during iterative refinement.
Please contact NAG.

$ifail = 8$: Singular matrix encountered during iterative refinement.
Please check that your system is well posed.

W $ifail = 9$: During iterative refinement, the maximum number of iterations was reached.

W $ifail = 10$: During iterative refinement, convergence was achieved, but the residual is more than $100 \times machine precision$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

The accuracy should be close to machine precision for well conditioned boundary value problems.

Further Comments

The number of operations is of the order

n \log (n)

and the memory requirements are

O (n)

; thus the computation remains efficient and practical for very fine discretizations (very large values of

n

). Collocation methods will be faster for small problems, but the method of nag_ode_bvp_ps_lin_solve (d02ue) should be faster for larger discretizations.

Example

This example solves the third-order problem

4 U_{x x x} + 3 U_{x x} + 2 U_{x} + U = 2 \sin x - 2 \cos x

[- π / 2, π / 2]

subject to the boundary conditions

U [- π / 2] = 0

3 U_{x x} [- π / 2] + 2 U_{x} [- π / 2] + U [- π / 2] = 2

, and

3 U_{x x} [π / 2] + 2 U_{x} [π / 2] + U [π / 2] = - 2

using the Chebyshev integration formulation on a Chebyshev Gauss–Lobatto grid of order

16

Open in the MATLAB editor: d02ue_example

function d02ue_example


fprintf('d02ue example results\n\n');

n = int64(16);
a = -pi/2;
b =  pi/2;


% Set up boundary condition on left side of domain
y = [a, b];
% Set up Dirichlet condition using exact solution at x=a.
bmat = zeros(2, 3);
bmat(1, 1:2) = [1, 1];
bmat(2, 1:2) = [1, 1];
bvec = [cos(a) - sin(a), cos(b) - sin(b)];

% Set up problem definition
f = [1, 2, 3];

% Set up solution grid
[x, ifail] = d02uc(n, a, b);

% Set up problem right hand sides for grid and transform
f0 = -2*sin(x) - 2*cos(x);
[c, ifail] = d02ua(n, f0);

% Solve in coefficient space
[bmat, f, uc, resid, ifail] = d02ue(n, a, b, c, bmat, y, bvec, f);

% Evaluate solution and derivatives on Chebyshev grid
u = zeros(n+1, 3);
for q=0:2
  [u(:, q+1),  ifail] = d02ub(n, a, b, int64(q), uc(:, q+1));
end

% Print Solution
fprintf('\nNumerical solution U and its first two derivatives\n');
fprintf('      x          U          Ux         Uxx\n');
fprintf('%10.4f %10.4f %10.4f %10.4f\n', [x u]');

d02ue example results


Numerical solution U and its first two derivatives
      x          U          Ux         Uxx
   -1.5708    -0.0000     1.0000     0.0000
   -1.5406     0.0302     0.9995    -0.0302
   -1.4512     0.1193     0.9929    -0.1193
   -1.3061     0.2616     0.9652    -0.2616
   -1.1107     0.4440     0.8960    -0.4440
   -0.8727     0.6428     0.7661    -0.6428
   -0.6011     0.8247     0.5656    -0.8247
   -0.3064     0.9534     0.3017    -0.9534
   -0.0000     1.0000     0.0000    -1.0000
    0.3064     0.9534    -0.3017    -0.9534
    0.6011     0.8247    -0.5656    -0.8247
    0.8727     0.6428    -0.7661    -0.6428
    1.1107     0.4440    -0.8960    -0.4440
    1.3061     0.2616    -0.9652    -0.2616
    1.4512     0.1193    -0.9929    -0.1193
    1.5406     0.0302    -0.9995    -0.0302
    1.5708    -0.0000    -1.0000    -0.0000

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox