Exploiting the nature of numbers - an arithmetic-analytical approach to solutions of problems in basic and applied sciences (Q1119613)

The tradition of the old Vedic (Indian) mathematics (8-12 A.D.) furnished simple arithmetic solutions in several steps for some class of equations. An analogous arithmetical-analytical approach is proposed for linear and nonlinear algebraic equations as well as for some differential equations, in order to find general integer solutions. So for instance the problem \(y''=\alpha y\) \((y=y_ 0\), \(x=X\); \(y'=0\), \(x=0)\), describing the reaction-diffusion systems, yields the solution \(y=\ell +0\), \(\ell =-,\pm 1,\pm 2,...,\) \(y'=m+0\), \(m=0,\pm 1,\pm 2,...\) where \(\ell\) and m are arbitrary integers whose i-values increments satisfy the relation \(\alpha \ell_ i\Delta \ell_ i=m_ i\Delta m_ i.\) Validating this relation successively one obtains recursive relations giving the arithmetic- analytical solution. The same method works for any linear or nonlinear equation, the general relation for the variation \(\Delta\) x being obtained in one step from the given equation. Another example is the system of differential equations governing the deterministic chaos [\textit{O. E. Rössler}, Z. Naturforsch., Teil A 31, 1168 (1976)], for which a discrete solution is obtained in terms of an arbitrary integer parameter. This approach presents some interest for solving problems in applied sciences, involving digital calculations.