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► Kruskal, he developed the “direct method” for determining symmetry solutions of
partial differential equations in
New similarity reductions of the Boussinesq equation (with M.
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§19.29 Reduction of General Elliptic Integrals
►
§19.29(i) Reduction Theorems
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►
§19.29(ii) Reduction to Basic Integrals
…
►The
reduction of
is carried out by a relation derived from
partial fractions and by use of two recurrence relations.
…
Partial fractions provide a
reduction to integrals in which
has at most one nonzero component, and these are then reduced to basic integrals by the recurrence relations.
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§19.14 Reduction of General Elliptic Integrals
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►It then improves the classical method by first applying Hermite
reduction to (
19.2.3) to arrive at integrands without multiple poles and uses implicit full
partial-fraction decomposition and implicit root finding to minimize computing with algebraic extensions.
The choice among 21 transformations for final
reduction to Legendre’s normal form depends on inequalities involving the limits of integration and the zeros of the cubic or quartic polynomial.
A similar remark applies to the transformations given in
Erdélyi et al. (1953b, §13.5) and to the choice among explicit
reductions in the extensive table of
Byrd and Friedman (1971), in which one limit of integration is assumed to be a branch point of the integrand at which the integral converges.
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