off echo;
lisp(depl!*:=nil)$ % clearing of all dependences
setcrackflags()$
lisp <<
write"-------------------------------------------------------"$terpri()$
write"This file is supposed to provide an automatic test of"$terpri()$
write"the program APPLYSYM. On the other hand the application"$terpri()$
write"of APPLYSYM is an interactive process, therefore the"$terpri()$
write"interested user should inspect the example described"$terpri()$
write"in APPLYSYM.TEX which demonstrates the application"$terpri()$
write"of symmetries to integrate a 2nd order ODE."$terpri()$
write"Here the program QUASILINPDE for integrating first"$terpri()$
write"order quasilinear PDE is demonstrated."$terpri()$terpri()$
write"The following equation comes up in the elimination"$terpri()$
write"of resonant terms in normal forms of singularities"$terpri()$
write"of vector fields (C.Herssens, P.Bonckaert, Limburgs"$terpri()$
write"Universitair Centrum/Belgium, private communication)."$terpri()$
write"-------------------------------------------------------"$terpri()$
>>$
depend w,x,y,z$
QUASILINPDE( df(w,x)*x+df(w,y)*y+2*df(w,z)*z-2*w-x*y, w, {x,y,z} )$
nodepend w,x,y,z$
lisp <<terpri()$
write"-------------------------------------------------------"$terpri()$
write"Comment:"$terpri()$
write"The result means that w is defined implicitly through "$terpri()$
write" "$terpri()$
write" - log(z)*x*y + 2*w sqrt(z)*x sqrt(z)*y "$terpri()$
write"0 = ff(---------------------,-----------,-----------) "$terpri()$
write" z z z "$terpri()$
write" "$terpri()$
write"with an arbitrary function ff of 3 arguments. As the PDE"$
terpri()$
write"was linear, the arguments of ff are such that we can "$terpri()$
write"solve for w: "$terpri()$
write" "$terpri()$
write" sqrt(z)*x sqrt(z)*y "$terpri()$
write"w = log(z)*x*y/2 + z*f(-----------,-----------)"$terpri()$
write" z z "$terpri()$
write""$terpri()$
write"with an arbitrary function f of 2 arguments."$terpri()$terpri()$
write"-------------------------------------------------------"$terpri()$
write"The following PDEs are taken from E. Kamke,"$terpri()$
write"Loesungsmethoden und Loesungen von Differential-"$terpri()$
write"gleichungen, Partielle Differentialgleichungen"$terpri()$
write"erster Ordnung, B.G. Teubner, Stuttgart (1979)."$terpri()$
>>$
write"------------------- equation 1.4 ----------------------"$
lisp(depl!*:=nil);
depend z,x,y$
QUASILINPDE( x*df(z,x)-y, z, {x,y})$
write"------------------- equation 2.5 ----------------------"$
lisp(depl!*:=nil);
depend z,x,y$
QUASILINPDE( x**2*df(z,x)+y**2*df(z,y), z, {x,y})$
write"------------------- equation 2.6 ----------------------"$
lisp(depl!*:=nil);
depend z,x,y$
QUASILINPDE( (x**2-y**2)*df(z,x)+2*x*y*df(z,y), z, {x,y})$
write"------------------- equation 2.7 ----------------------"$
lisp(depl!*:=nil);
depend z,x,y$
QUASILINPDE( (a0*x-a1)*df(z,x)+(a0*y-a2)*df(z,y), z, {x,y})$
write"------------------- equation 2.14 ---------------------"$
lisp(depl!*:=nil);
depend z,x,y$
QUASILINPDE( a*df(z,x)+b*df(z,y)-x**2+y**2, z, {x,y})$
write"------------------- equation 2.16 ---------------------"$
lisp(depl!*:=nil);
depend z,x,y$
QUASILINPDE( x*df(z,x)+y*df(z,y)-a*x, z, {x,y})$
write"------------------- equation 2.20 ---------------------"$
lisp(depl!*:=nil);
depend z,x,y$
QUASILINPDE( df(z,x)+df(z,y)-a*z, z, {x,y})$
write"------------------- equation 2.21 ---------------------"$
lisp(depl!*:=nil);
depend z,x,y$
QUASILINPDE( df(z,x)-y*df(z,y)+z, z, {x,y})$
write"------------------- equation 2.22 ---------------------"$
lisp(depl!*:=nil);
depend z,x,y$
QUASILINPDE( 2*df(z,x)-y*df(z,y)+z, z, {x,y})$
write"------------------- equation 2.23 ---------------------"$
lisp(depl!*:=nil);
depend z,x,y$
QUASILINPDE( a*df(z,x)+y*df(z,y)-b*z, z, {x,y})$
write"------------------- equation 2.24 ---------------------"$
lisp(depl!*:=nil);
depend z,x,y$
QUASILINPDE( x*(df(z,x)-df(z,y))-y*df(z,y), z,{x,y})$
write"------------------- equation 2.25 ---------------------"$
lisp(depl!*:=nil);
depend z,x,y$
QUASILINPDE( x*df(z,x)+y*df(z,y)-az, z, {x,y})$
write"------------------- equation 2.26 ---------------------"$
lisp(depl!*:=nil);
depend z,x,y$
QUASILINPDE( x*df(z,x)+y*df(z,y)-z+x**2+y**2-1, z, {x,y})$
write"------------------- equation 2.39 ---------------------"$
lisp(depl!*:=nil);
depend z,x,y$
QUASILINPDE( a*x**2*df(z,x)+b*y**2*df(z,y)-c*z**2, z, {x,y})$
write"------------------- equation 2.40 ---------------------"$
lisp(depl!*:=nil);
depend z,x,y$
QUASILINPDE( x*y**2*df(z,x)+2*y**3*df(z,y)-2*(y*z-x**2)**2, z,
{x,y})$
write"------------------- equation 3.12 ---------------------"$
lisp(depl!*:=nil);
depend w,x,y,z$
QUASILINPDE( x*df(w,x)+(a*x+b*y)*df(w,y)+(c*x+d*y+f*z)*df(w,z), w,
{x,y,z})$
write"------------------------ end --------------------------"$
lisp(depl!*:=nil);
end;