d02ua:: Ordinary Differential EquationsIntegrators for Stiff Ordinary Differential Systems (NAG Toolbox)

nag_ode_bvp_ps_lin_coeffs (d02ua) obtains the Chebyshev coefficients of a function discretized on Chebyshev Gauss–Lobatto points. The set of discretization points on which the function is evaluated is usually obtained by a previous call to nag_ode_bvp_ps_lin_cgl_grid (d02uc).

Syntax

Description

nag_ode_bvp_ps_lin_coeffs (d02ua) computes the coefficients

c_{j}

, for

j = 1, 2, \dots, n + 1

, of the interpolating Chebyshev series

\frac{1}{2} c_{1} T_{0} (\bar{x}) + c_{2} T_{1} (\bar{x}) + c_{3} T_{2} (\bar{x}) + \dots + c_{n + 1} T_{n} (\bar{x}),

which interpolates the function

f (x)

evaluated at the Chebyshev Gauss–Lobatto points

{\bar{x}}_{r} = - \cos ((r - 1) π / n), r = 1, 2, \dots, n + 1 .

Here

T_{j} (\bar{x})

denotes the Chebyshev polynomial of the first kind of degree

j

with argument

\bar{x}

defined on

[- 1, 1]

. In terms of your original variable,

x

say, the input values at which the function values are to be provided are

x_{r} = - \frac{1}{2} (b - a) \cos (π (r - 1) / n) + \frac{1}{2} (b + a), r = 1, 2, \dots, n + 1, ​

where

b

and

a

are respectively the upper and lower ends of the range of

x

over which the function is required.

References

Parameters

Compulsory Input Parameters

Optional Input Parameters

Output Parameters

Error Indicators and Warnings

Accuracy

Further Comments

Example

function d02ua_example


fprintf('d02ua example results\n\n');

% On [0,4], Solve u + ux = f0(x); u(0) = 0
% where f) is such that u(x) = sin(10xcos^2x).
% Set up Chebyshev grid on [a,b]
a = 0;
b = 4;
n = int64(128);
[x, ifail] = d02uc(n, a, b);

% Get Chebyshev coeficients on grid for f0(x) = 1.
z = 10*x.*cos(x).^2;
f0 = sin(z) - 10*cos(x).^2.*cos(10*x.*cos(x).^2).*(2*x.*tan(x)-1);
[f0_c, ifail] = d02ua(n, f0);

% Set up problem definition for  u + ux = f0 [(1 1).(u ux) = f0]
f = [1, 1];
% subject to u(a) = 0  [(1 0).(u ux)(a) = 0]
y = [a];
B = [1, 0];
beta = 0;

% Solve in coefficient space using f0_c for rhs.
[B, f, uc, resid, ifail] = d02ue(...
                                 n, a, b, f0_c, B, y, beta, f);

% Transform solution and derivative back to real space.
[u,  ifail] = d02ub(...
                    n, a, b, int64(0), uc(:, 1));
[ux, ifail] = d02ub(...
                    n, a, b, int64(1), uc(:, 2));

maxerr = max(abs(u - sin(z)));
fprintf('With n = %4d, maximum error in solution = %13.2e\n',n,maxerr);

% Plot solution
fig1 = figure;
plot(x,u,x,ux);
title('Solution of u + u_x = f_0 on [0,4]');
xlabel('x');
ylabel('u(x) and u_x(x)');
legend('u','u_x','Location','Northwest');

NAG Toolbox: nag_ode_bvp_ps_lin_coeffs (d02ua)

▸▿ Contents

Purpose