Liouville-Green asymptotic approximation for a class of matrix differential equations and semi-discretized partial differential equations |
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Authors: | Renato Spigler Marco Vianello |
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Affiliation: | a Dipartimento di Matematica, Università di “Roma Tre”, Largo S. Leonardo Murialdo 1, 00146 Roma, Italy b Dipartimento di Matematica Pura e Applicata, Università di Padova, Via Belzoni 7, 35135 Padova, Italy |
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Abstract: | A Liouville-Green (or WKB) asymptotic approximation theory is developed for the class of linear second-order matrix differential equations Y″=[f(t)A+G(t)]Y on [a,+∞), where A and G(t) are matrices and f(t) is scalar. This includes the case of an “asymptotically constant” (not necessarily diagonalizable) coefficient A (when f(t)≡1). An explicit representation for a basis of the right-module of solutions is given, and precise computable bounds for the error terms are provided. The double asymptotic nature with respect to both t and some parameter entering the matrix coefficient is also shown. Several examples, some concerning semi-discretized wave and convection-diffusion equations, are given. |
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Keywords: | Liouville-Green asymptotics WKB asymptotics Matrix differential equations Semi-discretized partial differential equations |
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