Scilab Function

bvode - boundary value problems for ODE

### Calling Sequence

  `[z]=bvode(points,ncomp,m,aleft,aright,zeta,ipar,ltol,tol,fixpnt,...`
  `fsub1,dfsub1,gsub1,dgsub1,guess1)`

### Parameters

- `z` The solution of the ode evaluated on the mesh given by points
- `points` an array which gives the points for which we want the solution
- `ncomp` number of differential equations (ncomp <= 20)
- `m` a vector of size `ncomp`. `m(j)` gives the order of the j-th differential equation
- `aleft` left end of interval
- `aright` right end of interval
- `zeta` `zeta(j)` gives j-th side condition point (boundary point). must have

`zeta(j) <= zeta(j+1)`

all side condition points must be mesh points in all meshes used, see description of `ipar(11)` and `fixpnt` below.

- `ipar` an integer array dimensioned at least 11. a list of the parameters in `ipar` and their meaning follows some parameters are renamed in bvode; these new names are given in parentheses.

- `ipar(1)` ( = nonlin )

- `= 0` if the problem is linear
- `= 1 if the problem is nonlinear`

- `ipar(2)` = number of collocation points per subinterval (= k ) where

`max m(i) <= k <= 7 .`

if `ipar(2)=0` then bvode sets

`k = max ( max m(i)+1, 5-max m(i) )`

- `ipar(3)` = number of subintervals in the initial mesh ( = n ). if `ipar(3) = 0` then bvode arbitrarily sets `n = 5`.
- `ipar(4)` = number of solution and derivative tolerances. ( = ntol ) we require

`0 < ntol <= mstar.`

- `ipar(5)` = dimension of `fspace` ( = ndimf ) a real work array. its size provides a constraint on nmax. choose ipar(5) according to the formula
- `ipar(6)` = dimension of ispace ( = ndimi )an integer work array. its size provides a constraint on nmax, the maximum number of subintervals. choose `ipar(6)` according to the formula
- `ipar(7)` output control ( = iprint )

- `= -1` for full diagnostic printout
- `= 0` for selected printout
- `= 1` for no printout

- `ipar(8)` ( = iread )

- `= 0` causes bvode to generate a uniform initial mesh.
- `= xx` Other values are not implemented yet in Scilab

- `= 1` if the initial mesh is provided by the user. it is defined in fspace as follows: the mesh

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 covergence (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)` = number of fixed points in the mesh other than `aleft` and `aright`. ( = nfxpnt , the dimension of `fixpnt`) the code requires that all side condition points other than `aleft` and `aright` (see description of zeta ) be included as fixed points in `fixpnt`.

- `ltol` an array of dimension `ipar(4)`. `ltol(j) = l` specifies that the j-th tolerance in tol controls the error in the l-th component of `z(u)`. also require that
- `tol` an array of dimension `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

if `v(x)` is the approximate solution vector.

- `fixpnt` an array of dimension `ipar(11)`. it contains the points, other than `aleft` and `aright`, which are to be included in every mesh.
- `externals` The function `fsub,dfsub,gsub,dgsub,guess` are Scilab externals i.e. functions (see syntax below) or the name of a Fortran subroutine (character string) with specified calling sequence or a list. An external as a character string refers to the name of a Fortran subroutine. The Fortran coded function interface to bvode are specified in the file `fcol.f`.

- `fsub` name of subroutine for evaluating

at a point x in `(aleft,aright)`. it should have the heading `[f]=fsub(x,z)` where `f` is the vector containing the value of `fi(x,z(u))` in the i-th component and

is defined as above under purpose .

- `dfsub` name of subroutine for evaluating the Jacobian of `f(x,z(u))` at a point x. it should have the heading `[df]=dfsub (x , z )` where `z(u(x))` is defined as for `fsub` and the (`ncomp`) by (`mstar`) array df should be filled by the partial derivatives of f, viz, for a particular call one calculates
- `gsub` name of subroutine for evaluating the i-th component of

at a point `x = zeta(i)` where

`1<=i<=mstar.`

it should have the heading`[g]=gsub (i , z)` where `z(u)` is as for fsub, and i and g=gi are as above. note that in contrast to f in fsub , here only one value per call is returned in g.

- `dgsub` name of subroutine for evaluating the i-th row of the Jacobian of `g(x,u(x))`. it should have the heading `[dg]=dgsub (i , z )` where `z(u)` is as for fsub, i as for gsub and the mstar-vector `dg` should be filled with the partial derivatives of g, viz, for a particular call one calculates
- `guess` name of subroutine to evaluate the initial approximation for `z(u(x))` and for `dmval(u(x))`= vector of the mj-th derivatives of `u(x)`. it should have the heading `[z,dmval]= guess (x )` note that this subroutine is used only if `ipar(9) = 1`, and then all `mstar` components of z and ncomp components of dmval should be specified for any x,

`aleft <= x <= aright .`

### Description

this package solves a multi-point boundary value problem for a mixed order system of ode-s given by

the boundary points satisfy

the orders mi of the differential equations satisfy

`1<=m(i)<=4`.

### Examples

    deff('df=dfsub(x,z)','df=[0,0,-6/x**2,-6/x]')
    deff('f=fsub(x,z)','f=(1 -6*x**2*z(4)-6*x*z(3))/x**3')
    deff('g=gsub(i,z)','g=[z(1),z(3),z(1),z(3)];g=g(i)')
    deff('dg=dgsub(i,z)',['dg=[1,0,0,0;0,0,1,0;1,0,0,0;0,0,1,0]';
                          'dg=dg(i,:)'])
    deff('[z,mpar]=guess(x)','z=0;mpar=0')// unused here

    deff('u=trusol(x)',[   //for testing purposes
       'u=0*ones(4,1)';
       'u(1) =  0.25*(10*log(2)-3)*(1-x) + 0.5 *( 1/x   + (3+x)*log(x) - x)'
       'u(2) = -0.25*(10*log(2)-3)       + 0.5 *(-1/x^2 + (3+x)/x      + log(x) - 1)'
       'u(3) = 0.5*( 2/x^3 + 1/x   - 3/x^2)'
       'u(4) = 0.5*(-6/x^4 - 1/x/x + 6/x^3)'])

    fixpnt=0;m=4;
    ncomp=1;aleft=1;aright=2;
    zeta=[1,1,2,2];
    ipar=zeros(1,11);
    ipar(3)=1;ipar(4)=2;ipar(5)=2000;ipar(6)=200;ipar(7)=1;
    ltol=[1,3];tol=[1.e-11,1.e-11];
    res=aleft:0.1:aright;
    z=bvode(res,ncomp,m,aleft,aright,zeta,ipar,ltol,tol,fixpnt,...
     fsub,dfsub,gsub,dgsub,guess)
    z1=[];for x=res,z1=[z1,trusol(x)]; end;
    z-z1

### See Also

`fort` `link` `external` `ode` `dassl`

### Author

u. ascher, department of computer science, university of british; columbia, vancouver, b. c., canada v6t 1w5; g. bader, institut f. angewandte mathematik university of heidelberg; im neuenheimer feld 294d-6900 heidelberg 1 ; ; Fortran subroutine colnew.f