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Zbl 1192.65111
Simos, T.E.
Exponentially and trigonometrically fitted methods for the solution of the Schrödinger equation. (English)
[J] Acta Appl. Math. 110, No. 3, 1331-1352 (2010). ISSN 0167-8019; ISSN 1572-9036/e

Summary: We compare two methodologies for the development of exponentially and trigonometrically fitted methods. One is based on the exact integration of functions of the form: $$\{1,x,x ^{2},\dots ,x ^{p }, \exp (\pm wx),x \exp (\pm wx),\dots ,x^{m} \exp (\pm wx)\},$$ and the second is based on the exact integration of functions of the form: $$\{1,x,x ^{2},\dots ,x ^{p }, \exp (\pm wx), \exp (\pm 2wx),\dots , \exp (\pm mwx)\}.$$ The above functions are used in order to improve the efficiency of the classical methods of any kind for the numerical solution of ordinary differential equations of the form of the Schrödinger equation. We mention here that the above sets of exponential functions are the two most common sets of exponential functions for the development of the special methods for the efficient solution of the Schrödinger equation. It is first time in the literature in which the efficiency of the above sets of functions is studied and compared with the approximate solution of the Schrödinger equation. We present the error analysis of the above two approaches for the numerical solution of the one-dimensional Schrödinger equation. Finally, numerical results for the resonance problem of the radial Schrödinger equation are presented.

MSC 2000:: *65L10 Boundary value problems for ODE (numerical methods)
34B05 Linear boundary value problems of ODE
34L40 Particular ordinary differential operators

Keywords: multistep methods; hybrid methods; exponential fitting; trigonometric fitting; error analysis; numerical results; resonance problem; radial Schrödinger equation

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Zentralblatt MATH Berlin [Germany]