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New formulae for approximate solution of one-dimensional wave equation. (Russian) Zbl 0693.34008

The author considers the differential equation \(\frac{d^2\psi}{dt^2} + k^2(x)\psi = 0\) where the function k(x) is assumed differentiable. Setting \(d\psi/dx = \phi\), \(d\xi/dx = k\) the above equation is transformed into the system \(d\psi/d\xi = (1/k)\phi, \quad d\phi/d\xi = -k\psi\), and if one introduces the functions p and q by conditions \(\psi\phi = ip\), \(\phi/\psi = iq\) the above system is transformed into the nonlinear system \[ \frac{1}{q} \frac{dq}{d\xi} = i (\frac{k}{q} -\frac{q}{k}), \frac{1}{p} \frac{dp}{d\xi} = i(\frac{k}{q} + \frac{q}{k}) \] which has the advantage that the first equation of it involves only one of the unknown functions, namely q. The author observes that if \(| (1/k^2)(dk/dx)| \ll 1\) then this equation can be approximated by equations: \[ \frac1q \frac{dq}{d\xi} = 2i \ln \frac{k}{q}, \quad \frac1q \frac{dq}{d\xi} = 2i(\frac{k}{q}-1) \text{\quad and\quad} \frac1q \frac{dq}{d\xi} = 2i(1-\frac{q}{k}), \] the solution of which can be presented by explicit formulas.
Reviewer: L.Janos

MSC:

34A45 Theoretical approximation of solutions to ordinary differential equations
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
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