## equations.txt     20220419
# definition of equations E6, SE6, E5, SE5, E4, SE4, ST4,
#  Z4, E3, SE3, Z3, E3a, E43, E32, E2

with(DEtools):
z:=x*dx:

## Equation E6 =E6(e1,e2,e3,e4,e5,e6,e7,e8,e9)
s11:=e1+e2+e3:
s12:=e4+e5+e6:
s13:=e7+e8+e9:
s21:=e1*e2+e1*e3+e2*e3:
s22:=e4*e5+e4*e6+e5*e6:
s23:=e7*e8+e7*e9+e8*e9:
s31:=e1*e2*e3:
s32:=e4*e5*e6:
s33:= e7*e8*e9:
s:=-(s11+s12+s13-6)/3:

T13:=-3:
T12:=-9+s11-2*s13: 
T11:=-8+1/3*s11^2+2/3*s11*s13-1/3*s12^2+1/3*s13^2+s11-5*s13-s21+s22-2*s23:
T23:=3: 
T22:=18+s13-2*s11:
T21:=35-1/3*s11^2-2/3*s11*s13+1/3*s12^2-1/3*s13^2-7*s11+5*s13+2*s21-s22+s23:

T10:=(-5+2/3*s11-s21+s31+s22-s32-14/3*s13-5*s23-3*s33
  +1/3*s11^2+2/3*s11*s13-1/3*s12^2+1/3*s13^2-1/3*s11*s21
  +2/3*s11*s23+1/3*s12*s22-2/3*s13*s21+2/3*s13*s22
  +1/3*s13*s23+2/27*s11^3+2/9*s11^2*s13-2/27*s12^3-2/9*s12^2*s13)/2:

T20:=-T10+19+1/27*s11^3+1/9*s11^2*s13-1/9*s11*s12^2+1/9*s11*s13^2
  -2/27*s12^3-1/9*s12^2*s13+1/27*s13^3-2/3*s11^2-4/3*s11*s13
  +1/3*s11*s22+2/3*s12^2+1/3*s22*s12-2/3*s13^2+1/3*s22*s13
  -5*s11+4*s13+3*s21-2*s22-s31-s32-s33:

T0:=mult(z+s,z+1+s,z+2+s,z+e7,z+e8,z+e9,[dx,x]):
T1:=mult(z+2+s,z+1+s,(T13*mult(z,z,z,[dx,x])+T12*mult(z,z,[dx,x])+T11*z+T10),[dx,x]):
T2:=mult(z+2+s,(T23*mult(z,z,z,[dx,x])+T22*mult(z,z,[dx,x])+T21*z+T20),[dx,x]):
T3:=mult(-z-3+e1,-z-3+e2,-z-3+e3,[dx,x]):
E6:=T0+mult(T1,dx,[dx,x])+mult(T2,dx,dx,[dx,x])+mult(T3,dx,dx,dx,[dx,x]):

# remark
B1:=T13*mult(z,z,z,[dx,x])+T12*mult(z,z,[dx,x])+T11*z+T10:
B2:=T23*mult(z,z,z,[dx,x])+T22*mult(z,z,[dx,x])+T21*z+T20:

## Equation SE6 =SE6(a,b,c,g,p,q,r)
etop6:={e1=p+r+1,e2=a+c+p+r+2,e3=2*a+2*c+g+p+r+3,e4=q+r+1,
       e5=b+c+q+r+2,e6=2*b+2*c+g+q+r+3,e7=-2*c-p-q-r-1,
       e8=-a-b-2*c-p-q-r-g-2,e9=-2*a-2*b-2*c-p-q-r-g-3,s=-r}:
SE6:=subs(etop6,E6):

## Equation E5=E5(e1,e2,e3,e4,e5,e6,e7,e8)
E5:=rightdivision(subs(e9=0,E6),dx,[dx,x])[1]:

## Equation SE5=SE5(a,b,c,b,p,q)
etop5:={e1=-2*a-2*b-2*c-g-q-2, e2=-a-2*b-c-g-q-1, e3=-2*b-q,
        e4=-2*a-2*b-2*c-g-p-2, e5=-b-2*a-c-g-p-1, e6=-2*a-p,
        e7=2*a+2*b+g+2, e8=a+b+1}:
SE5:=subs(etop5,E5):

## Eqation E4=E4(e1,e2,e3,e4,e5,e6,e7)
t11:=e1+e2:
t12:=e3+e4:
t13:=e5+e6+e7+e8:
t21:=e1*e2:
t22:=e3*e4:
t23:=e5*e6+e5*e7+e5*e8+e6*e7+e6*e8+e7*e8:
t3:=e6*e7*e8+e5*e7*e8+e5*e6*e8+e5*e6*e7:
Q12:=e1+e2-e5-e6-e7-e8-5:
Q11:=3*(e1+e2)-e1*e2+e3*e4-e5*e6-e5*e7-e5*e8-e6*e7-e6*e8-e7*e8-8:
Q10:=(-4*t11^3+ 4*t12^3 -(6*t12 - 27)*t11^2+(6*t11- 27)*t12^2
      +9*(t21-2*t22+t23+4)*t11-9*(t22-2*t21+t23-2)*t12
      - 81*t21 + 81*t22 - 27*t23 - 27*t3- 135)/54:
Q0:=mult(z+e5,z+e6,z+e7,z+e8,[dx,x]):
Q1:=-2*mult(z,z,z,[dx,x])+Q12*mult(z,z,[dx,x])+Q11*z+Q10:
Q2:=mult(z-e1+2, z-e2+2,[dx,x]):
E4:= Q0+mult(Q1,dx,[dx,x])+mult(Q2,dx,dx,[dx,x]):
# remark e1+e2+e3+e4+e5+e6+e7+e8=4

## Equation SE4=SE4(a,b,c,g,q)
etop4:={e1=-2*b-q-1, e2=-a-2*b-c-g-q-2, e3=-2*c-q-1, e4=-a-2*c-b-g-q-2,
        e5=q+1, e6=b+c+q+2, e7=2*b+2*c+g+q+3, e8=2*a+2*b+2*c+g+q+4}:
SE4:=subs(etop4, E4):

## Equation Z4=Z4(A0,A1,A2,A3,k)
etoZk:={e1=1/2-A0-k, e2=1/2+A0-k, e3=1/2-A1-k, e4=1/2+A1-k,
        e5=1/2-A2+k, e6=1/2+A2+k, e7=1/2-A3+k, e8=1/2+A3+k}:
Z4:=subs(etoZk,E4):

## Equation ST4=ST4(e1,e2,e3,e4,e5,e6)
s4:=(3-e1-e2-e3-e4-e5-e6)/2:
V0:=mult(z+s4+1, z+s4, z+e5, z+e6,[dx,x]):
V1:=mult(z+s4+1, -2*mult(z,z,[dx,x])+(e1+e2-e5-e6-4)*z
        +1/4*((e6-e5)^2-(e3-e4)^2+(e1-e2)^2+2*(e1+e2-3)*(e5+e6+1)-1),[dx,x]):
V2:=mult(z+2-e1, z+2-e2, [dx,x]):
ST4:=V0+mult(V1,dx,[dx,x])+mult(V2,dx,dx,[dx,x]):

## Equation 4E3(a0,a1,a2,a3;b1,b2,b3) denoted E43 
E43:=mult(z+a0,z+a1,z+a2,z+a3,[dx,x])-mult(z+b1,z+b2,z+b3,dx,[dx,x]):

## Equation E3=E3(e1, ..., e6)
a00:=(-4*(e1+e2-e3-e4)^3-27*e5*e6*e7+9*(e1+e2-e3-e4)*(e5*e6+e5*e7+e6*e7-2)
      +9*e1*e2*(e1+e2-1)+18*(e1+e2-1)*(e3^2+e3*e4+e4^2)
      -9*e3*e4*(e3+e4-1)-18*(e3+e4-1)*(e1^2+e1*e2+e2^2))/54:
Sn:=mult(z+e5,z+e6,z+e7,[dx,x]):
S0:=-2*mult(z,z,z,[dx,x])+(2*e1+2*e2+e3+e4-3)*mult(z,z,[dx,x])
    +(-e1*e2+(e3-1)*(e4-1)-e5*e6-(e5+e6)*e7)*z+a00:
S1:=mult(z-e1+1,z-e2+1,[dx,x]):
E3:=x*Sn+S0+mult(S1,dx,[dx,x]):
#remark e7=3-(e1+e2+e3+e4+e5+e6);
 
## Equation SE3=SE3(a,b,c,g)
etop3:={e1=a+c+1, e2=2*e1+g, e3=b+c+1, e4=2*e3+g,
        e5=-2*c, e6=-(a+b+2*c+g+1), e7=2*e6+g-e5}:
SE3:=subs(etop3,subs(etop3, E3)):
#remark: a00=c*(2*a+2*c+1+g)*(2*a+2*b+2*c+2+g)


## Equation Z3(A0,A1,A2,A3)
etoA:={e1=-A0-A2, e2=A0-A2, e3=-A1-A2, 
       e4=A1-A2, e5=2*A2+1, e6=A2+A3+1,e7=1+A2-A3}:
Z3:=subs(etoA, E3):
# a00=(2*A3 + 1)*(A0^2 - A1^2 - A2^2 + A3^2 - 2*A2 - 1)/2

## Equation E3a(e1,e2,e5,e6) #remark e7=3-(e1+e2+e3+e4+e5+e6);
E3t:=subs(e7=3-(e1+e2+e3+e4+e5+e6),E3):  # e7 is eliminated

E3a:=subs({e3=e1,e4=e2},E3t):

## Equation E3b(e1,e3,e5,e6)
E3b:=subs({e2=-e3-e5-2*e6+3-e1,e4=e3-e5+e6},E3t): 

## Equation E3c(e1,e3,e4,e6)
E3c:=subs({e5=1-e1-e3-e4,e2=2*e1+e3+e4},E3t):

## Equation E3d(e1,e3,e4,e6)
E3d:=subs({e5=e1+e6,e2=3/2-e1-1/2*e3-1/2*e4-3/2*e6},E3t):

## Equation 3E2(a0,a1,a2;b1,b2) denoted by E32
E32:=mult(z+a0,z+a1,z+a2,[dx,x])-mult(z+b1,z+b2,dx,[dx,x]):

## Equation E2=Gauss(a,b,c)
E2:=mult(z+a,z+b,[dx,x])-mult(z+c,dx,[dx,x]);
# remark e1=1-c, e2=c-a-b, e3=a, e4=b