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Numerical Methods for Schrödinger-Type Problems

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Trends in Industrial and Applied Mathematics

Part of the book series: Applied Optimization ((APOP,volume 72))

Abstract

Several physical phenomena are modeled by initial-boundary value problems which can be formulated as Schrödinger-type systems of partial differential equations. In this paper, two classes of problems of this kind, Schrödinger equations, which arise in various areas of physics, and certain vibration problems from civil and mechanical engineering, are considered. A survey of numerical methods for solving linear and nonlinear problems in one and several space variables is presented, with special attention being devoted to the parabolic wave equation, the cubic Schrödinger equation, and to fourth order parabolic equations arising in vibrating beam and plate problems. Recently developed finite element methods for solving Schrödinger-type systems are also outlined.

This work was supported in part by national Science Foundation grant DMS-9805827

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  1. Department of Mathematical and Computer Sciences, Colorado School of Mines, Golden, Colorado, 80401, USA

    Graeme Fairweather & Morrakot Khebchareon

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  1. Department of Mathematical Sciences, King Fand University of Petroleum and Minerals, Dhahran, Saudi Arabia

    Abul Hasan Siddiqi

  2. Institute of Applied Mathematics, University of Erlangen, Erlangen, Germany

    Michal Kočvara

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Fairweather, G., Khebchareon, M. (2002). Numerical Methods for Schrödinger-Type Problems. In: Siddiqi, A.H., Kočvara, M. (eds) Trends in Industrial and Applied Mathematics. Applied Optimization, vol 72. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0263-6_10

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