colnew

colnew_appsln (x, zlen, fspace, ispace) — Function

Return a list of solution values from colnew_expert results.

The function arguments are:

x — List of x-coordinates to calculate solution. zlen — mstar, the length of the solution list z fspace — List fspace returned from colnew_005fexpert. ispace — List ispace returned from colnew_005fexpert.

See also: colnew_expert.

colnew_expert (ncomp, m, aleft, aright, zeta, ipar, ltol, tol, fixpnt, ispace, fspace, iflag, fsub, dfsub, gsub, dgsub, guess) — Function

colnew_expert solves mixed-order systems of boundary-value problems (BVPs) in ordinary differential equations (ODEs) using a numerical collocation method.

colnew_expert returns the list [iflag, fspace, ispace]. iflag is an error flag. Lists fspace and ispace contain the state of the solution and can be: used by colnew_appsln to calculate solution values at arbitrary points in the solution domain; and passed back to colnew_expert to restart the solution process with different arguments.

The function arguments are:

ncomp — Number of differential equations (ncomp ≤ 20) m — Integer list of length ncomp.
$m_j$ is the order of the $j$-th differential equation, with $1 \le m_j \le 4$ and $m^* = \sum_j m_j \le 40.$ aleft — Left end of interval aright — Right end of interval zeta — zeta $(\zeta)$ is a real list of length $m^*.$ $\zeta_j$ is the $j$-th boundary or side condition point. The list $\zeta$ must be ordered, with
$\zeta_j \le \zeta_{j+1}.$ All side condition points must be mesh points in all meshes used, see description of ipar[11] and fixpnt below. ipar — A integer list of length 11. The parameters in ipar are: - * ipar[1]

0, if the problem is linear

1, if the problem is nonlinear

ipar[2] ( = $k$ ) Number of collocation points per subinterval , where $\max_j m_j \le k \le 7.$ If ipar[2]=0 then colnew sets $k = \max\left(\max_i m_i + 5, 5 - \max_i m_i\right).$

ipar[3] ( = $n$ ) Number of subintervals in the initial mesh. If ipar[3] = 0 then colnew arbitrarily sets $n = 5$.

ipar[4] ( = ntol ) Number of solution and derivative tolerances. Require $0 < {\rm ntol} < m^*.$

ipar[5] ( = ndimf ) The length of list fspace. Its size provides a constraint on nmax. Choose ipar[5] according to the formula ${\rm ipar[5]} \ge {\rm nmax} \cdot {\rm nsizef}$ where ${\rm nsizef} = 4 + 3m^* + (5 + {\rm kd})\times{\rm kdm} + (2m^* - {\rm nrec})\times 2 m^*.$

ipar[6] ( = ndimi ) The length of list ispace. Its size provides a constraint on nmax, the maximum number of subintervals. Choose ipar[6] according to the formula ${\rm ipar[6]} = {\rm nmax}\times {\rm nsizei}$ where ${\rm nsizei} = 3 + {\rm kdm}$ with $$\eqalign{ {\rm kdm} &= {\rm kd} + m^* \cr {\rm kd} &= k + {\rm ncomp} \cr {\rm nrec} &= {\it number, of, right, end, boundary, conditions} \cr }$$ $$\eqalign{ {\rm kdm} &= {\rm kd} + m^* \cr {\rm kd} &= k + {\rm ncomp} \cr {\rm nrec} &= {\it number, of, right, end, boundary, conditions} \cr }$$ .

ipar[7] ( = iprint ) output control

-1, for full diagnostic printout

0, for selected printout

1, for no printout

ipar[8] ( = iread )

0, causes colnew to generate a uniform initial mesh.

1, if the initial mesh is provided by the user. it is defined in fspace as follows: the mesh aleft=x[1]<x[2]< … <x[n]<x[n+1]=aright will occupy fspace[1], …, fspace[n+1]. the user needs to supply only the interior mesh points fspace[j] = x[j], j = 2, …, n.

2, if the initial mesh is supplied by the user as with ipar[8]=1, and in addition no adaptive mesh selection is to be done.

ipar[9] ( = iguess )

0, if no initial guess for the solution is provided

1, if an initial guess is provided by the user in subroutine guess.

2, if an initial mesh and approximate solution coefficients are provided by the user in fspace. (the former and new mesh are the same).

3, if a former mesh and approximate solution coefficients are provided by the user in fspace, and the new mesh is to be taken twice as coarse; i.e.,every second point from the former mesh.

4, if in addition to a former initial mesh and approximate solution coefficients, a new mesh is provided in fspace as well. (see description of output for further details on iguess = 2, 3, and 4.)

ipar[10]

0, if the problem is regular

1, if the first relax factor is =rstart, and the nonlinear iteration does not rely on past convergence (use for an extra sensitive nonlinear problem only).

2, if we are to return immediately upon (a) two successive nonconvergences, or (b) after obtaining error estimate for the first time.

ipar[11] ( = nfxpnt , the dimension of fixpnt) The number of fixed points in the mesh other than aleft and aright. The code requires that all side condition points other than aleft and aright (see description of zeta) must be included as fixed points in fixpnt. ltol — A list of length ntol=ipar[4]. ltol[j]=k specifies that the j-th tolerance in tol controls the error in the k-th component of z(u). The list ltol must be ordered with $1 ≤ ltol[1] < ltol[2] < … < ltol[ntol] ≤ mstar$. tol — An list of length ntol=ipar[4]. tol[j] is the error tolerance on the ltol[j]-th component of z(u). Thus, the code attempts to satisfy for j=1,…,ntol on each subinterval $abs(z(v)-z(u))[k] ≤ tol(j)*abs(z(u))[k]+tol(j)$ if v(x) is the approximate solution vector. fixpnt — An list of length ipar[11]. It contains the points, other than aleft and aright, which are to be included in every mesh. All side condition points other than aleft and aright (see zeta) be included as fixed points in fixpnt. ispace — An integer work list of length ipar[6]. fspace — A real work list of length ipar[5]. fsub — fsub is a function f(x,z1,…,z[mstar]) which realizes the system of ODEs. It returns a list of ncomp values, one for each ODE. Each value is the highest order derivative in each ode in terms of of x,z1,…,z[mstar] . dfsub — dfsub is a function df(x,z1,…,z[mstar]) for evaluating the Jacobian of f. gsub — Name of subroutine gsub(i,z1,z2,…,z[mstar]) for evaluating the i-th component of the boundary value function g(z1,…,z[mstar]). The independent variable x is not an argument of g. The value x=zeta[i] must be substituted in advance. dgsub — Name of subroutine dgsub(i,z1,…,z[mstar]) for evaluating the i-th row of the Jacobian of g(z1,…,z[mstar]). guess — Name of subroutine to evaluate the initial approximation for (u(x)) and for dmval(u(x))= vector of the mj-th derivatives of u(x). This subroutine is needed only if using ipar(9) = 1.

The return value of colnew_expert is the list [iflag, fspace, ispace], where:

iflag — The mode of return from colnew_expert.

= 1 for normal return

= 0 if the collocation matrix is singular.

= -1 if the expected no. of subintervals exceeds storage specifications.

= -2 if the nonlinear iteration has not converged.

= -3 if there is an input data error. fspace — A list of floats of length ipar[5]. ispace — A list of integers of length ipar[6].

colnew_appsln uses fspace and ispace to calculate solution values at arbitrary points. The lists can also be used to restart the solution process with modified meshes and parameters.

See also: colnew_appsln.