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Theory and Modern Applications

Table 1 Legendre polynomials

From: Application of Legendre polynomials based neural networks for the analysis of heat and mass transfer of a non-Newtonian fluid in a porous channel

n

\(L_{n}(\eta )\)

0

1

1

η

2

\(\frac{1}{2} (3 \eta ^{2}-1 )\)

3

\(\frac{1}{2} (5 \eta ^{3}-3 \eta )\)

4

\(\frac{1}{8} (35 \eta ^{4}-30 \eta ^{2}+3 )\)

5

\(\frac{1}{8} (63 \eta ^{5}-70 \eta ^{3}+15 \eta )\)

6

\(\frac{1}{16} (231 \eta ^{6}-315 \eta ^{4}+105 \eta ^{2}-5 )\)

7

\(\frac{1}{16} (429 \eta ^{7}-693 \eta ^{5}+315 \eta ^{3}-35 \eta )\)

8

\(\frac{1}{128} (6435 \eta ^{8}-12{,}012 \eta ^{6}+6930 \eta ^{4}-1260 \eta ^{2}+35 )\)

9

\(\frac{1}{128} (12{,}155 \eta ^{9}-25{,}740 \eta ^{7}+18{,}018 \eta ^{5}-4620 \eta ^{3}+315 \eta )\)

10

\(\frac{1}{256} (46{,}189 \eta ^{10}-109{,}395 \eta ^{8}+90{,}090 \eta ^{6}-30{,}030 \eta ^{4}+3465 \eta ^{2}-63 )\)