Computation for Formulation 14.16 on page 153

While Version 5.1 fails to compute the unique coefficients for Formulation 14.16, it is capable of verifying that the coefficients found with Version 3.0 are such that Formulation 14.16 is valid.

The version number of Mathematica used to evaluate this notebook:

$VersionNumber

aAA[
        zet_]  =  f'[zet]^2*
          a[f[zet]] + (f'''[zet]/(2*f'[zet])) - (3/
              4) (f''[zet]/f'[zet])^2

The preceding represents  \(TraditionalForm\`\(\(a\^ ** \)\_\(m, 0, 2\)\)(ζ)\)  and we have chosen  A  to suggest asterisk.

We write

PAA[zet_]  =  f'[zet]^p*P[f[zet]]

for (14.20).  In particular, we have

D[PAA[zet],  zet]

for (14.21) and

Collect[ 
      D[PAA[zet], {zet, 2}],  {P''[f[zet]],  P'[f[zet]], 
        P[f[zet]]}, Simplify ]

for (14.22).  Similarly, we introduce

QAA[zet_]  =  f'[zet]^q*Q[f[zet]]

We introduce

v[0]  =  1

v[1]  =   -((3 (5 + 2 q))/p)

v[2]  =  ((15 (2 + q) (5 + 2 q))/(p (1 + 2 p)))

v[3]  =  -((10 (2 + q) (3 + 2 q) (5 + 
                  2 q))/(p (1 + p) (1 + 
                  2 p)))

v[4]  =  ((15 (1 + q) (2 + q) (3 + 2 q) (5 + 2 q))/(p (1 + p) (1 + 2 p) (3 + 2 p)))

v[5]  =  -((3 (1 + q) (2 + q) (1 + 2 q) (3 + 
                  2 q) (5 + 2 q))/(p (1 + p) (2 + 
                  p) (1 + 2 p) (3 + 2 p)))

v[6]  =  ((q (1 + q) (2 + q) (1 + 2 q) (3 + 2 q) (5 + 2 q))/(p (1 + p) (2 + p) (1 + 2 p) (3 + 2 p) (5 + 2 p)))

v[7]  =  -((20 (17 + 4 p + 15 q + 3 p q + 
                  3 (q^2)))/(1 + 2 p))

v[8]  =  ((20 (5 + 2 q) (9 + 27 p + 6 (p^2) + 10 q + 30 p q + 6 (p^2) q + 2 (q^2) + 6 p (q^2)))/(p (1 + p) (1 + 2 p)))

v[9]  =  -((60 (2 + q) (5 + 2 q) (12 + 19 p + 
                  4 (p^2) + 19 q + 30 p q + 6 (p^2) q + 4 (q^2) + 
                  6 p (q^2)))/(p (1 + p) (1 + 2 p) (3 + 
                  2 p)))

v[10]  =  ((20 (2 + q) (3 + 2 q) (5 + 2 q) (9 + 10 p + 2 (p^2) + 27 q + 30 p q + 6 (p^2) q + 6 (q^2) + 6 p (q^2)))/(p (1 + p) (2 + p) (1 + 2 p) (3 + 2 p)))

v[11]  =  -((20 q (1 + q) (2 + q) (3 + 
                  2 q) (5 + 2 q) (17 + 15 p + 3 (p^2) + 4 q + 
                  3 p q))/(p (1 + p) (2 + p) (1 + 
                  2 p) (3 + 2 p) (5 + 2 p)))

v[12]  =  -((10 (-57 + 9 p + 6 (p^2) - 90 q + 12 p q + 
                  8 (p^2) q - 48 (q^2) - 8 (q^3)))/((1 + p) (1 + 
                  2 p)))

v[13]  =  (30 (5 + 2 q) (-3 - 13 p + 12 (p^2) + 
                4 (p^3) - 7 q - 30 p q + 24 (p^2) q + 8 (p^3) q - 
                9 (q^2) - 36 p (q^2) - 2 (q^3) - 
                8 p (q^3)))/(p (1 + p) (1 + 2 p) (3 + 
                2 p))

v[14]  =  -(30 (2 + q) (5 + 2 q) (3 + 7 p + 
                  9 (p^2) + 2 (p^3) + 13 q + 30 p q + 36 (p^2) q + 
                  8 (p^3) q - 12 (q^2) - 24 p (q^2) - 4 (q^3) - 
                  8 p (q^3)))/(p (1 + p) (2 + p) (1 + 
                2 p) (3 + 2 p))

v[15]  =  ((10 q (2 + q) (3 + 2 q) (5 + 2 q) (57 + 90 p + 48 (p^2) + 8 (p^3) - 9 q - 12 p q - 6 (q^2) - 8 p (q^2)))/(p (1 + p) (2 + p) (1 + 2 p) (3 + 2 p) (5 + 2 p)))

v[16]  =  -((6 (156 + 22 p + 24 (p^2) + 8 (p^3) + 400 q + 
                  55 p q + 60 (p^2) q + 20 (p^3) q + 355 (q^2) + 
                  140 (q^3) + 20 (q^4)))/((1 + p) (1 + 
                  2 p) (3 + 2 p)))

v[17]  =  (6 (5 + 2 q) (12 + 55 p + 35 (p^2) + 20 (p^3) + 
                4 (p^4) + 55 q + 250 p q + 175 (p^2) q + 100 (p^3) q + 
                20 (p^4) q + 35 (q^2) + 175 p (q^2) + 20 (q^3) + 
                100 p (q^3) + 4 (q^4) + 20 p (q^4)))/(p (1 + 
                p) (2 + p) (1 + 2 p) (3 + 
                2 p))

v[18]  =  -(6 q (2 + q) (5 + 2 q) (156 + 400 p + 
                  355 (p^2) + 140 (p^3) + 20 (p^4) + 22 q + 55 p q + 
                  24 (q^2) + 60 p (q^2) + 8 (q^3) + 
                  20 p (q^3)))/(p (1 + p) (2 + p) (1 + 
                2 p) (3 + 2 p) (5 + 2 p))

v[19]  =  (1/((1 + 2 p) (3 + 2 p))) (16 (822 + 
                392 p + 46 (p^2) + 2025 q + 930 p q + 105 (p^2) q + 
                1530 (q^2) + 555 p (q^2) + 45 (p^2) (q^2) + 450 (q^3) + 
                90 p (q^3) + 45 (q^4)))

v[20]  =  -(16 (5 + 2 q) (144 + 600 p + 570 (p^2) + 
                  180 (p^3) + 18 (p^4) + 600 q + 2500 p q + 
                  2375 (p^2) q + 750 (p^3) q + 75 (p^4) q + 570 (q^2) + 
                  2375 p (q^2) + 2110 (p^2) (q^2) + 570 (p^3) (q^2) + 
                  45 (p^4) (q^2) + 180 (q^3) + 750 p (q^3) + 
                  570 (p^2) (q^3) + 90 (p^3) (q^3) + 18 (q^4) + 75 p (q^4) + 
                  45 (p^2) (q^4)))/(p (1 + p) (2 + 
                p) (1 + 2 p) (3 + 2 p))

v[21]  =  (16 q (1 + q) (2 + q) (5 + 
                2 q) (822 + 2025 p + 1530 (p^2) + 450 (p^3) + 
                45 (p^4) + 392 q + 930 p q + 555 (p^2) q + 90 (p^3) q + 
                46 (q^2) + 105 p (q^2) + 45 (p^2) (q^2)))/(p (1 + 
                p) (2 + p) (1 + 2 p) (3 + 2 p) (5 + 
                2 p))

v[22]  =  -(2 (-84 + 25 p + 35 (p^2) + 20 (p^3) + 
                  4 (p^4) - 375 q + 150 p q + 210 (p^2) q + 
                  120 (p^3) q + 24 (p^4) q - 675 (q^2) - 510 (q^3) - 
                  180 (q^4) - 24 (q^5)))/((1 + p) (2 + 
                p) (1 + 2 p) (3 + 2 p))

v[23]  =  (2 q (5 + 2 q) (84 + 375 p + 675 (p^2) + 
                510 (p^3) + 180 (p^4) + 24 (p^5) - 25 q - 150 p q - 
                35 (q^2) - 210 p (q^2) - 20 (q^3) - 120 p (q^3) - 4 (q^4) - 
                24 p (q^4)))/(p (1 + p) (2 + p) (1 + 
                2 p) (3 + 2 p) (5 + 2 p))

v[24]  =  (8 (-924 - 250 p + 760 (p^2) + 370 (p^3) + 
                44 (p^4) - 4050 q - 735 p q + 3810 (p^2) q + 
                1805 (p^3) q + 214 (p^4) q - 7110 (q^2) - 4695 p (q^2) + 
                1965 (p^2) (q^2) + 1170 (p^3) (q^2) + 120 (p^4) (q^2) - 
                4980 (q^3) - 4430 p (q^3) - 220 (p^2) (q^3) + 
                120 (p^3) (q^3) - 1440 (q^4) - 1320 p (q^4) - 
                120 (p^2) (q^4) - 144 (q^5) - 120 p (q^5)))/((1 + 
                p) (2 + p) (1 + 2 p) (3 + 
                2 p))

v[25]  =  -(8 q (5 + 2 q) (924 + 4050 p + 7110 (p^2) + 
                  4980 (p^3) + 1440 (p^4) + 144 (p^5) + 250 q + 735 p q + 
                  4695 (p^2) q + 4430 (p^3) q + 1320 (p^4) q + 
                  120 (p^5) q - 760 (q^2) - 3810 p (q^2) - 
                  1965 (p^2) (q^2) + 220 (p^3) (q^2) + 120 (p^4) (q^2) - 
                  370 (q^3) - 1805 p (q^3) - 1170 (p^2) (q^3) - 
                  120 (p^3) (q^3) - 44 (q^4) - 214 p (q^4) - 
                  120 (p^2) (q^4)))/(p (1 + p) (2 + 
                p) (1 + 2 p) (3 + 2 p) (5 + 
                2 p))

v[26]  =  -(2 q (3 + p + q) (20 + 39 p + 62 (p^2) + 
                  36 (p^3) + 8 (p^4) + 39 q - 26 p q - 12 (p^2) q - 
                  8 (p^3) q + 62 (q^2) - 12 p (q^2) + 8 (p^2) (q^2) + 
                  36 (q^3) - 8 p (q^3) + 8 (q^4)))/((1 + 
                p) (2 + p) (1 + 2 p) (3 + 2 p) (5 + 
                2 p))

v[27]  =  (8 q (3 + p + q) (1720 + 2604 p + 1882 (p^2) + 
                696 (p^3) + 88 (p^4) + 2604 q + 2414 p q + 1113 (p^2) q + 
                452 (p^3) q + 60 (p^4) q + 1882 (q^2) + 1113 p (q^2) - 
                32 (p^2) (q^2) + 696 (q^3) + 452 p (q^3) + 88 (q^4) + 
                60 p (q^4)))/((1 + p) (2 + p) (1 + 
                2 p) (3 + 2 p) (5 + 2 p))

v[28]  =  (8 q (3 + p + q) (-1380 - 1566 p - 
                228 (p^2) + 216 (p^3) + 48 (p^4) - 1566 q - 2181 p q - 
                702 (p^2) q + 132 (p^3) q + 40 (p^4) q - 228 (q^2) - 
                702 p (q^2) - 492 (p^2) (q^2) - 40 (p^3) (q^2) + 216 (q^3) + 
                132 p (q^3) - 40 (p^2) (q^3) + 48 (q^4) + 
                40 p (q^4)))/((1 + p) (2 + p) (1 + 
                2 p) (3 + 2 p) (5 + 2 p))

v[29]  =  -((960 q (1 + q) (2 + q) (3 + p + 
                  q) (4 + p + q) (5 + p + q))/((1 + 
                  2 p) (3 + 2 p) (5 + 2 p)))

so that (14.48) yields

\(\(VAA[
          zet_] = \((Sum[
                v[i]*D[PAA[zet], {zet, i}]*D[QAA[zet], {zet, 6 - i}], {i, 0, 
                  6}] + aAA[zet]*
                Sum[\ v[7 + i]*D[PAA[zet], {zet, i}]*
                    D[QAA[zet], {zet, 4 - i}], {i, 0, 4}] + 
              D[aAA[zet], zet]*
                Sum[\ v[12 + i]*D[PAA[zet], {zet, i}]*
                    D[QAA[zet], {zet, 3 - i}], {i, 0, 3}]
 + 
              D[aAA[zet], {zet, 2}]*
                Sum[v[16 + i]*D[PAA[zet], {zet, i}]*
                    D[QAA[zet], {zet, 2 - i}], {i, 0, 2}] + \((aAA[zet])\)^2*
                Sum[\ v[19 + i]*D[PAA[zet], {zet, i}]*
                    D[QAA[zet], {zet, 2 - i}], {i, 0, 2}] + 
              D[aAA[zet], {zet, 3}]*
                Sum[\ v[22 + i]*D[PAA[zet], {zet, i}]*
                    D[QAA[zet], {zet, 1 - i}], {i, 0, 1}] + 
              aAA[zet]*D[aAA[zet], zet]*
                Sum[\ v[24 + i]*D[PAA[zet], {zet, i}]*
                    D[QAA[zet], {zet, 1 - i}], {i, 0, 1}] + 
              v[26]*D[aAA[zet], {zet, 4}]*\ PAA[zet]*QAA[zet] + 
              v[27]*aAA[zet]*D[aAA[zet], {zet, 2}]*\ PAA[zet]*QAA[zet] + 
              v[28]*D[aAA[zet], zet]^2*PAA[zet]*QAA[zet] + 
              v[29]*aAA[zet]^3*\ PAA[zet]*QAA[zet])\) // Expand;\)\)

Length[VAA[zet]]

and

V[
          z_] = (Sum[v[i]*D[P[z], {z, i}]*D[Q[z], {z, 6 - i}], {i, 0, 6}] + 
              a[z]*Sum[ 
                  v[7 + i]*D[P[z], {z, i}]*D[Q[z], {z, 4 - i}], {i, 0, 4}] + 
              D[a[z], z]*
                Sum[ v[12 + i]*D[P[z], {z, i}]*D[Q[z], {z, 3 - i}], {i, 0, 
                    3}] + D[a[z], {z, 2}]*
                Sum[ v[16 + i]*D[P[z], {z, i}]*D[Q[z], {z, 2 - i}], {i, 0, 
                    2}] + (a[z])^2*
                Sum[ v[19 + i]*D[P[z], {z, i}]*D[Q[z], {z, 2 - i}], {i, 0, 
                    2}] + D[a[z], {z, 3}]*
                Sum[ v[22 + i]*D[P[z], {z, i}]*D[Q[z], {z, 1 - i}], {i, 0, 
                    1}] + a[z]*D[a[z], z]*
                Sum[ v[24 + i]*D[P[z], {z, i}]*D[Q[z], {z, 1 - i}], {i, 0, 
                    1}] + v[26]*D[a[z], {z, 4}]* P[z]*Q[z] + 
              v[27]*a[z]*D[a[z], {z, 2}]*P[z]*Q[z] + 
              v[28]*D[a[z], z]^2*P[z]*Q[z] + v[29]*a[z]^3* P[z]*Q[z]) // 
          Expand;

Date[]

differ  =   Together[ 
        VAA[zet]  -  f'[zet]^(p + q + 6)*V[f[zet]] ]

Date[]

In view of differ = 0, we see that Version 5.1 has verified that Formulation 14.16 is satisfied by the coefficients v[0], v[1], ..., v[29] that were computed by means of Version 3.0.