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A family of Numerov-type exponentially fitted predictor-corrector methods for the numerical integration of the radial Schrödinger equation

R.M. Thomas, T.E. Simos, G.V. Mitsou
1996 Journal of Computational and Applied Mathematics  
A family of predictor-corrector exponential Numerov-type methods is developed for the numerical integration of the one-dimensional Schr6dinger equation.  ...  The new methods are very simple and integrate more exponential functions than both the well-known fourth-order Numerov-type exponentially fitted methods and the sixth algebraic order Runge-Kutta-type methods  ...  Acknowledgements The authors wish to thank Prof. C.T.H. Baker, Dr. J.P. Coleman, Prof. I. Gladwell and an anonymous referee for their valuable comments and suggestions.  ... 
doi:10.1016/0377-0427(94)00126-x fatcat:gtyi2twv2rf4jlhemnczddegfq

A family of four-step exponentially fitted predictor-corrector methods for the numerical integration of the Schrödinger equation

T.E. Simos
1995 Journal of Computational and Applied Mathematics  
A family of predictor-corrector exponential four-step methods is developed for the numerical integration of the one-dimensional Schrodinger equation.  ...  Numerical results indicate that the new method is much more accurate than other exponentially fitted methods.  ...  Raptis and Allison [9] have derived a Numerov-type exponentially fitted method.  ... 
doi:10.1016/0377-0427(93)e0274-p fatcat:uor7a7xlbvgxzapwpw7l34kptq

A family of four-step exponential fitted methods for the numerical integration of the radial Schrödinger equation

T.E. Simos, G.V. Mitsou
1994 Computers and Mathematics with Applications  
A family of exponential four-step methods is developed for the numerical integration of the one-dimensional Schrijdinger equation.  ...  Numerical results indicate that the new methods are much more accurate than other exponentially fitted methods.  ...  family), Method M VI: New exponential-fitted method (Case III of the family).  ... 
doi:10.1016/0898-1221(94)00125-1 fatcat:qbjf5mgqhjb5hopqf5yne6tuqm

Two Optimized Symmetric Eight-Step Implicit Methods for Initial-Value Problems with Oscillating Solutions [article]

G.A. Panopoulos, Z.A. Anastassi, T.E. Simos
2008 arXiv   pre-print
The methods are constructed to solve numerically the radial time-independent Schr\"odinger equation with the use of the Woods-Saxon potential.  ...  We measure the efficiency of the methods and conclude that the new method with infinite order of phase-lag is the most efficient of all the compared methods and for all the problems solved.  ...  Mitsou, A family of four-step exponential fitted methods for the numerical integration of the radial Schrödinger equation, Computers and Mathematics with Applications 28 41-50(1994). 83. T.E.  ... 
arXiv:0811.2470v1 fatcat:xwbqb4wvjzh25kmlrhqav7qz6i

A New Methodology for the Development of Numerical Methods for the Numerical Solution of the Schrödinger Equation [article]

Z.A. Anastassi, D.S. Vlachos, T.E. Simos
2008 arXiv   pre-print
In the present paper we introduce a new methodology for the construction of numerical methods for the approximate solution of the one-dimensional Schr\"odinger equation.  ...  The new methodology is based on the requirement of vanishing the phase-lag and its derivatives. The efficiency of the new methodology is proved via error analysis and numerical applications.  ...  The New Family of Numerov-Type Hybrid Methods -Construction of the New Methods First Method of the Family We introduce the following family of methods to integrate y ′′ = f (x, y) : y n = y n − a 0 h  ... 
arXiv:0811.2473v1 fatcat:vbtf4tqtwvhwvlkaayps7t7wc4

Multiderivative methods of eighth algebraic order with minimal phase-lag for the numerical solution of the radial Schrödinger equation

D.P. Sakas, T.E. Simos
2005 Journal of Computational and Applied Mathematics  
Multiderivative methods with minimal phase-lag are introduced in this paper, for the numerical solution of the one-dimensional Schrödinger equation.  ...  Numerical application of the newly introduced method to the resonance problem of the one-dimensional Schrödinger equation shows its efficiency compared with other similar well-known methods of the literature  ...  behavior than the exponentially fitted method of Raptis and Allison [23] , the Numerov's method and the explicit Numerov-type method of Chawla [6] . • The explicit Numerov-type method with minimal phase-lag  ... 
doi:10.1016/j.cam.2004.06.013 fatcat:krws4mgm5bchvczlp5yvdaiiqm

A family of hybrid exponentially fitted predictor-corrector methods for the numerical integration of the radial Schrödinger equation

R.M. Thomas, T.E. Simos
1997 Journal of Computational and Applied Mathematics  
A family of hybrid, exponentially fitted, predictor-corrector methods is developed for the numerical integration of the one-dimensional Schr6dinger equation.  ...  The new methods are of algebraic order six, they are very simple and integrate more exponential functions than both the well-known fourth-order Numerov-type exponentially fitted methods and the Runge-Kutta-type  ...  Acknowledgements The authors wish to thank an anonymous referee for his careful reading of the manuscript and his fruitful comments and suggestions.  ... 
doi:10.1016/s0377-0427(97)00188-x fatcat:oglh2nrkqvfsdflibupebprdky

High Order Multistep Methods with Improved Phase-Lag Characteristics for the Integration of the Schrödinger Equation [article]

D.S. Vlachos, Z.A. Anastassi, T.E. Simos
2008 arXiv   pre-print
In this work we introduce a new family of twelve-step linear multistep methods for the integration of the Schr\"odinger equation.  ...  The efficiency of the new family of methods is proved via error analysis and numerical applications.  ...  Mitsou, A family of Numerov-type exponentially fitted predictor-corrector methods for the numerical integration of the radial Schrödinger equation, Journal of Computational and Applied Mathematics 67(2  ... 
arXiv:0811.2476v1 fatcat:nlgvsgn6yfe5hjow2fw4mtak6a

A Family of Runge-Kutta Methods with Zero Phase-Lag and Derivatives for the Numerical Solution of the Schrödinger Equation and Related Problems [article]

Z.A. Anastassi, D.S. Vlachos, T.E. Simos
2008 arXiv   pre-print
We construct a family of two new optimized explicit Runge-Kutta methods with zero phase-lag and derivatives for the numerical solution of the time-independent radial Schr\"odinger equation and related  ...  The numerical results show the superiority of the new technique of nullifying both the phase-lag and its derivatives.  ...  Simos, A family of P-stable exponentially-fitted methods for the numerical solution of the Schrödinger equation, J. Math. Chem 25 (1) 65-84 (1999) 52. T.E.  ... 
arXiv:0811.2469v1 fatcat:enjrsdo5nzdhjbjraxvzqn6xeu

An explicit Numerov-type method for second-order differential equations with oscillating solutions

Hans Van de Vyver
2007 Computers and Mathematics with Applications  
In this paper a new explicit Numerov-type method is introduced. The construction is based on a modification of a sixth-order explicit Numerov-type method recently developed by Tsitouras [Ch.  ...  Numerical results obtained for well-known test problems show the efficiency of the new method.  ...  Acknowledgments The author is very grateful to an anonymous referee whose valuable comments and suggestions improved this paper. This research was supported by "Grant 0T/04/21 of Onderzoeksfonds K.U.  ... 
doi:10.1016/j.camwa.2006.06.012 fatcat:l5ller5bvbajvedbfx7mvt7qzy

A new two-step hybrid method for the numerical solution of the Schrödinger equation

A. Konguetsof
2009 Journal of Mathematical Chemistry  
With this paper, a new algorithm is developed for the numerical solution of the one-dimensional Schrödinger equation. The new method uses the minimum order of the phase-lag and its derivatives.  ...  Error analysis and the numerical results illustrate the efficiency of the new algorithm.  ...  1 For the solution of the Schrödinger equation the frequency of the exponential fitting is equal to the frequency of the scalar test equation.  ... 
doi:10.1007/s10910-009-9606-5 fatcat:x7f3tiynzzd2zosssr4mykp5cm

High Order Phase Fitted Multistep Integrators for the Schrödinger Equation with Improved Frequency Tolerance [article]

D.S. Vlachos, Z.A. Anastassi, T.E. Simos
2008 arXiv   pre-print
In this work we introduce a new family of 14-steps linear multistep methods for the integration of the Schrödinger equation.  ...  The efficiency of the new family of methods is proved via error analysis and numerical applications.  ...  Mitsou, A family of Numerov-type exponentially fitted predictor-corrector methods for the numerical integration of the radial Schrödinger equation, Journal of Computational and Applied Mathematics 67(2  ... 
arXiv:0811.2478v1 fatcat:ojdxhjayhrbptdhykkfnd7e7ny

Volterra type integral equation method for the radial Schrödinger equation: Single channel case

Sheon-Young Kang
2004 Computers and Mathematics with Applications  
A new Volterra type integral equation method for the numerical solution of the single channel radial Schr5dinger equation is investigated.  ...  The method, carried out in configuration space, is based on the conversion of differential equations into a system of Volterra type integral equations together with the application of a spectral type Clenshaw-Curtis  ...  The Volterra type integral equation method described here provides an alternative method for the solution of the radial Schrhdinger equation which gives high accuracy at a cost comparable to that of the  ... 
doi:10.1016/j.camwa.2004.08.004 fatcat:pjd2qdpt5fcqxcbjuv7q2ccxti

Present state-of-the-art in exponential fitting. A contribution dedicated to Liviu Ixaru on his 70th birthday

Beatrice Paternoster
2012 Computer Physics Communications  
The standard monograph in this area is the book Exponential fitting by Ixaru and Vanden Berghe (Kluwer, Boston -Dordrecht -London, 2004) but a fresh look on things is necessary because many new contributions  ...  With no claim that our investigation is exhaustive we consider various directions of interest, try to integrate the new contributions in a natural, easy to follow way, and also detect some open problems  ...  Shortly before, Raptis and Allison had published a paper, [19] , based on Lyche's theory [20] , where the famous Numerov method was adapted by exponential fitting to become tuned on the Schrödinger equation  ... 
doi:10.1016/j.cpc.2012.06.013 fatcat:vurgeeq7j5gfxkk665kfs5fcti

An exponentially fitted eighth-order method for the numerical solution of the Schrödinger equation

T.E. Simos
1999 Journal of Computational and Applied Mathematics  
An eighth algebraic order exponentially ÿtted method is developed for the numerical integration of the Schr odinger equation.  ...  Numerical and theoretical results indicate that the new method is much more e cient than other classical and exponentially ÿtted methods.  ...  Acknowledgements The author wishes to thank an anonymous referee for his careful reading of the manuscript and his fruitful comments and suggestions.  ... 
doi:10.1016/s0377-0427(99)00109-0 fatcat:qsgni56o35bwdbml7k6acav3rm
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