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dsolvexpde.old
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dsolvexpde.old
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uniqueName := proc ( un )
global _uniqueCounter;
if not assigned ( _uniqueCounter[un] ) then
RETURN ( FAIL );
fi;
_uniqueCounter[un] := _uniqueCounter[un] + 1;
un[_uniqueCounter[un]];
end;
#
#
resetUniqueName := proc ( uc, un )
global _uniqueCounter;
_uniqueCounter[un] := uc;
end;
#
#
resetUniqueName ( 0, GUNIQ );
#
#
genUniqueName := proc ()
local res;
res := uniqueName ( GUNIQ );
res := cat ( op (0, res), op(res) );
end;
#
#
#
pdenormal := proc ( e )
dsimplify ( e, symbolic );
end;
#
#
#
`type/xpdeconst` := proc ( e, cnl )
if type ( e, name ) then
if type ( e, indexed ) then
if type ( op (0, e ), name ) and type ( op ( e ), constant ) and
member ( op ( 0, e ), cnl ) then
RETURN ( true );
fi;
fi;
fi;
RETURN ( false );
end;
#
#
#
`type/xpdeconstfun` := proc ( e, cnl )
if type ( e, functionx (name) ) then
if type ( op ( 0, e ), indexed ) then
if type ( op ( op (0, e) ), constant ) and
type ( op ( 0, op ( 0, e ) ), name ) and
member ( op ( 0, op ( 0, e ) ), cnl ) then
RETURN ( true );
fi;
fi;
fi;
RETURN ( false );
end;
#
#
getxpdeconst := proc ( e )
local idts,
idtsc, idtsf,
idtsnams;
if nargs > 1 then
idtsnams := args[2];
else
idtsnams := [[_CC], [_FF]];
fi;
idts := [op(indets ( e ))];
idtsc := select ( type, idts, name );
idtsc := select ( type, idts, indexed );
idtsc := select ( (xx,yy) -> member(op ( 0, xx ), yy ), idts, idtsnams[1] );
idtsf := select ( type, idts, functionx (name) );
idtsf := select ( xx-> type ( op(0,xx), indexed), idts );
idtsf := select ( (xx,yy)->has ( op(0, op(0, xx ) ), yy ), idtsf, idtsnams[2] );
[idtsc, idtsf];
end;
#
#
#
xpdeconstsl := proc ( cns, constnams, newnam )
local sl,
i, j, k,
cc,
lc;
j := 1;
sl := [];
for i from 1 to nops ( constnams ) do
cc := constnams[i];
lc := select ( has, cns, cc );
lc := map ( xx -> eval(condev2 ( type ( xx, functionx (name) ), 'op ( 0, xx )', 'xx' )),
lc );
lc := sort ( lc, (xx,yy) -> evalb ( op(xx) < op (yy) ) );
for k from 1 to nops ( lc ) do
sl := [op(sl), lc[k] = newnam[j]];
j := j + 1;
od;
od;
sl;
end;
#
#
#
xpdeconstnormal := proc ( e, constnams )
local cns,
sl1,
sl2,
e1;
cns := getxpdeconst ( e, constnams );
sl1 := xpdeconstsl ( cns[1], constnams[1], '_CC' );
sl2 := xpdeconstsl ( cns[2], constnams[2], '_FF' );
e1 := subs ( sl1, subs ( sl2, e ) );
e1;
end;
#
#
#
#
makesolbasis := proc ( loes )
local cns,
res1,
i, j,
sl;
cns := getxpdeconst ( loes );
cns := [op(cns[1]), op(cns[2])];
res1 := [];
for i from 1 to nops ( cns ) do
sl := eval(condev2 ( has ( cns[1], _CC ), 'cns[1] = 1', 'cns[1] = cns[1]' ));
sl := sl, seq ( cns[j] = 0, j=2..nops ( cns ) );
res1 := [ op(res1), subs ( [sl], loes ) ];
cns := [op(cns[2..-1]), cns[1]];
od;
res1;
end;
#
#
#
xpdegensol := proc ( solbasis, knownsols )
local fn,
ord,
vars,
sol,
res;
fn := lhs ( solbasis[1][1] );
vars := [op(fn)];
ord := nops ( solbasis );
if hasoptname ( [args[3..-1]], inhom ) then
solbasis[1];
else
for sol in solbasis do
res := funclincomb ( rhs ( sol[1] ), knownsols, vars, ord );
if res = FAIL then
RETURN ( sol );
fi;
od;
RETURN ( FAIL );
fi;
end;
#
#
#
vectfform := proc ( loes, fs, vB, cnstn, funn )
local fs1,
cnstl, funl,
rsl,
resl, sl,
i;
fs1 := subs ( loes, fs );
cnstl := select ( has, [op(indets ( loes ))], cnstn );
cnstl := sort ( cnstl, (x,y)-> evalb( op(x) <= op(y) ) );
funl := select ( has, [op(indets ( loes ))], funn );
funl := sort ( funl, (x,y)-> evalb( op(op(0,x)) <= op(op(0,y)) ) );
cnstl := [op(cnstl), op (funl)];
fs1 := convert ( zip ( (x,y)->x*y, fs1, vB ), `+` );
rsl := [0 $ (nops ( cnstl ) - 1), 1 ];
resl := [];
for i from 1 to nops ( cnstl ) do
rsl := [rsl[-1], op(rsl[1..-2])];
sl := zip ( (x,y)->x=y, cnstl, rsl );
sl := map ( proc ( x,y) option operator;
if has ( lhs(x), y ) and rhs(x) = 1 then
lhs(x)=lhs(x);
else
x;
fi; end, sl, funn );
resl := [op(resl), normal ( eval ( value ( subs ( sl, fs1 ) ) ) ) ];
od;
resl;
end;
#
#
#
xpdesolve := proc ( eql, fnl )
local res, ccl, ffl,
vB, vfeq,
optlist,
orderopt,
i, j,
sl,
res1,
cns;
global pdenormal, pdelexorder, pdesolvesysXopt;
resetUniqueName ( 0, `_CC` );
resetUniqueName ( 0, `_FF` );
optlist := select ( type, [args[3..-1]], equation );
orderopt := select ( (xx,yy)->lhs(xx)=yy, optlist, order );
if nops ( orderopt ) = 1 then
orderopt := rhs ( orderopt[1] );
pdelexorder := orderopt;
else
pdelexorder := lexOrderA;
fi;
pdesolvesysXopt := {};
if has ( [args[3..-1]], rational ) then
pdesolvesysXopt := pdesolvesysXopt union {rational};
fi;
if has ( [args[3..-1]], hypexp ) then
pdesolvesysXopt := pdesolvesysXopt union {hypexp};
fi;
res := pdesolvesysX ( eql, fnl );
ccl := select ( ( x, y )-> has ( x, y), indets ( res ), _CC );
ccl := select ( type, ccl, xpdeconst ( [_CC] ) );
ffl := select ( ( x, y )-> has ( x, y), indets ( res ), _FF );
ffl := select ( type, ffl, xpdeconstfun ( [_FF] ) );
resetUniqueName ( 0, `_DD` );
resetUniqueName ( 0, `_GG` );
ccl := map ( x-> x = uniqueName (`_DD`), ccl );
ffl := map ( x-> x = uniqueName (`_GG`)(op(x)), ffl );
res := subs ( _GG= _FF, subs ( ffl, res ) );
res := subs ( _DD= _CC, subs ( ccl, res ) );
if has ( [args[3..-1]], basis ) then
res := makesolbasis ( res );
fi;
if has ( [args[3..-1]], vectf ) then
vfeq := select ( x->has ( x, vectf ), [args[3..-1]] )[1];
if type ( vfeq, equation ) then
vB := rhs ( vfeq );
else
vB := [p, q];
fi;
res := vectfform ( res, fnl, vB, '_CC', '_FF' );
fi;
res;
end;
pdesolvesysX := proc ( eql, fnl )
local i, vars, lexord,
oldcotes,
jbeql, jbeql1,
zeroeq,
sl, slc, slcns,
odecnsl,
res, erg, loes, loesv,
fnakt,
nn,
aktfun, fun,
mindeq, mindeql, mindeql0, mindeql1, mindeql2, mindeqll, mindeqlfn,
varszero, varsremain,
varsremfl,
dits, dits1,
zerobed, zerovars, zerovars1,
varsode, mindeqode, odeloes, odeloes1,
neweql,
fnl1,
constlist, constlistnew,
dummyf, dummysols,
ndeg, var1,
funname, constfunl, nondeql, fnx, vrs;
global pdelexorder, pdesolvesysXopt, _xpderestdeqs;
if nops ( fnl ) = 0 then
if nops ( eql ) = 0 then
RETURN ( [] );
fi;
fi;
vars := dgetvars ( fnl );
print ( fnl );
if nops ( eql ) = 0 then
RETURN ( map ( (x,vv)->x=uniqueName(`_FF`)(op(vv)), fnl, vars ) );
fi;
############################## experimental code
jbeql := eql;
mindeql2 := select ( (x,y,z)->type(x, isode(y,z) ), jbeql, vars, fnl );
mindeql2 := select ( (x,y)-> not type(x, elemdiff(y)), mindeql2, fnl );
for mindeq in mindeql2 do
# mindeq := mindeql2[1];
jbeql := remove ( (x,y)->x=y, jbeql, mindeq );
dits := dindets ( mindeq, fnl );
aktfun := select ( (x,y)->has(y,x), fnl, dits ) ;
aktfun := aktfun[1];
varsode := [op(aktfun)];
dits1 := map ( difflist, dits );
var1 := `union` ( op ( map ( x-> {op(x[2..-1])}, dits1 ) ) );
zerobed := select ( (x, y) -> type ( x, elemdiff ( y ) ), jbeql, aktfun );
zerovars := map ( x-> op ( 2, x ), zerobed );
zerovars1 := [op({op(var1),op(zerovars)})];
ndeg := max ( op ( map ( x->nops(x)-1, dits1 ) ) );
dummyf := GG( op (var1) );
mindeqode := subs ( aktfun = dummyf, mindeq );
if not has ( pdesolvesysXopt, [rational, hypexp] ) then
odeloes := dsolve ( mindeqode, dummyf, explicit = true );
if has ( [odeloes], [int,Int] ) then odeloes:= op([]); fi;
if nops([odeloes]) = 0 then
odeloes := dsolve ( mindeqode, dummyf, [linear], explicit);
if has ( [odeloes], [int,Int] ) then odeloes:= op([]); fi;
fi;
if nops ( [odeloes] ) > 0 then
break;
else
next;
fi;
elif has ( pdesolvesysXopt, hypexp ) then
vrs := pruneleft ( (a,b)->a<>b, vars, op(var1) );
odeloes := expratlode ( mindeqode, dummyf, vars );
if nops ( [odeloes] ) > 0 then
if nops ( odeloes ) = 2 and type (odeloes[1], list) then
# this is the case for an inhomogeneous equation
odeloes := dummyf = add ( _C||i * odeloes[1][i], i=1..nops(odeloes[1]) ) +
odeloes[2];
else
odeloes := dummyf = add ( _C||i * odeloes[i], i=1..nops(odeloes) );
fi;
break;
else
next;
fi;
else
# odeloes := dsolve ( mindeqode, dummyf, explicit = true );
odeloes := ratlode ( mindeqode, dummyf );
if nops ( [odeloes] ) > 0 then
odecnsl := select ( has, odeloes[2], _C );
slcns := [seq ( odecnsl[i] = _C||i, i = 1..nops ( odecnsl ) )];
odeloes := subs ( slcns, odeloes[1] );
odeloes := dummyf = odeloes;
break;
else
next;
fi;
fi;
od;
if nops ( [odeloes] ) > 0 and
type ( odeloes, equation ) and lhs ( odeloes ) = dummyf then
# solution found
varsremain := remove ( has, varsode, zerovars1 );
odeloes1 := odeloes;
constlist := [seq ( _C||i, `i` = 1..ndeg )];
constlist := select ( (x,y)->has(y,x), constlist, odeloes1 );
constlistnew := [ seq ( uniqueName (`_CC`), `i` = 1..nops ( constlist ) ) ];
slc := zip ( (x,y)->x=y, constlist, constlistnew );
odeloes1 := subs ( slc, odeloes1 );
constlist := constlistnew;
fnl1 := [];
varsremfl := false;
neweql := [];
if nops ( varsremain ) > 0 or true then
varsremfl := true;
if nops ( constlist ) < ndeg then
print ( `solutions incomplete` );
fi;
funname := op ( 0, aktfun );
constfunl := [seq ( cat(funname, `_`, i )(op(vars)),
`i` = 1..nops ( constlist ) )];
neweql := outer ( [constfunl, zerovars1 ], (x,y)->diff(x,y) );
sl := zip ( (x,y)-> x=y, constlist, constfunl );
odeloes1 := subs ( sl, odeloes1 );
fnl1 := constfunl;
fi;
odeloes1 := subs ( dummyf = aktfun, odeloes1 );
jbeql1 := jbeql;
jbeql1 := pdenormal ( eval ( value ( subs ( odeloes1, jbeql1 ) ) ) );
jbeql1 := select ( x -> x <> 0, jbeql1 );
jbeql1 := [op( jbeql1), op(neweql) ];
if nops ( jbeql1 ) = 0 then
# setcotes ( fnl, oldcotes );
RETURN ( [odeloes1] );
fi;
fnl1 := [op(remove ( has, fnl, aktfun )), op(fnl1) ];
# setcotes ( fnl, oldcotes );
res := pdesolvesysX ( jbeql1, fnl1 );
if varsremfl then
dummysols := select ( (x,y)->has(lhs(x),y), res, constfunl );
res := remove ( (x,y)->has(lhs(x),y), res, constfunl );
res := subs ( dummysols, [ odeloes1, op(res) ] );
res := pdenormal ( res );
RETURN ( res );
else
erg := [ odeloes1, op(res) ];
RETURN ( erg );
fi;
fi;
############################# end experimental code
lexord := pdelexorder ( fnl );
oldcotes := setcotes ( fnl, lexord );
jbeql := makeJanetBase ( eql, fnl );
jbeql := csimpJanetBase ( jbeql, fnl );
if _insolvJanet = true then
RETURN ( [] );
fi;
zeroeq := select ( x->type(x,functionx (name)), jbeql );
if nops ( zeroeq ) <> 0 then
sl := map ( x -> x = 0, zeroeq );
jbeql1 := remove ( (x,y)->member(x,y), jbeql, zeroeq );
fnl1 := remove ( member, fnl, zeroeq );
jbeql1 := pdenormal ( eval ( subs ( sl, jbeql1 ) ) );
setcotes ( fnl, oldcotes );
res := pdesolvesysX ( jbeql1, fnl1 );
erg := [op(res), op(sl )];
RETURN ( erg );
fi;
# dzerothorder returns [eq, ht, rest] or FAIL
mindeql0 := map ( (x,f) -> dzerothorder ( x, f ), jbeql, fnl );
mindeql0 := select ( x->x <> FAIL, mindeql0 );
# mindeql0 contains zeroth order equations, that allow to express a function
# directly
if nops ( mindeql0 ) > 0 then
jbeql1 := select ( (x,y)->not member ( x, y ), jbeql, map ( x->x[1], mindeql0) );
loes := map ( x->x[2]= -x[3], mindeql0 );
jbeql1 := pdenormal ( eval ( subs ( loes, jbeql1 ) ) );
jbeql1 := select ( x->x<>0, jbeql1 );
fnl1 := select ( (x,y) -> not member ( x, y ), fnl, map ( lhs, loes ) );
setcotes ( fnl, oldcotes );
erg := pdesolvesysX ( jbeql1, fnl1 );
erg := [op(loes), op(erg)];
RETURN ( erg );
fi;
# if mindeql0 is empty then we have to go further:
#
# mindeql contains the odes in jbeql
#
mindeql := select ( (x,y,z)->type(x, isode(y,z) ), jbeql, vars, fnl );
mindeql1 := select ( (x,y)->type(x, elemdiff(y)), mindeql, fnl );
# contains the elementary differential monomials, i.e. diff(xi,x), diff(eta,x) etc.
mindeql2 := select ( (x,y)-> not type(x, elemdiff(y)), mindeql, fnl );
# contains the others, potential equations to be solved
nondeql := select ( x->not has ( x, [diff,Diff] ), jbeql );
if nops(mindeql) = 0 then
print ( jbeql );
ERROR ( `couldn't reduce to ode` );
fi;
if true then
# special routine to take care of functions that have as equations only elemdiffs
# and as such can be trivially solved.
mindeqll := map ( (f,x) -> select ( has, x, f ), fnl, mindeql1 );
# mindeqll is a list of list, which sorts mindeql1 by functions in fn
loes := [];
# now we go function by function
fnx := [];
for mindeqlfn in mindeqll do
if nops ( mindeqlfn ) = 0 then next; fi;
fnakt := op(1, mindeqlfn[1]);
nn := nops ( select ( has, jbeql, fnakt ) );
# nn is the number of ocurrences of the current function in the list
# of all equations ( jbeql )
# nops ( mindeqlfn ) is the number of elemdiff monomial equations for the
# current function
# if they are equal then we can solve :
#
if nn = nops ( mindeqlfn ) then
fnx := [op(fnx), fnakt];
varszero := `union` (op ( map ( x-> { op(2,x) }, mindeqlfn ) ) );
varsremain := remove ( member, vars, varszero );
if nops ( varsremain ) > 0 then
loes := [op(loes), fnakt = uniqueName ( `_FF` ) (op(varsremain)) ];
else
loes := [op(loes), fnakt = uniqueName(`_CC`) ];
fi;
fi;
od;
if nops ( loes ) <> 0 then
jbeql1 := pdenormal ( eval ( subs ( loes, jbeql ) ) );
jbeql1 := select ( x->x<>0, jbeql1 );
fnl1 := select ( (x,y) -> not member ( x, y ), fnl, map ( lhs, loes ) );
setcotes ( fnl, oldcotes );
erg := pdesolvesysX ( jbeql1, fnl1 );
erg := [op(loes), op(erg)];
RETURN ( erg );
fi;
# end of special routine
fi;
if nops ( mindeql2 ) = 0 then
# then only elemdiffs are in mindeql
# sort by function name
mindeqll := map ( (f,x) -> select ( has, x, f ), fnl, mindeql );
loes := [];
fnx := [];
for mindeqlfn in mindeqll do
if nops ( mindeqlfn ) = 0 then next; fi;
varszero := `union` (op ( map ( x-> { op(2,x) }, mindeqlfn ) ) );
if varszero = {op(vars) } then
fun := op(1, mindeqlfn[1]);
fnx := [op(fnx), fun];
loes := [op(loes), fun = uniqueName (`_CC`) ];
fi;
od;
if nops ( loes ) = 0 then
_xpderestdeqs := jbeql;
print ( `Residual equations = `, _xpderestdeqs );
setcotes ( fnl, oldcotes );
RETURN ( [] );
fi;
jbeql1 := pdenormal ( eval ( subs ( loes, jbeql ) ) );
jbeql1 := select ( x->x<>0, jbeql1 );
fnl1 := select ( (x,y) -> not member ( x, y ), fnl, map ( lhs, loes ) );
setcotes ( fnl, oldcotes );
erg := pdesolvesysX ( jbeql1, fnl1 );
erg := [op(loes), op(erg)];
RETURN ( erg );
fi;
for mindeq in mindeql2 do
# mindeq := mindeql2[1];
jbeql := remove ( (x,y)->x=y, jbeql, mindeq );
dits := dindets ( mindeq, fnl );
aktfun := select ( (x,y)->has(y,x), fnl, dits ) ;
aktfun := aktfun[1];
varsode := [op(aktfun)];