Name
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 havezeta(j) <= zeta(j+1)all side condition points must be mesh points in all meshes used, see description of
ipar(11)andfixpntbelow.- ipar
an integer array dimensioned at least 11. a list of the parameters in
iparand their meaning follows some parameters are renamed in bvode; these new names are given in parentheses.- ipar(1)
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)=0then bvode setsk = max ( max m(i)+1, 5-max m(i) )- ipar(3)
= number of subintervals in the initial mesh ( = n ). if
ipar(3) = 0then bvode arbitrarily setsn = 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(5)>=nmax*nsizefwhere
nsizef=4+3*mstar+(5+kd)*kdm+(2*mstar-nrec)*2*mstar.- 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(6)>=nmax*nsizeiwhere
nsizei=3 + kdmwithkdm=kd+mstar;kd=k*ncomp;nrec=number of right end boundary conditions.- 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 pointsfspace(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
aleftandaright. ( = nfxpnt , the dimension offixpnt) the code requires that all side condition points other thanaleftandaright(see description of zeta ) be included as fixed points infixpnt.
- ltol
an array of dimension
ipar(4).ltol(j) = lspecifies that the j-th tolerance in tol controls the error in the l-th component ofz(u). also require that:1 <= ltol(1) < ltol(2) < ... < ltol(ntol) <= mstar- tol
an array of dimension
ipar(4).tol(j)is the error tolerance on theltol(j)-th component ofz(u). thus, the code attempts to satisfy forj=1:ntolon each subinterval
if
v(x)is the approximate solution vector.- fixpnt
an array of dimension
ipar(11). it contains the points, other thanaleftandaright, which are to be included in every mesh.- externals
The function
fsub,dfsub,gsub,dgsub,guessare 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 filefcol.f.- fsub
name of subroutine for evaluating
at a point x in
(aleft,aright). it should have the heading[f]=fsub(x,z)wherefis the vector containing the value offi(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 )wherez(u(x))is defined as forfsuband 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
g(x,z(u(x))) = g (zeta(i),z(u(zeta(i))))at a pointx = zeta(i)where1<=i<=mstar.it should have the heading
[g]=gsub(i, z)wherez(u)is as forfsub, andiandg=giare as above. Note that in contrast tofinfsub, here only one value per call is returned ing.- 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)wherez(u)is as for fsub, i as for gsub and the mstar-vectordgshould 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 fordmval(u(x))= vector of the mj-th derivatives ofu(x). it should have the heading[z,dmval]= guess (x )note that this subroutine is used only ifipar(9) = 1, and then allmstarcomponents 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
where
is the exact solution vector
is a (generally) nonlinear function of
z(u)=z(u(x)).
is a (generally) nonlinear function used to represent a boundary condition.
the boundary points satisfy
aleft <= zeta(1) <= .. <= zeta(mstar) <=
aright.
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
//define trusol for testing purposes
deff('u=trusol(x)',['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