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Extra resources for A solution to a problem of Fermat, on two numbers of which the sum is a square and the sum of their squares is a biquadrate, inspired by the Illustrious La Grange

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90) k—» — oo See Eq. 80b). Since ui and u 2 must be fixed-points of Eq. 91) y2 - ( u i +u2)y + uxu2 = 0. 92) where Eq. 93, In summary, the asymptotic values of yk determine both the traveling wave velocity and the integration constant A. Note that c is exactly that given by the ODE. 3], the following ex­ pression is obtained for the solution to Eq. 95)

is an arbitrary constant. 97) 40 Nonstandard Finite Difference Schemes force (/i) to satisfy the constraint 4>(h) <—, h > 0. 98) u2 A particular explicit functional form for cj>(h) that satisfies Eqs.

122). The scheme of Eq. 123) was constructed by making the following discrete replacements in Eq. 121) U2 = 2u 2 _ u 2 ^ { u l + i f + { u l i ? «+l)2 + _ ^km+l+<-^ «-l? 125) One way to reason why these structural forms are used is to realize that Eq. 121) is invariant under x -> —x. The discrete analogue is to have the difference equation invariant under (fc 4- 1) ** (fc — 1); see references [34; 35]. 127) Note that if u^ is non-negative, then u^1"1 will also be non-negative if 1 - 2R > 0. 128) Nonstandard Finite Difference Schemes 44 The selection of the equality gives the following functional relation between the step-sizes (Ax)2 At = ¥=2- .

49] R. E. Mickens, "Relation between the time and space step-sizes in nonstandard finite-difference schemes for the Fisher equation," Numerical Methods for Partial Differential Equations 13 (1997), 51-55. [50] R. E. Mickens, "Nonstandard finite difference schemes for reaction-diffusion equations," Numerical Methods for Partial Differential Equations 15 (1999), 201-214. [51] S.

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