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reductions of partial differential equations

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1: 32.13 Reductions of Partial Differential Equations
§32.13 Reductions of Partial Differential Equations
2: Peter A. Clarkson
 Kruskal, he developed the “direct method” for determining symmetry solutions of partial differential equations in New similarity reductions of the Boussinesq equation (with M. …
3: 19.29 Reduction of General Elliptic Integrals
§19.29 Reduction of General Elliptic Integrals
§19.29(i) Reduction Theorems
§19.29(ii) Reduction to Basic Integrals
The reduction of I ( 𝐦 ) 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 𝐦 has at most one nonzero component, and these are then reduced to basic integrals by the recurrence relations. …
4: 19.14 Reduction of General Elliptic Integrals
§19.14 Reduction of General Elliptic Integrals
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. …