naginterfaces.library.ode.bvp_​ps_​lin_​solve

naginterfaces.library.ode.bvp_ps_lin_solve(a, b, c, bmat, y, bvec, f)

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

For full information please refer to the NAG Library document for d02ue

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/d02/d02uef.html

Parameters

afloat

$a$, the lower bound of domain $[a,b]$.

bfloat

$b$, the upper bound of domain $[a,b]$.

cfloat, array-like, shape $(n+1)$

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 bvp_ps_lin_coeffs().

bmatfloat, array-like, shape $(m,m+1)$

$bmat[\text{i}-1,\text{j}+1-1]$ must contain the coefficients $B_{\text{i},\text{j}+1}$, for $\text{j}=0,1,\dots ,m$, for $\text{i}=1,2,\dots ,m$, in the problem formulation of Notes.

yfloat, array-like, shape $\left(m\right)$

The points, $y_{\text{i}}$, for $\text{i}=1,2,\dots ,m$, where the boundary conditions are discretized.

bvecfloat, array-like, shape $\left(m\right)$

The values, ${\beta }_{\text{i}}$, for $\text{i}=1,2,\dots ,m$, in the formulation of the boundary conditions given in Notes.

ffloat, array-like, shape $(m+1)$

The coefficients, $f_{\text{j}}$, for $\text{j}=1,2,\dots ,m+1$, in the formulation of the linear boundary value problem given in Notes. The highest order term, $f[m+1-1]$, needs to be nonzero to have a well posed problem.

Returns

bmatfloat, ndarray, shape $(m,m+1)$

The coefficients have been scaled to form an equivalent problem defined on the domain $[-1,1]$.

ffloat, ndarray, shape $(m+1)$

The coefficients have been scaled to form an equivalent problem defined on the domain $[-1,1]$.

ucfloat, ndarray, shape $(n+1,m+1)$

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 $uc[0:n+1,0]$ contain the coefficients representing the solution $U\left(x_{\text{i}}\right)$, for $\text{i}=0,1,\dots ,n$. $uc[0:n+1,\text{j}]$ contains the coefficients representing the $\text{j}$th derivative of $U$, for $\text{j}=1,2,\dots ,m$.

residfloat

The maximum residual resulting from substituting the solution vectors returned in $uc$ into the equations representing the linear boundary value problem and associated boundary conditions.

Raises

NagValueError

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 8$.

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $\text{n}$ is even.

(errno $2$)

On entry, $a=⟨value⟩$ and $b=⟨value⟩$.

Constraint: $a<b$.

(errno $3$)

On entry, $f\left[m\right]=0.0$.

(errno $6$)

On entry, $m=⟨value⟩$.

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

(errno $7$)

Internal error while unpacking matrix during iterative refinement.

Please contact NAG.

(errno $8$)

Singular matrix encountered during iterative refinement.

Please check that your system is well posed.

Warns

NagAlgorithmicWarning

(errno $9$)

During iterative refinement, the maximum number of iterations was reached.

$\text{Number of iterations}=⟨value⟩$ and $\text{residual achieved}=⟨value⟩$.

(errno $10$)

During iterative refinement, convergence was achieved, but the residual is more than $100\times \text{machine precision}$. $\text{Residual achieved on convergence}=⟨value⟩$.

Notes

bvp_ps_lin_solve solves the constant linear coefficient ordinary differential problem

$$\sum _0^m{f}_{j+1}\frac{d^{j}u}{d{x}^j}=f\left(x\right)\text{,}x\in [a,b]$$

subject to a set of $m$ linear constraints at points $y_{\text{i}}\in [a,b]$, for $\text{i}=1,2,\dots ,m$:

$$\sum _0^m{B}_{i,j+1}{\left(\frac{d^{j}u}{{dx}^j}\right)}_{(x=y_i)}={\beta }_i\text{,}$$

where $1\le m\le 4$, $B$ is an $m\times (m+1)$ matrix of constant coefficients and ${\beta }_i$ are constants. The points $y_i$ are usually either $a$ or $b$.

The function $f\left(x\right)$ 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 bvp_ps_lin_cgl_grid()); the coefficients are normally obtained by a previous call to bvp_ps_lin_coeffs(). 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 bvp_ps_lin_cgl_vals() 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 bvp_ps_lin_grid_vals().

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