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Theory and Modern Applications

Table 1 Some solutions of auxiliary equation of Eq. 8

From: Oblique explicit wave solutions of the fractional biological population (BP) and equal width (EW) models

ρ

p

q

ϕ(ϑ)

1

\(-(1+m^{2})\)

\(m^{2}\)

sn(ϑ)

\(1-m^{2}\)

\(2 m^{2}-1\)

\(-m^{2}\)

cn(ϑ)

\(m^{2}-1\)

\(2-m^{2}\)

-1

dn(ϑ)

\(\frac{1}{4}\)

\(\frac{1-2 m^{2}}{2}\)

\(\frac{1}{4}\)

ns(ϑcs(ϑ) or \(\frac{sn(\vartheta)}{1 \pm cn(\vartheta)}\)

\(\frac{1-m^{2}}{4}\)

\(\frac{1+m^{2}}{2}\)

\(\frac{1-m^{2}}{4}\)

nc(ϑsc(ϑ) or \(\frac{cn(\vartheta)}{1 \pm sn(\vartheta)}\)

1

\(2-m^{2}\)

\(1-m^{2}\)

sc(ϑ)

\(1-m^{2}\)

\(2-m^{2}\)

1

cs(ϑ)