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 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.     

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.     

A family of explicit linear six-step methods with vanished phase-lag and its first derivative

A family of explicit linear sixth algebraic order six-step methods with vanished phase-lag and its first derivative is obtained in this paper. The investigation of the above family of methods contains:

theoretical study of the new family of methods and

computational study of the new family of methods.

The theoretical study of the above mentioned family of methods contains:

1. the development of the method,
2. the computation of the local truncation error,
3. the comparative local truncation error analysis. The comparison is taken place between the new family of methods with the corresponding method with constant coefficients and
4. the stability analysis of the new family of methods. The stability analysis is taken place using test equation with different frequency than the frequency of the test equation used for the phase-lag analysis of the methods.

The application of the new family of linear six-step sixth algebraic order methods to the resonance problem of the one-dimensional time independent Schrödinger equation is used for the computational study of the new family of methods. The result of the above mentioned theoretical and computational investigation is that the new proposed family of linear explicit schemes are computationally and theoretically more effective than other well known methods for the approximate solution of the radial Schrödinger equation and related initial or boundary value problems with periodic and/or oscillating solutions.     

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...     

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 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 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...     

An adapted explicit hybrid four-step method for the numerical solution of perturbed oscillators

In this paper, the construction of an adapted explicit hybrid four-step method for the numerical integration of perturbed oscillators is investigated. This four-step method is based on the algorithm of Scheifele which is obtained by refining the classical method of power series. The local truncation error, phase properties and linear stability of the new method are analyzed. Numerical experiments are reported to show the high accuracy and efficiency of the new method when it is compared with some high-quality methods recently proposed in the literature.     

Analytic first and second derivatives of the energy in the fragment molecular orbital method combined with molecular mechanics

Analytic first and second derivatives of the energy are developed for the fragment molecular orbital method interfaced with molecular mechanics in the electrostatic embedding scheme at the level of Hartree-Fock and density functional theory. The importance of the orbital response terms is demonstrated. The role of electrostatic embedding upon molecular vibrations is analyzed, comparing force field and quantum mechanical treatments for an ionic liquid and a solvated protein. The method is applied for 100 protein conformations sampled in molecular dynamics (MD) to take into account the complexity of a flexible protein structure in solution, and a good agreement with experimental data is obtained: Frequencies from an experimental infrared (IR) spectrum are reproduced within 17 cm⁻¹ .     

A new method of synthesizing aminosugar derivatives

Conclusions A new method of synthesizing aminosugar derivatives based on the condensation reaction of isonitriles with the ammonium salts of carbohydrates is proposed.Translated from Izvestiya Akademii Nauk SSSR, Seriya Khimicheskaya, No. 5, pp. 1168–1169, May, 1969.     

Molecular first hyperpolarizabilities of retinal and its derivatives

The hyper-Rayleigh scattering technique (HRS) has been used to determine the molecular first hyperpolarizabilities β of retinal, retinoic acid, vitamin A acetate and a retinal Schiff base in its protonated and unprotonated form. The former is responsible for the linear and non-linear optical properties of bacteriorhodopsin. A previous examination of these compounds by HRS at 1064 nm yielded surprisingly high β-values. Here, we report HRS measurements performed at 1064, 1300 and 1500 nm. The derived hyperpolarizabilities are self-consistent with the two-state model for all three wavelengths, but they are one order of magnitude lower than those reported previously.     

A new eighth order exponentially fitted explicit Numerov-type method for solving oscillatory problems

In this work we derive a new family of seven stages, eighth order, exponential fitted, Numerov (i.e. two-step) type methods. A particular variable coefficient method is constructed. The numerical results show that it outperforms exponential fitted two step hybrid method found in the literature sharing similar characteristics. Moreover, the presented method outperforms Numerov type methods with minimized phase-lag for problems in consideration share a non dominant linear part.     

A new synthetic route to phomoidride B and its derivatives

We have developed a new synthetic route to phomoidride B, which could also be applied to the synthesis of phomoidride B derivatives using Pd-catalyzed coupling reaction of a thiolester with an organozinc reagent. In addition, direct construction of the maleic anhydride moiety has been achieved by a Pd-catalyzed carbonylation reaction.     

A new access to efaroxan and its 5-amino derivatives

The formal total synthesis of racemic efaroxan, a 2-disubstituted 2,3-dihydrobenzofuran having interesting therapeutic properties, and the preparation of some 5-amino derivatives were achieved using a convergent approach. The key intermediates of both syntheses were obtained after a [3+2] cycloaddition reaction between two easily accessible partners.