module spde; % Determine Lie symmetries of partial differential eqns.
% Author: Fritz Schwarz.
%*******************************************************************$
% $
% This is the REDUCE package SPDE for determining $
% Lie symmetries of partial differential equations $
% Version of November 1986 $
% $
% $
% Fritz Schwarz $
% GMD Institut F1 $
% Postfach 1240 $
% 5205 St. Augustin $
% West Germany $
% $
% Tel. 02241-142782 $
% EARN Id. DBNGMD21.GF1002 $
%*******************************************************************$
algebraic operator x,u,xi,eta,c,xi!*,eta!*$
algebraic operator deq,dx,du,gl,gen,sder,rule$
share pclass,mm,nn$
global'(pclass mm nn depl!* num!-cgen num!-dgen)$
lisp(pclass:=mm:=nn:=num!-cgen:=num!-dgen:=0)$
lisp(operator simpsys,result,prsys,prsys!*)$
fluid '(!*list kord!*)$
fluid'(uhf dfsub csub czero rdep !*rational)$
fluid'(list!-m list!-deq list!-pq)$
%symbolic procedure prload$
% begin
% if not getd 'solve1 then load solve1,solvetab,quartic;
% if not getd 'depend1 then load depend;
% if not getd 'ratfunpri then load ratprin;
% end$
symbolic procedure prload; nil;
%*******************************************************************$
% Auxiliary RLISP procedures $
%*******************************************************************$
symbolic procedure ordp(u,v)$
%Modified ordering function which orders kernels with CAR parts;
%DF, ETA, XI and C ahead of anything else;
if null u then null v else if null v then t else
if eq(u,'df) or eq(u,'eta) and not eq(v,'df)
or eq(u,'xi) and not(eq(v,'df) or eq(v,'eta))
or eq(u,'c) and not(eq(v,'df) or eq(v,'eta) or eq(v,'xi)) then t else
if eq(u,'eta) and eq(v,'df)
or eq(u,'xi) and (eq(v,'df) or eq(v,'eta))
or eq(u,'c) and (eq(v,'df) or eq(v,'eta) or eq(v,'xi))
or eq(v,'df) or eq(v,'eta) or eq(v,'xi) or eq(v,'c) then nil else
if atom u then if atom v then
if numberp u then numberp v and not u<v else
if numberp v then t else orderp(u,v) else nil else
if atom v then t else
if car u=car v then ordp(cdr u,cdr v) else ordp(car u,car v)$
symbolic procedure makeset u$
if not u then nil else
if member(car u,cdr u) then makeset cdr u else
car u . makeset cdr u$
symbolic procedure lastmem u$
if cdr u then lastmem cdr u else car u$
symbolic procedure xmember(u,v)$ reverse member(u,reverse v)$
symbolic procedure sacar(a,u)$
if atom u then nil else
if eq(a,car u) and cdr u then list u else
append(sacar(a,car u),sacar(a,cdr u))$
symbolic procedure scar(a,u)$
if atom u then nil else if a=car u then u else
scar(a,car u) or scar(a,cdr u)$
symbolic procedure inter(u,v);
if not u then nil else
if member(car u,v) then
(car u) . inter(cdr u,v) else inter(cdr u,v)$
symbolic procedure compl(u,v)$
if not u then nil else if member(car u,v) then
compl(cdr u,v) else car u . compl(cdr u,v)$
symbolic procedure vlist u$
%U is list of items, returns U with all integers omitted;
if not u then nil else
if numberp car u then vlist cdr u else (car u) . vlist cdr u$
symbolic procedure delnil u$
%U is list, returns U with all occurences of nil deleted;
if not u then nil else
if car u then (car u) . delnil cdr u else delnil cdr u$
symbolic procedure prlist u$
%U is list of items, returns list of all pairs in U;
if not u then nil else
if pairp car u then (car u) . prlist cdr u else prlist cdr u$
symbolic procedure appends(u,v,w)$ append(u,append(v,w))$
symbolic procedure propa(fn,u)$
%FN is predicate of a single argument, U a list;
%Returns T if predicate is true for all elements of U;
begin scalar ind;
ind:=t;
while ind and u do <<ind:=apply(fn,list car u); u:=cdr u>>;
return ind;
end$
symbolic procedure sortx(fn,u)$
begin scalar v,w;
while u do<<v:=maxmem(fn,u); u:=delete(v,u); w:=v . w>>;
return w;
end$
symbolic procedure maxmem(fn,u)$
%FN is function of a single argument, U a list;
%Returns element of U for which FN is maximal;
begin scalar v;
v:=car u;
foreach x in cdr u do
if greaterp(apply(fn,list x),apply(fn,list v)) then v:=x;
return v;
end$
symbolic procedure maxl u$
%U is list of integers, returns largest element of U;
if not u then -10000 else max(car u,maxl cdr u)$
symbolic procedure suml u$
%U is list of integers, returns sum of all elements;
if not u then 0 else plus2(car u,suml cdr u)$
symbolic procedure spde!-subsetp(u,v)$
%U and V are list representing sets;
%Returns T if set U is subset of V;
if not u then t else
member(car u,v) and spde!-subsetp(cdr u,v)$
symbolic procedure product!-set2(u,v)$
%U and V are lists representing sets, returns list representing;
%product set of sets represented by U and V;
begin scalar w;
foreach x in u do foreach y in v do w:=list(x,y) . w;
return w;
end$
symbolic procedure leqgrt(l,i,j)$
i leq j and eqn(l,i) or i geq add1 j$
symbolic procedure fidep u$
assoc(u,depl!*) and cdr assoc(u,depl!*)$
symbolic procedure mkdep u$
foreach x in cdr u do depend1(car u,x,t)$
symbolic procedure rmdep u$
<<rmsubs(); foreach x in cdr u do depend1(car u,x,nil)>>$
symbolic procedure blanks l;
begin scalar u;
u := '(!");
for k:=1:l do u:='! . u;
return compress('!" . u)
end$
symbolic procedure terpri2$ <<terpri(); terpri()>>$
%*******************************************************************$
% Auxiliary procedures for manipulating standard forms $
%*******************************************************************$
symbolic procedure lcf u$ not domainp u and lc u$
symbolic procedure minus!-f u$
%U is s.f., returns T if lnc U is negative;
minusf numr simp reval u$
lisp operator minus!-f$
symbolic procedure lengthn u$
if not u then 0 else
if numberp car u then plus(sub1 car u,lengthn cdr u) else
plus(1,lengthn cdr u)$
symbolic procedure degreef(u,v)$
%U is s.f., V kernel, returns degree of V in U;
if domainp u then 0 else
if mvar u=v then ldeg u else
max(degreef(lc u,v),degreef(red u,v))$
symbolic procedure lengthf u$
%U is prefix s.f., returns printlength for U;
if not u then 0 else
if atom u then flatsizec u else
if eqcar(u,'plus)
then plus(times(3,sub1 length cdr u),lengthf cdr u) else
if eqcar(u,'times) or eqcar(u,'minus)
then plus(sub1 length cdr u,lengthf cdr u) else
if eqcar(u,'quotient) then
if !*rational then add1 add1 max(flatsizec cadr u,flatsizec caddr u)
else add1 plus(flatsizec cadr u,flatsizec caddr u) else
if eqcar(u,'expt) then add1 flatsizec cadr u else
if eqcar(u,'dx) or eqcar(u,'du) then plus(flatsizec cadr u,4) else
if eqcar(u,'xi) or eqcar(u,'eta) or eqcar(u,'c)
or eqcar(u,'x) or eqcar(u,'u) then times(2,length u) else
if eqcar(u,'df) then plus(4,lengthf cadr u,lengthf cddr u) else
plus(lengthf car u,lengthf cdr u)$
lisp operator lengthf$
symbolic procedure diford u$ lengthn cddr u$
symbolic procedure adiff(u,v)$
%U is kernel with CAR part DF, V is kernel;
%Returns U integrated with respect to V;
if not member(v,u) then u else
if length u=3 and member(v,u) then cadr u else
if not cdr member(v,u)
or not numberp cadr member(v,u) then delete(v,u) else
if cadr member(v,u)=2 then
append(xmember(v,u),cddr member(v,u)) else
append(xmember(v,u),(sub1 cadr member(v,u)) . cddr member(v,u))$
symbolic procedure sub!-int!-df u$
%U is kernel with CAR part INT, returns integrated kernel if CADR;
%part of U is DF and integration variable occurs as argument of DF;
if eqcar(cadr u,'df) and member(lastmem u,cadr u) then
adiff(cadr u,lastmem u) else u$
symbolic procedure subintf u$
%U is s.f., performs all integrations which may be done;
%by cancellation of corresponding differentiation;
begin
foreach x in makeset sacar('int,u) do
u:=subst(sub!-int!-df x,x,u);
return numr simp prepf u;
end$
symbolic procedure monop u$
%Returns T if u is monomial;
domainp u or not red u and monop lc u$
symbolic procedure solvef(u,v)$ car solve0(prepf u,v)$
symbolic procedure comfacn u$ lnc ckrn u$
symbolic procedure remfacn u$ quotf(u,lnc ckrn u)$
%*******************************************************************$
% Procedures for manipulating l.d.f.'s, U is always l.d.f. $
% in this section $
%*******************************************************************$
symbolic procedure ldf!-mvar u$
%Returns function argument of mvar U;
(if eqcar(x,'df) then cadr x else x) where x=mvar u;
symbolic procedure ldf!-fvar u$
%Returns all function arguments occuring in U;
makeset foreach x in u collect ldt!-tvar x$
symbolic procedure ldf!-fvar!-part(u,v)$
%V is function xi(i), eta(alpha) or c(k), returns l.d.f. of those;
%terms in U with ldt-tvar x equal to V, overall factors not removed;
begin scalar w;
foreach x in u do if eq(ldt!-tvar x,v) then w:=x . w;
return reverse w;
end$
symbolic procedure ldf!-dep!-var u$
%Returns all variables x(i) or u(alpha) which occur as;
%arguments of XI, ETA or C;
begin scalar v;
foreach x in u do if assoc(ldt!-tvar x,depl!*) then
v:=append(cdr assoc(ldt!-tvar x,depl!*),v);
return makeset v;
end$
symbolic procedure ldf!-pow!-var u$
%Returns all variables x(i) or u(alpha) which occur as powers;
begin scalar v,z;
foreach x in u do v:=append(v,kernels tc x);
foreach y in prlist makeset v do
if eqcar(y,'x) or eqcar(y,'u) then z:=y . z;
return makeset z;
end$
symbolic procedure ldf!-deg(u,v)$
%V is kernel x(i) or u(alpha), returns degree of U in V;
maxl foreach x in u collect degreef(tc x,v)$
symbolic procedure ldf!-spf!-var u$
%Returns all variables x(i) or u(alpha) which occur as;
%arguments of any other kernel than xi, eta or c;
begin scalar v,z;
foreach x in u do v:=append(v,kernels tc x);
foreach y in prlist makeset v do
if not eqcar(y,'x) and not eqcar(y,'u) then
z:=appends(sacar('x,cdr y),sacar('u,cdr y),z);
return makeset z;
end$
symbolic procedure ldf!-all!-var u$
%Returns all variables x(i) or u(alpha) which occur in U;
makeset appends(ldf!-dep!-var u,ldf!-pow!-var u,ldf!-spf!-var u)$
symbolic procedure ldf!-sep!-var u$
%Returns all variables w.r.t. which U may be separated;
compl(compl(ldf!-pow!-var u,ldf!-dep!-var u),ldf!-spf!-var u)$
symbolic procedure ldf!-int!-var u$
%Returns all variables w.r.t. which U may be integrated;
if eqcar(mvar u,'df) then
begin scalar v;
v:=ldf!-all!-var u;
while v and u do
<<v:=compl(v,compl(ldt!-dep car u,ldt!-dfvar car u)); u:=cdr u>>;
return v;
end$
symbolic procedure ldf!-int u$
%U is l.d.f, returns U with all possible integrations performed;
%or unchanged if integration is not possible;
begin scalar v,w,z,test; integer nfun;
a:
test:=nil;
w:=ldf!-int!-var u;
nfun:=find!-nfun();
foreach x in w do
if not smember('int,z:=caadr algebraic int(lisp prepf u,x))
or not smember('int,z:=subintf z) then
<<v:=!*a2k list('c,nfun:=nfun+1); test:=t;
mkdep(v . delete(x,ldf!-all!-var u));
u:=addf(z,!*k2f v)>>;
if test then go to a;
return u;
end$
symbolic procedure ldf!-df!-diff u$
%Returns list of all df-kernels which may be obtained;
%from U by differentiation or nil;
begin scalar dfvar,dfsub,v,w,z0,z; integer n0,nmax;
v:=compl(ldf!-dep!-var u,ldf!-spf!-var u);
if not v then return;
w:=foreach x in v collect list(x,add1 ldf!-deg(u,x));
nmax:=maxl foreach x in w collect cadr x;
while (n0:=n0+1) leq nmax and not(z0:=nil) do
<<foreach x in w do if cadr x geq n0 then z0:=(car x) . z0;
z:=z0 . z>>;
z:=reverse z;
dfvar:=foreach x in car z collect list x;
foreach x in cdr z do dfvar:=
append(dfvar,foreach y in dfvar collect car product!-set2(x,y));
foreach x in dfvar do
begin scalar p,q;
p:=x; q:=u;
while p and q and red q do
<<q:=ldf!-simp numr difff(q,car p); p:=cdr p>>;
if pairp q and not red q and eqcar(mvar q,'df) then
dfsub:=(mvar q) . dfsub;
end;
return makeset dfsub;
end$
symbolic procedure ldf!-sub!-var u$
%Returns function w.r.t. which U may be resolved;
begin scalar v,w,z;
w:=ldf!-all!-var u;
foreach x in u do if not v and not eqcar(z:=tvar x,'df)
and monop tc x and spde!-subsetp(w,ldt!-dep x)
and not smember(z,delete(x,u)) then v:=z;
return v;
end$
symbolic procedure ldf!-simp u$
%Returns l.d.f. form of U;
if not u then nil else
if not red u then numr simp prepf !*k2f mvar u else
begin scalar v;
v:=numr simp prepf u;
if not domainp v then v := quotf(v,cdr comfac v);
return absf v
end$
symbolic procedure ldf!-sep u$
%Returns list of l.d.f. into which U has been separated;
begin scalar v; integer k;
if not(v:=ldf!-sep!-var u) then return list u;
foreach x in v do u:=subst(list('ux,1,k:=k+1),x,u);
return foreach x in coeff!-all(u,'ux) collect
ldf!-simp numr simp prepf x;
end$
symbolic procedure ldf!-subf0 u$
%Returns U with CZERO substituted;
ldf!-simp delnil foreach x in u collect ldt!-subt0 x$
%*******************************************************************$
% Procedures for manipulating l.d.t.'s, U is always l.d.t. $
% in this section $
%*******************************************************************$
symbolic procedure ldt!-tvar u$
%U is l.d.t., returns function argument of tvar U;
(if eqcar(x,'df) then cadr x else x) where x=tvar u$
symbolic procedure ldt!-dfvar u$
%U is l.d.t., returns variables w.r.t. which tvar u is derived or nil;
(if eqcar(x,'df) then vlist cddr x else nil) where x=tvar u$
symbolic procedure ldt!-dep u$
%U is l.d.t., returns list of variables x or y which occur as;
%arguments LDT-tvar u;
(if x then cdr x else nil) where x=assoc(ldt!-tvar u,depl!*)$
symbolic procedure ldt!-subt0 u$
%U is l.d.t., returns U if LDT-tvar u is not on czero;
if not member(ldt!-tvar u,czero) then u else nil$
%*******************************************************************$
% Procedures for constructing the determining system $
%*******************************************************************$
symbolic procedure cresys u$
begin scalar r,v,w,lgl,lsub,depl!*!*,list!-sder;
remprop('df,'kvalue); remprop('df,'klist);
remprop('c,'kvalue); remprop('c,'klist);
prload();
rmsubs();
depl!*:=nil;
if car u then
list!-deq:=foreach x in u collect assoc(x,get(car x,'kvalue)) else
list!-deq:=get('deq,'kvalue);
if eqn(length list!-deq,1) then
begin scalar p;
p:=maxmem(function length,makeset sacar('u,list!-deq));
p:=mk!*sq !*k2q p;
list!-sder:=list list(list('sder, cadaar list!-deq),p);
end else if car u then
list!-sder:=foreach x in list!-deq collect
assoc(list('sder,cadar x),get('sder,'kvalue)) else
list!-sder:=get('sder,'kvalue);
if not list!-deq then rederr
"Differential equations not defined";
if not list!-sder then rederr
"Substitutions for derivatives not defined";
mm:=find!-m list!-deq; nn:=find!-n list!-deq;
list!-m:=
makeset foreach x in sacar('u,list!-deq) collect cadr x;
for k:=1:nn do<<w:=!*a2k list('xi,k) . w; v:=!*a2k list('x,k) . v>>;
for k:=1:mm do if member(k,list!-m) then
<<w:=!*a2k list('eta,k) . w; v:=!*a2k list('u,k) . v>>;
for k:=1:nn do r:=(!*a2k list('dx,k)) . r;
for k:=1:mm do r:=(!*a2k list('du,k)) . r;
for k:=1:mm do depl!*!*:=(!*a2k list('eta,k) . v) . depl!*!*;
for k:=1:nn do depl!*!*:=(!*a2k list('xi,k) . v) . depl!*!*;
depl!*:=depl!*!*;
kord!*:=reverse r;
foreach x in list!-sder do
lsub:=((mvar caadr cadr x) . prepsq caar solvef(caadr cadr assoc
(list('deq,cadar x),list!-deq),mvar caadr cadr x)) . lsub;
foreach x in list!-deq do
begin scalar s,z,lx,lu;
z:=caadr cadr x;
lx:=makeset sacar('x,z);
lu:=makeset sacar('u,z);
foreach y in lx do s:=addf(s,
multf(!*k2f !*a2k list('xi,cadr y),numr simp prepsq difff(z,y)));
foreach y in lu do if length y=2 then
s:=addf(s,multf
(!*k2f !*a2k list('eta,cadr y),numr simp prepsq difff(z,y))) else
s:=addf(s, multf(numr zeta!* cdr y,numr simp prepsq difff(z,y)));
s:=numr subf(s,lsub);
s:=numr subf(s,lsub);
lgl:=append(coeff!-all(s,'u),lgl);
end;
uhf:=list(makeset lgl,foreach x in reverse w collect !*k2q x);
end$
lisp rlistat'(cresys)$
symbolic procedure totder(u,i)$
begin scalar z,v,w;
v:=car difff(u,!*a2k list('x,i));
z:=makeset sacar('u,u);
for k:=1:mm do if member(k,list!-m) then
z:=(!*a2k list('u,k)) . z;
foreach x in makeset z do w:=addf(w,
multf(!*k2f !*a2k append(x,list i),car difff(u,x)));
return numr simp prepf addf(v,w);
end$
symbolic procedure zeta!* u$
if not get('deq,'kvalue) and (eqn(mm,0) or eqn(nn,0)) then
rederr"Number of variables not defined" else
if length u geq 3 then
begin scalar v,w;
prload();
if eqn(nn,0) then nn:=find!-n list!-deq;
v:=totder(numr zeta!* reverse cdr reverse u,car reverse u);
for s:=1:nn do w:=addf(w,
multf(!*k2f !*a2k('u . append(reverse cdr reverse u,list s)),
totder(!*k2f !*a2k list('xi,s),car reverse u)));
return simp prepsq(addf(v,negf w) ./ 1);
end else
begin scalar v,w;
prload();
if eqn(nn,0) then
<<nn :=find!-n list!-deq; mm:=find!-m list!-deq>> else
begin scalar p,z;
for k:=1:mm do z:=cons(k,z);
for k:=1:nn do p:=(!*a2k list('x,k)) . p;
for k:=1:mm do p:=(!*a2k list('u,k)) . p;
for k:=1:nn do mkdep((!*a2k list('xi,k)) . p);
for k:=1:mm do mkdep((!*a2k list('eta,k)) . p);
list!-m:=z;
end;
v:=totder(!*k2f !*a2k list('eta,car u),cadr u);
for s:=1:nn do w:=addf(w,
multf(!*k2f !*a2k list('u,car u,s),
totder(!*k2f !*a2k list('xi,s),car reverse u)));
return simp prepsq(addf(v,negf w) ./ 1);
end$
symbolic procedure simpu u$
!*p2q mksp(('u . (car u . reverse ordn cdr u)),1)$
put('u,'simpfn,'simpu)$
put('zeta,'simpfn,'zeta!*)$
symbolic procedure coeff!-all(u,v)$
begin scalar z;
list!-pq:=nil;
splitrec(u,v,1,nil);
foreach x in list!-pq do
z:=(ldf!-simp numr simp prepf cdr x) . z;
return makeset z;
end$
symbolic procedure splitrec(u,v,p,q)$
if domainp u then
begin scalar y;
p:=multf(u,p);
if y:=assoc(q,list!-pq) then
rplacd(y,addf(cdr y,p)) else list!-pq:=(q . p) . list!-pq;
end else
begin
if eqcar(mvar u,v) and length mvar u greaterp 2
then splitrec(lc u,v,p,(lpow u) . q)
else splitrec(lc u,v,!*t2f(lpow(u) .* p),q);
if red u then splitrec(red u,v,p,q);
end$
symbolic procedure find!-m u$
maxl makeset foreach x in sacar('u,u) collect cadr x$
symbolic procedure find!-n u$
begin scalar vx,vu,wx,wu;
vx:=makeset sacar('x,u);
vu:=makeset sacar('u,u);
foreach x in vx do wx:=(cadr x) . wx;
foreach x in vu do if length x geq 3 then
wu:=append(cddr x,wu);
return max(maxl wx,maxl wu);
end$
%*******************************************************************$
% Procedures for solving the determining system $
%*******************************************************************$
symbolic procedure rule0$
%Searches for equations of the form C(I)=0 and stores them on CZERO;
if uhf then foreach x in car uhf do
if not red x and not eqcar(mvar x,'df)
then czero:=(mvar x) . czero$
symbolic procedure rule1$
%Searches for equations of the form DF(function,variable)=0;
%and stores it on the list RDEP;
if uhf and car uhf then
begin scalar dfsub;
foreach x in car uhf do
if not red x and eqcar(mvar x,'df) and eqn(diford mvar x,1)
then rdep:=(mvar x) . rdep;
if rdep then return t;
end$
symbolic procedure rule1!-diff$
%Searches for equations of the form DF(function,variable)=0;
%which may be obtained by a single differentiation and stores it on;
%the list RDEP;
if uhf and car uhf then
begin scalar u,v,z;
foreach x in car uhf do if(z:=ldf!-df!-diff x) then
u:=append(z,u);
foreach x in u do if eqn(diford x,1) then v:=x . v;
rdep:=makeset v;
if rdep then return t;
end$
symbolic procedure rulec l$
%Searches for equations of length L which may be solved for a;
%function and stores the corresponding rules on CSUB;
if uhf and car uhf then
begin scalar v;
foreach u in car uhf do if leqgrt(length u,l,4)
and (v:=ldf!-sub!-var u) and not smember(v,csub)
and not inter(foreach x in csub collect car x,ldf!-fvar u)
then csub:=(v . prepsq caar solvef(u,v)) . csub;
if csub then return t;
end$
symbolic procedure ruledf l$
%Searches for equations of the form DF(function,derivative list)=0;
%the derivative beeing of order L and stores the resulting;
%substitution polynomial on CSUB;
if uhf and car uhf then
begin scalar dfsub;
foreach x in car uhf do
if not red x and eqcar(mvar x,'df) and eqn(diford mvar x,l)
and not smember(ldf!-mvar x,dfsub) then dfsub:=(mvar x) . dfsub;
csub:=foreach x in dfsub collect(cadr x) . crepol x;
if csub then return t;
end$
symbolic procedure ruledf!-diff l$
%Searches for all equations of the form;
%DF(function,derivative list)=0 which may be obtained by;
%differentiation, picks out those of order L and stores;
%the corresponding substitution polynomial on CSUB;
if uhf and car uhf then
begin scalar v,dfsub;
foreach u in car uhf do v:=append(v,ldf!-df!-diff u);
if not(v:=makeset v) then return;
foreach x in v do if eqn(diford x,l) then dfsub:=x . dfsub;
if not dfsub then return;
csub:=((cadar dfsub) . crepol car dfsub) . csub;
if csub then return t;
end$
symbolic procedure rule!-int l$
%Searches for an equation of length L which may be solved for a;
%function after beeing integrated and stores the corresponding;
%rule on CSUB;
if uhf and car uhf then
begin scalar v,w;
foreach u in car uhf do if not csub and leqgrt(length u,l,4)
and (v:=ldf!-sub!-var(w:=ldf!-int u))
then csub:=list(v . prepsq caar solvef(w,v));
if csub then return t;
end$
symbolic procedure simpsys0$
%Removes variable which are stored on list CZERO;
begin scalar u,v;
if pclass=2 then<<write"Entering SIMPSYS0"; terpri2()>>;
u:=delnil foreach x in car uhf collect ldf!-subf0 x;
v:=foreach x in cadr uhf collect ldf!-subf0 numr x ./ denr x;
uhf:=list(makeset u,v);
if pclass=1 then
begin
terpri2();
if eqn(length czero,1) then
write"Substitution" else write"Substitutions";
terpri();
foreach x in czero do
algebraic write (lisp aeval x),":=0";
terpri();
end;
if pclass=2 then<<write"CZERO:="; prettyprint czero; terpri()>>;
czero:=nil;
if pclass=2 then<<write"Leaving SIMPSYS0"; terpri2()>>;
end$
symbolic procedure simpsys!-rdep$
%Removes dependencies which are stored on list RDEP;
begin scalar u,v;
if pclass=2 then<<write"Entering SIMPSYS!-RDEP"; terpri2()>>;
foreach x in rdep do rmdep cdr x;
u:=makeset delnil foreach x in car uhf collect ldf!-simp x;
v:=foreach x in cadr uhf collect simp prepsq x;
uhf:=list(u,v);
if pclass=1 then
begin
terpri();
write"Dependencies removed"; terpri2();
foreach x in rdep do
<<maprin cadr x; prin2!*" independent of ";
maprin caddr x; terpri!* t;>>;
terpri();
end;
if pclass=2 then<<write"RDEP:='"; prettyprint rdep; terpri()>>;
if pclass=2 then<<write"Leaving SIMPSYS!-RDEP"; terpri2()>>;
end$
symbolic procedure simpsys!-sep$
%Performs all possible separations;
if uhf and car uhf then
begin scalar u,v,test;
if pclass=2 then<<write"Entering SIMPSYS!-SEP"; terpri2()>>;
foreach x in car uhf do
if eqn(length(v:=ldf!-sep x),1) then u:=x . u else
begin
u:=append(v,u);
if pclass=1 or pclass=2 then
begin scalar z; integer l;
terpri();
l:=length car uhf-length member(x,car uhf)+1;
write"Equation ",l," separated into the terms";
terpri();
if pclass=1 then for k:=1:length v do
begin
z:=prepf nth(v,k);
!*list := lengthf z geq 50;
algebraic write"Term ",k," ",z;
end;
if pclass=2 then foreach y in v do prettyprint y;
end;
test:=t;
end;
!*list := nil;
if test then uhf:=list(reverse makeset u,cadr uhf);
if pclass=2 then<<write"Leaving SIMPSYS!-SEP"; terpri2()>>;
end$
symbolic procedure simpsys!-sub$
%Performs all substitutions which are stored on CSUB;
if uhf and car uhf then
begin scalar u,v;
if pclass=2 then<<write"Entering SIMPSYS!-SUB"; terpri2()>>;
if pclass=1 then prrule csub;
if pclass=2 then<<write"CSUB:='"; prettyprint csub; terpri()>>;
u:=makeset delnil foreach x in car uhf collect
ldf!-simp numr subf(x,csub);
v:=foreach x in cadr uhf collect subsq(x,csub);
uhf:=list(u,v);
csub:=nil;
if pclass=2 then<<write"Leaving SIMPSYS!-SUB"; terpri2()>>;
end$
symbolic procedure simpsys$
if not uhf then
rederr"The determining system is not defined" else
if not car uhf then
rederr"The determining system completely solved" else
begin scalar u,v; integer nfun;
prload();
u:=makeset delnil foreach x in car uhf collect ldf!-simp x;
v:=foreach x in cadr uhf collect simp prepsq x;
uhf:=list(u,v);
mark0:
if pclass=1 then<<prsys!*"Entering main loop">> else
if pclass=2 then prtlist"Entering main loop";
czero:=csub:=rdep:=nil;
simpsys!-sep();
rule0();
if czero then<<simpsys0(); go to mark0>>;
if rule1() or rule1!-diff() then<<simpsys!-rdep(); go to mark0>>;
if ruledf 2 or rulec 2 or rule!-int 2 or ruledf!-diff 2
or ruledf 3 or rulec 3 or rule!-int 3 or ruledf!-diff 3
or ruledf 4 or rulec 4 or rule!-int 4 or ruledf!-diff 4
or ruledf 5 or rulec 5 or rule!-int 5 or ruledf!-diff 5
then <<simpsys!-sub(); go to mark0>>;
if car uhf then
<<write"Determining system is not completely solved";
terpri2(); prsys!*"The remaining equations are";
if not zerop(nfun:=find!-nfun()) then
write"Number of functions is ",nfun>>;
end$
symbolic procedure crepol u$
begin scalar l1,f; integer pow,nfun;
nfun:=find!-nfun();
l1:=cdr assoc(car(u:=cdr u),depl!*);
while (u:=cdr u) do
begin scalar v;
v:=car u;
if length u=1 or not numberp cadr u then pow:=1 else
<<pow:=cadr u; u:=delete(pow,u);>>;
for k:=1:pow do
begin scalar w;
w:=!*a2k list('c,nfun:=nfun+1);
mkdep(w . delete(v,l1));
if k=1 then f:=w . f;
if k=2 then f:=list('times,w,v) . f;
if k geq 3 then
f:=list('times,w,list('expt,v,k-1)) . f;
end;
end;
return append('(plus),f);
end$
%*************************************************************$
% Procedures for analysing the result $
%*************************************************************$
symbolic procedure cpar u$
begin scalar v;
v:=makeset appends(sacar('xi,u),sacar('eta,u),sacar('c,u));
foreach x in v do if not assoc(x,depl!*) then v:=delete(x,v);
return v;
end$
symbolic procedure makeset!-c!-x u$
if not u then nil else
if member!-c!-x(car u,cdr u) then makeset!-c!-x cdr u else
car u . makeset!-c!-x cdr u$
symbolic procedure member!-c!-x(u,v)$
if not v then nil else
if equal!-c!-x(u,car v) then v else member!-c!-x(u,cdr v)$
symbolic procedure equal!-c!-x(u,v)$
begin scalar p,q;
p:=scar('c,u) or scar('xi,u) or scar('eta,u);
q:=scar('c,v) or scar('xi,v) or scar('eta,v);
return equal(subst('cxx,p,u),subst('cxx,q,v));
end$
symbolic procedure numgen$ length get('gen,'kvalue)$
symbolic operator numgen$
symbolic procedure gengen$
begin scalar u,z,cgen,dgen; integer ngen;
remprop('gen,'kvalue); remprop('gen,'klist);
foreach x in cadr uhf do u:=append(ldf!-fvar numr x,u);
foreach x in makeset u do
begin scalar v,w;
w:=nil ./ 1;
if assoc(x,depl!*) then
v:=foreach y in cadr uhf collect
simp prepsq(ldf!-fvar!-part(numr y,x) ./denr y) else
v:=foreach y in cadr uhf collect
simp prepsq((lcf ldf!-fvar!-part(numr y,x)) ./denr y);
for k:=1:nn do if numr nth(v,k) then
w:=addsq(multsq(nth(v,k),!*k2q !*a2k list('dx,k)),w);
for k:=1:mm do if numr nth(v,nn+k) then
w:=addsq(multsq(nth(v,nn+k),!*k2q !*a2k list('du,k)),w);
if assoc(x,depl!*) then
cgen:=(absf remfacn numr simp prepf numr w) . cgen else
dgen:=(absf remfacn numr simp prepf numr w) . dgen;
end;
dgen:=makeset dgen; cgen:=makeset!-c!-x cgen;
num!-dgen:=length dgen; num!-cgen:=length cgen;
for k:=1:nn do if member(z:=!*k2f !*a2k list('dx,k),dgen) then
<<setk(list('gen,ngen:=add1 ngen),prepf z); dgen:=delete(z,dgen)>>;
for k:=1:mm do if member(z:=!*k2f !*a2k list('du,k),dgen) then
<<setk(list('gen,ngen:=add1 ngen),prepf z); dgen:=delete(z,dgen)>>;
dgen:=sortx(function length,dgen);
foreach x in dgen do setk(list('gen,ngen:=add1 ngen),prepf x);
cgen:=sortx(function length,cgen);
foreach x in cgen do setk(list('gen,ngen:=add1 ngen),prepf x);
end$
symbolic operator gengen$
algebraic procedure comm(a,b)$
begin scalar z;
if (lisp length list!-deq)=0 then
<<write"Differential equations not defined"; return nil>>;
z:= (for k:=1:nn sum df(a,dx k)*df(b,x k)-df(b,dx k)*df(a,x k))
+(for k:=1:mm sum df(a,du k)*df(b,u k)-df(b,du k)*df(a,u k))$
return z;
end$
algebraic procedure result$
begin integer l;
if (l:=lisp length list!-deq)=1 then
write"The differential equation" else
write"The differential equations";
for j:=1:l do
begin scalar z; integer i,k;
lisp(z:=car cadadr nth(list!-deq,j));
i:=lisp cadar nth(list!-deq,j);
k:=lisp lengthf prepf z;
symbolic(!*list := k>40);
write"DEQ(",i,"):=",lisp prepf z;
end;
!*list := nil;
if (lisp length car uhf) neq 0 then
prsys!*"The determining system is not completely solved" else
<<lisp gengen(); prgen(); comm!-tab()>>;
end$
%*************************************************************$
% Procedures for displaying the output $
%*************************************************************$
symbolic procedure prsys!* u$
if uhf and car uhf then
<<terpri(); write u; terpri(); prsys(); terpri()>>$
symbolic procedure prsys$
begin scalar v;
terpri();
remprop('gl,'kvalue); remprop('gl,'klist);
for k:=1:length car uhf do
begin scalar z; integer l;
z:=prepf nth(car uhf,k);
l:=lengthf prepf nth(car uhf,k);
!*list := l>50;
algebraic write"GL(",k,"):=",z;
setk(list('gl,k),z);
end;
terpri2();
write"The remaining dependencies";
terpri2();
v:=makeset
appends(sacar('xi,car uhf),sacar('eta,car uhf),sacar('c,car uhf));
foreach x in v do write!-dep x;
!*list := nil;
end$
symbolic procedure prrule u$
begin
terpri2();
if eqn(length u,1) then
write"Substitution" else write"Substitutions";
terpri2();
foreach x in u do
<<maprin car x; prin2!*" = "; maprin cdr x; terpri!* t;>>;
terpri();
foreach x in u do foreach y in sacar('c,cdr x) do write!-dep y;
end$
symbolic procedure prtlist u$
<<write u; terpri2(); write"DEPL!*:='"; prettyprint depl!*;
write"UHF:='"; prettyprint uhf>>$
symbolic procedure write!-df!-sub$
if get('df,'kvalue) then
begin scalar w;
w:=get('df,'kvalue);
remprop('df,'kvalue);
terpri();
if length w=1 then write"Constraint" else write"Constraints";
terpri2();
foreach x in w do
begin scalar u,v;
u:=car x;
v:=cadadr x;
algebraic write lisp u,":=",lisp prepsq v;
terpri2();
end;
put('df,'kvalue,w);
end$
algebraic procedure prgen$
begin scalar lcpar;
for k:=1:nn do <<order dx k; factor dx k>>;
for k:=1:mm do factor du k$
lisp(lcpar:=cpar get('gen,'kvalue));
write"The symmetry generators are";
for k:=1:numgen() do
if (lisp lengthf reval list('gen,k)) leq 60 then
<<symbolic(!*list := nil); write"GEN(",k,"):=",gen k>> else
begin scalar z; integer r,s,nt; operator gen!*;
nt:=lisp length(z:=numr simp reval list('gen,k));
r:=lisp maxl foreach x in z collect abs comfacn list x;
if r=1 then r:=0 else r:=lisp flatsizec r;
for l:=1:nt do gen!* l:=lisp prepf list nth(z,l);
for l:=1:nt do
begin
symbolic(!*list := lengthf prepf tc nth(z,l) geq 56);
s:=lisp abs comfacn list nth(z,l);
if r=0 then s:=0 else
if s=1 then s:=-1 else s:=lisp flatsizec s;
if l=1 then write"GEN(",k,"):=",lisp blanks(r-s+1),gen!* 1 else
if minus!-f gen!* l=t then
write lisp blanks(r-s+6),gen!* l else
write lisp blanks(r-s+6)," + ",gen!* l;
end;
clear gen!*;
symbolic(!*list := nil);
end;
if (lisp length lcpar) neq 0 then
<<write"The remaining dependencies"; lisp terpri()>>;
for k:=1:(lisp length lcpar) do
<<lisp write!-dep nth(lcpar,k);>>;
if (lisp length lcpar) neq 0 then lisp terpri();
lisp write!-df!-sub();
end$
algebraic procedure comm!-tab$
if (lisp num!-dgen) geq 2 then
begin integer nd; scalar v;
nd:=lisp num!-dgen;
write"The non-vanishing commutators of the finite subgroup";
for i:=1:nd-1 do for j:=(i+1):nd do
if(v:=comm(gen i,gen j)) neq 0 then
if (lisp lengthf reval v) leq 60 then
<<symbolic(!*list := nil); write"COMM(",i,",",j,"):= ",v>> else
begin integer r,s,nt; scalar z; operator gen!*;
nt:=lisp length(z:=numr simp reval v);
r:=lisp maxl foreach x in z collect abs comfacn list x;
if r=1 then r:=0 else r:=lisp flatsizec r;
for i:=1:nt do gen!* i:=lisp prepf list nth(z,i);
for l:=1:nt do
begin
symbolic(!*list := lengthf reval list('gen!*,l) geq 63);
s:=lisp abs comfacn list nth(z,l);
if r=0 then s:=0 else
if s=1 then s:=-1 else s:=lisp flatsizec s;
if l=1 then
write"COMM(",i,",",j,"):=",lisp blanks(r-s+1),gen!* 1 else
if minus!-f gen!* l=t then
write lisp blanks(r-s+9),gen!* l else
write lisp blanks(r-s+9)," + ",gen!* l;
end;
clear gen!*;
end;
symbolic(!*list := nil);
end$
symbolic procedure write!-dep u$
if assoc(reval u,depl!*) then
begin scalar v;
v:=cdr assoc(u,depl!*);
write car u,"(",cadr u,") depends on ";
write caar v,"(",cadar v,")";
foreach x in cdr v do write",",car x,"(",cadr x,")";
terpri2();
end$
symbolic operator write!-dep$
symbolic procedure find!-nfun$
if not get('c,'klist) then 0 else
maxl makeset foreach x in get('c,'klist) collect cadar x$
endmodule;
end;