Fwd: [maxima - Help] FEM Weak form & Maxima (Integration by Parts)

integration by parts requires that the integrand be
decomposed into a product of u*dv   where both u and v are
functions of x.
There are many ways of doing that.  If you have an algorithm
for decomposing a product correctly for your application, then
you are set.

If you do not know how to do it, then maybe the reason Maxima
doesn't have it "built in"  is that no one else knows how to do
it algorithmically.

It is possible to come up with heuristics for integration by parts,
but as you may be aware, in general there are prospects for
this technique to go off in the wrong direction, or loop.

RJF


----- Original Message -----
From: Robert Dodier <robert_dodier at yahoo.com>
Date: Thursday, April 6, 2006 10:01 am
Subject: Fwd: [maxima - Help] FEM Weak form & Maxima (Integration by Parts)

> hello,
> 
> i'm forwarding this message which was posted in the
> maxima forum at SF.
> (https://sourceforge.net/forum/message.php?msg_id=3671925)
> please direct followups to the mailing list instead of the forum.
> to ck: you'll probably need to register in order to post
> to the mailing list. sorry for the confusion.
> see http://maxima.sf.net/maximalist.html .
> 
> robert dodier
> 
> --- "SourceForge.net" <noreply at sourceforge.net> wrote:
> 
> > To: noreply at sourceforge.net
> > From: "SourceForge.net" <noreply at sourceforge.net>
> > Subject: [maxima - Help] FEM Weak form & Maxima (Integration by
> > Parts)
> > Date: Wed, 05 Apr 2006 20:01:09 -0700
> > 
> > 
> > Read and respond to this message at: 
> > https://sourceforge.net/forum/message.php?msg_id=3671925
> > By: ck_
> > 
> > 1) I'd like to be able to generate the weak form of an ODE/PDE
> > entirely in Maxima.
> > I'm a good part of the way there, but the partial integration by
> > parts step
> > is a road block.  Ideas?
> > 
> > sf: diff(u(x),x,2) - 4.0*u(x) = 0;
> > sf1: expand(v(x)*sf);
> > term_list(X) := block([], makelist(part(X,1,i), i,1,length(X)));
> > terms: term_list(sf1);
> > map( lambda([X], integrate(X,x,0,4.0)), terms );
> > 
> > /* Why cant Macsyma do integration by parts? */
> > 
> > wf1:integrate(diff(u(x),x)*diff(v(x),x),x,0,4) 
> > 	+ integrate(4.0*v(x)*u(x),x,0,4) - v(x)*diff(u(x),x);
> > dwf: wf1, 'diff(u(x),x,1)='sum(u(j)*diff(N(j,x),x,1),j,1,NN)$
> > dwf: dwf, 'diff(v(x),x,1)='sum(v(i)*diff(N(i,x),x,1),i,1,NN)$
> > dwf: dwf, u(x)='sum(u(j)*N(j,x),j,1,NN)$
> > dwf: dwf, v(x)='sum(v(i)*N(i,x),i,1,NN)$
> > 
> > 2) Once you have the weak form, it would be nice to have Maxima
> > generate the
> > stiffness matrices as well... but the following chunk fails.  
> What is
> > wrong?
> > 
> > /* Defining Shape Functions for a Quadratic Element */
> > N1(x):= ((x-x2)*(x-x3)) / ((x1-x2)*(x1-x3))$
> > N2(x):= ((x-x1)*(x-x3)) / ((x2-x1)*(x2-x3))$
> > N3(x):= ((x-x1)*(x-x2)) / ((x3-x1)*(x3-x2))$
> > dN1(x) := diff(N1(x),x)$
> > dN2(x) := diff(N2(x),x)$
> > dN3(x) := diff(N3(x),x)$
> > Nij_quad: [N1(x),N2(x),N3(x)]$
> > dNij_quad: map(lambda([X],diff(X,x,1)), Nij_quad)$
> > 
> > /* Defining Map for Quadratic Element Local Basis 
> >  *	x2z: x=(L/2)*z + x2$
> >  */
> > x2z: x=((x3-x1)/2)*z + x2$
> > 
> > /* Specific Kije for this ODE */
> > kij1e_quad(i,j) := 'integrate(dNij_quad[i]*dNij_quad[j],x,x1,x3)$
> > kij1e_quad_z(i,j) := changevar(kij1e_quad(i,j),x2z,z,x)$
> > kij2e_quad(i,j) := 'integrate(4.0*Nij_quad[i]*Nij_quad[j],x,x1,x3)$
> > kij2e_quad_z(i,j) := changevar(kij2e_quad(i,j),x2z,z,x)$
> > NN:3$
> > kij1e_q:
> > matrix([kij1e_quad_z(1,1),kij1e_quad_z(1,2),kij1e_quad_z(1,3)],
> >         	[kij1e_quad_z(2,1),kij1e_quad_z(2,2),kij1e_quad_z(2,3)],
> >         	[kij1e_quad_z(3,1),kij1e_quad_z(3,2),kij1e_quad_z(3,3)])$
> > kij1e_q: ev(kij1e_q, nouns)$
> > kij2e_q:
> > matrix([kij2e_quad_z(1,1),kij2e_quad_z(1,2),kij2e_quad_z(1,3)],
> >         	[kij2e_quad_z(2,1),kij2e_quad_z(2,2),kij2e_quad_z(2,3)],
> >         	[kij2e_quad_z(3,1),kij2e_quad_z(3,2),kij2e_quad_z(3,3)])$
> > kij2e_q: ev(kij2e_q, nouns)$
> > 
> > thanks all,
> > 
> > ck
> > 
> >
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