IMPROVEMENT ON THE WKB METHOD USING AN ALGEBRAIC FORMALISM

The equation d2u/dt2+w2(t)u=0 can be transformed to a system of two first-order differential equations dy/dt=My by different ways of parametrisation. The method of decoupled exponents of Wei and Norman (1964) yields a Riccati differential equation for the unknown exponents and an easily interpretable solution y. A solution with vanishing exponents in the parametrisation of Mulholland (1973) corresponds exactly to the WKB method. Therefore an approximate solution of the Riccati differential equation gives an improvement on the WKB.