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