An explicit linear six-step method with vanished phase-lag and its first derivative

An explicit linear sixth algebraic order six-step method with vanished phase-lag and its first derivative is constructed in this paper. We will study the method theoretically and computationally. Theoretical investigation contains the building of the method, the calculation of the local truncation error, the comparative error analysis of the new method with the method with constant coefficients and the stability analysis of the new method using scalar test equation with different frequency than the frequency of the scalar test equation used for the development of the method. Computational investigation contains the application of the new obtained linear six-step method to the resonance problem of the radial time independent Schrödinger equation. The theoretical and computational study lead us to the summary that the new proposed linear scheme is computationally and theoretically more efficient than other well known methods for the numerical solution of the Schrödinger equation and related periodic initial or boundary value problems.     

An explicit four-step method with vanished phase-lag and its first and second derivatives

In this paper we will develop an explicit fourth algebraic order four-step method with phase-lag and its first and second derivatives vanished. The comparative error and the stability analysis of the above mentioned paper is also presented. The new obtained method is applied on the resonance problem of the Schrödinger equationIn order in order to examine its efficiency. The theoretical and the computational results shown that the new obtained method is more efficient than other well known methods for the numerical solution of the Schrödinger equation and related initial-value or boundary-value problems with periodic and/or oscillating solutions.     

A family of ten-step methods with vanished phase-lag and its first derivative for the numerical solution of the Schrödinger equation

A family of high algebraic order ten-step methods is obtained in this paper. The new developed methods have vanished phase-lag (the first one) and phase-lag and its first derivative (the second one). We apply the new developed methods to the resonance problem of the radial Schrödinger equation. The efficiency of the new proposed methods is shown via error analysis and numerical applications.     

A new explicit hybrid four-step method with vanished phase-lag and its derivatives

A hybrid explicit sixth algebraic order four-step method with phase-lag and its first, second and third derivatives vanished is obtained in this paper. We present the development of the new method, its comparative error analysis and its stability analysis. The resonance problem of the Schrödinger equation, is used in order to study the efficiency of the new developed method. After the presentation of the theoretical and the computational results it is easy to see that the new constructed method is more efficient than other well known methods for the approximate solution of the Schrödinger equation and related initial-value or boundary-value problems with periodic and/or oscillating solutions.     

A family of eight-step methods with vanished phase-lag and its derivatives for the numerical integration of the Schr?dinger equation

A family of tenth algebraic order eight-step methods is constructed in this paper. For this family of methods, we require the phase-lag and its first, second and third derivatives to be vanished. Three alternative methods are proposed which satisfy the above requirements. An error analysis and a stability analysis is also investigated in this paper and a comparison with other methods is also studied. The new proposed methods are applied for the numerical solution of the one dimensional Schrödinger equation. The efficiency of the new methodology is proved via theoretical analysis and numerical applications.     

New three-stages symmetric six-step finite difference method with vanished phase-lag and its derivatives up to sixth derivative for second order initial and/or boundary value problems with periodical and/or oscillating solutions

In this paper, for the first time in the literature, we develop a symmetric three-stages six-step method with the following characteristics; the method

1. 1.
   is a symmetric hybrid (multistages) six-step method,
    
2. 2.
   is of three-stages,
    
3. 3.
   is of twelfth algebraic order,
    
4. 4.
   has vanished the phase-lag and
    
5. 5.
   has vanished the derivatives of the phase-lag up to order six.
    

A detailed theoretical, numerical and computational analysis is also presented. The above analyses consist of:

• the construction of the new six-step pair,
• the presentation of the computed local truncation error of the new six-step pair,
• the comparative error analysis of the new six-step pair with other six-step pairs of the same family which are:

  • the classical six-step pair of the family (i.e. the six-step pair with constant coefficients),
  • the recently proposed six-step pair of the same family with vanished phase-lag and its first derivative,
  • the recently proposed six-step pair of the same family with vanished phase-lag and its first and second derivatives,
  • the recently proposed six-step pair of the same family with vanished phase-lag and its first, second and third derivatives,
  • the recently proposed six-step pair of the same family with vanished phase-lag and its first, second, third and fourth derivatives and finally,
  • the recently proposed six-step pair of the same family with vanished phase-lag and its first, second, third, fourth and fifth derivatives
• the stability and the interval of periodicity analysis for the new obtained six-step pair and finally
• the investigation of the accuracy and computational efficiency of the new developed six-step pair for the solution of the Schrödinger equation.

The theoretical, numerical and computational achievements lead to the conclusion that the new produced three-stages symmetric six-step pair is more efficient than other known or recently developed finite difference pairs of the literature.

     

A new method with vanished phase-lag and its derivatives of the highest order for problems in quantum chemistry

Journal of Mathematical Chemistry - A new FiDif (finite difference) method with zeroing phase-lag and its derivatives up to order seven, for initial or boundary value problems with periodical...     

Two-step method with vanished phase-lag and its derivatives for problems in quantum chemistry: an economical case

Journal of Mathematical Chemistry - A new ECON2STEP (Economical Two-Step Method) method with vanished phase-lag and its derivatives up to order five is introduced in this paper, for initial or...     

A new hybrid two-step method with vanished phase-lag and its first and second derivatives for the numerical solution of the Schr?dinger equation and related problems

The maximization of the efficiency of a hybrid two-step method for the numerical solution of the radial Schr?dinger equation and related problems with periodic or oscillating solutions via the procedure of vanishing of the phase-lag and its derivatives is studied in this paper. More specifically, we investigate the vanishing of the phase-lag and its first and second derivatives and how this disappearance maximizes the efficiency of the hybrid two-step method.     

A new economical method with eliminated phase-lag and its derivative for problems in chemistry

Journal of Mathematical Chemistry - A new EcoFinDif (Economical Finite Difference) method with eliminated phase-lag and its first derivative, for periodical and/or oscillating initial or boundary...     

A new high order two-step method with vanished phase-lag and its derivatives for the numerical integration of the Schr?dinger equation

In this paper we develop a new hybrid method of high order with phase-lag and its first, second and third derivatives equal to zero. For the produced method we study its error and stability. We apply the newly obtained method to the Schr?dinger equation. The application shows the efficiency of the new produced method.     

A family of Runge-Kutta methods with zero phase-lag and derivatives for the numerical solution of the Schrödinger equation and related problems

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 Schrödinger equation and related ordinary differential equations with oscillating solutions. The numerical results show the superiority of the new technique of nullifying both the phase-lag and its derivatives.     

A generator of families of two-step numerical methods with free parameters and minimal phase-lag

This research work is oriented in the behaviour of oscillating systems. In order to study such problems, we deal with the solution of ordinary second order differential equations. A generator of families of numerical methods is developed in our effort to solve equations of that type. The families created have constant coefficients and free parameters. We calculate the free parameters taking into consideration the condition of minimal phase-lag. The new methods are applied to the problem of the time independent Schrödinger equation and their results are presented. We also examine the properties of stability and minimum local truncation error.     

A family of numerical multistep methods with three distinct schemes: explicit advanced step-point (EAS) methods and the EAS1 approach

In this work, for the first time in an article, we present in a comprehensive way the explicit advanced step-point (EAS) methods. The EAS methods is a family of methods designed for the numerical solution of non-stiff and mildly stiff initial value problems (IVPs) and comprises three distinct schemes: EAS1, EAS2 and EAS3. A thorough theoretical analysis of the EAS family of predictor–corrector methods is presented in terms of their accuracy and stability characteristics and requirements, as well as the rationale for creating the three distinct schemes mentioned above. In this paper we also examine in detail one of the three schemes, the EAS1 methods. EAS1 are assessed for the very first time, are meticulously studied and their superior regions of absolute stability are presented. Furthermore the computational efficiency of EAS1 is examined and comparative numerical results are presented with the use of a variable step, variable order EAS1 code. The numerical results provide good evidence that EAS1 could be seen as superior to the well established Adams methods for the numerical solution of mildly stiff initial value problems.     

Determination of lamivudine and zidovudine in binary mixtures using first derivative spectrophotometric, first derivative of the ratio-spectra and high-performance liquid chromatography-UV methods

Three methods are presented for the simultaneous determination of lamivudine and zidovudine. The first method depends on first derivative UV spectrophotometry, with zero-crossing and peak-to-base measurement. The first derivative amplitudes at 265.6 and 271.6 nm were selected for the assay of lamivudine and zidovudine, respectively. The second method depends on first derivative of the ratio-spectra by measurements of the amplitudes at 239.5 and 245.3 nm for lamivudine and 225.1 and 251.5 nm for zidovudine. Calibration graphs were established for 1-50 μg/ml for lamivudine and 2-100 μg/ml for zidovudine. In the third method (HPLC), a reversed-phase column with a mobile phase of methanol:water:acetonitrile (70:20:10 (v/v/v)) at 0.9 ml/min flow rate was used to separate both compounds with a detection of 265.0 nm. Linearity was obtained in the concentration range of 0.025-50 μg/ml for lamivudine and 0.15-50 μg/ml for zidovudine. All of the proposed methods have been extensively validated. These methods allow a number of cost and time saving benefits. The described methods can be readily utilized for analysis of pharmaceutical formulations. There was no significant difference between the performance of all of the proposed methods regarding the mean values and standard deviations. The described HPLC method showed to be appropriate for simultaneous determination of lamivudine and zidovudine in human serum samples.     

The variational explicit polarization potential and analytical first derivative of energy: Towards a next generation force field

A previous article proposed an electronic structure-based polarizable potential, called the explicit polarization (X-POL) potential, to treat many-body polarization and charge delocalization effects in polypeptides. Here, we present a variational version of the X-POL potential, in which the wave function of the entire molecular system is variationally optimized to yield the minimum total electronic energy. This allows the calculation of analytic gradients, a necessity for efficient molecular dynamics simulations. In this paper, the detailed derivations of the Fock matrix and analytic force are presented and discussed. The calculations involve a double self-consistent-field procedure in which the wave function of each fragment is self-consistently optimized in the presence of other fragments, and in addition the polarization of the entire system is self-consistently optimized. The variational X-POL potential has been implemented in the Chemistry at Harvard Molecular Mechanics (CHARMM) package and tested successfully for small model compounds.     

A new FinDiff numerical scheme with phase-lag and its derivatives equal to zero for periodic initial value problems

Journal of Mathematical Chemistry - A new Finite Difference complete in phase method with vanished derivatives of the Phase-lag up to order five for the initial value problems with periodical...     

High order multistep methods with improved phase-lag characteristics for the integration of the Schr?dinger equation

In this work we introduce a new family of 12-step linear multistep methods for the integration of the Schr?dinger equation. The new methods are constructed by adopting a new methodology which improves the phase lag characteristics by vanishing both the phase lag function and its first derivatives at a specific frequency. This results in decreasing the sensitivity of the integration method on the estimated frequency of the problem. The efficiency of the new family of methods is proved via error analysis and numerical applications. T. E. Simos is a highly cited researcher, active member of the European Academy of Sciences and Arts. Corresponding member of the European Academy of Sciences, corresponding member of European Academy of Arts, Sciences and Humanities.