naginterfaces.library.ode.bvp_​coll_​nlin_​interp

naginterfaces.library.ode.bvp_coll_nlin_interp(x, neq, mmax, comm)

bvp_coll_nlin_interp interpolates on the solution of a general two-point boundary value problem computed by bvp_coll_nlin_solve().

For full information please refer to the NAG Library document for d02ty

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/d02/d02tyf.html

Parameters

xfloat

$x$, the independent variable.

neqint

The number of differential equations.

mmaxint

The maximal order of the differential equations, $max\left(m_{\text{i}}\right)$, for $\text{i}=1,2,\dots ,neq$.

commdict, communication object, modified in place

Communication structure.

This argument must have been initialized by a prior call to bvp_coll_nlin_setup().

Returns

yfloat, ndarray, shape $(neq,mmax)$

$y[\text{i}-1,\text{j}]$ contains an approximation to $y_{\text{i}}^{\left(\text{j}\right)}\left(x\right)$, for $\text{j}=0,1,\dots ,m_{\text{i}}-1$, for $\text{i}=1,2,\dots ,neq$. The remaining elements of $y$ (where $m_i<mmax$) are initialized to $0.0$.

Raises

NagValueError

(errno $1$)

On entry, $x=⟨value⟩$.

Constraint: $x\le ⟨value⟩$.

(errno $1$)

On entry, $x=⟨value⟩$.

Constraint: $x\ge ⟨value⟩$.

(errno $1$)

On entry, $mmax=⟨value⟩$ and $max\left(\text{m}\right[i\left]\right)=⟨value⟩$ in bvp_coll_nlin_setup().

Constraint: $mmax=max\left(\text{m}\right[i\left]\right)$ in bvp_coll_nlin_setup().

(errno $1$)

The solver function did not produce any results suitable for interpolation.

(errno $1$)

The solver function does not appear to have been called.

(errno $1$)

On entry, $neq=⟨value⟩$ and $\text{neq}=⟨value⟩$ in bvp_coll_nlin_setup().

Constraint: $neq=\text{neq}$ in bvp_coll_nlin_setup().

Warns

NagAlgorithmicWarning

(errno $2$)

The solver function did not satisfy the error requirements.

Interpolated values should be treated with caution.

(errno $2$)

The solver function did not converge to a suitable solution.

A converged intermediate solution has been used.

Interpolated values should be treated with caution.

Notes

bvp_coll_nlin_interp and its associated functions (bvp_coll_nlin_setup(), bvp_coll_nlin_solve(), bvp_coll_nlin_contin() and bvp_coll_nlin_diag()) solve the two-point boundary value problem for a nonlinear mixed order system of ordinary differential equations

$$\begin{array}{c}\begin{array}{ccc}{y}_1^{\left(m_1\right)}\left(x\right)& =& f_1(x,y_1,y_1^{\left(1\right)},\dots ,y_1^{(m_1-1)},y_2,\dots ,y_n^{(m_n-1)})\\ y_2^{\left(m_2\right)}\left(x\right)& =& f_2(x,y_1,y_1^{\left(1\right)},\dots ,y_1^{(m_1-1)},y_2,\dots ,y_n^{(m_n-1)})\\ & ⋮& \\ y_n^{\left(m_n\right)}\left(x\right)& =& f_n(x,y_1,y_1^{\left(1\right)},\dots ,y_1^{(m_1-1)},y_2,\dots ,y_n^{(m_n-1)})\end{array}\end{array}$$

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 $y_i^{\left(m\right)}\left(x\right)$ is the $m$th derivative of the $i$th solution component. Hence $y_i^{\left(0\right)}\left(x\right)=y_i\left(x\right)$. The left boundary conditions at $a$ are defined as

$$g_i\left(z\left(y\left(a\right)\right)\right)=0\text{,}i=1,2,\dots ,p\text{,}$$

and the right boundary conditions at $b$ as

$${\overline{g}}_j\left(z\left(y\left(b\right)\right)\right)=0\text{,}j=1,2,\dots ,q\text{,}$$

where $y=(y_1,y_2,\dots ,y_n)$ and

$$z\left(y\left(x\right)\right)=(y_1\left(x\right),y_1^{\left(1\right)}\left(x\right),\dots ,y_1^{(m_1-1)}\left(x\right),y_2\left(x\right),\dots ,y_n^{(m_n-1)}\left(x\right))\text{.}$$

First, bvp_coll_nlin_setup() must be called to specify the initial mesh, error requirements and other details. Then, bvp_coll_nlin_solve() can be used to solve the boundary value problem. After successful computation, bvp_coll_nlin_diag() can be used to ascertain details about the final mesh and other details of the solution procedure, and bvp_coll_nlin_interp 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).

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