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 multistep method with optimal properties for second order differential equations

In the present paper an effective symmetric hybrid six-step method for the approximation of the solution of the Schrödinger equation and related problems is developed. More specifically, we will produced a method with the following properties: (1) is a symmetric hybrid six-step method, (2) is of twelfth algebraic order, (3) has three stages, (4) has eliminated phase-lag, (5) has eliminated the derivatives of the phase-lag up to order three. This numerical pair is obtained for the first time in the literature. We present a detailed analysis for the new obtained numerical scheme. More specifically we present:

• the construction of the new pair
• the computation of the local truncation error of the new numerical pair
• the comparison of the asymptotic form of the local truncation error of the new numerical scheme with the the classical pair of the family (i.e. scheme with constant coefficients), the recently developed scheme of the family with vanished phase-lag and its first derivative and the recently developed algorithm of the family with vanished phase-lag and its first and second derivatives
• the stability and interval of periodicity analysis and
• finally, the accuracy and computational efficiency of the new numerical pair for the solution of the Schrödinger equation.

The theoretical and numerical achievements which are presented in this paper, show the efficiency of the new numerical pair compared with other known or recently developed pairs of the literature.

     

High order four-step hybrid method with vanished phase-lag and its derivatives for the approximate solution of the Schrödinger equation

In this paper, we develop a new four-step hybrid method of sixth algebraic order with vanished phase-lag and its first and second derivatives. For the obtained method we study:

– its error and

– its stability

We apply the produced method to the Schrödinger equation in order to show its efficiency.     

An efficient six-step method for the solution of the Schrödinger equation

In this paper we develop an efficient six-step method for the solution of the Schrödinger equation and related problems. The characteristics of the new obtained scheme are:

• It is of twelfth algebraic order.
• It has three stages.
• It has vanished phase-lag.
• It has vanished its derivatives up to order two.
• All the stages of the scheme are approximations on the point \(x_{n+3}\).

This method is developed for the first time in the literature. A detailed theoretical analysis of the method is also presented. In the theoretical analysis, a comparison with the the classical scheme of the family (i.e. scheme with constant coefficients) and with recently developed algorithm of the family with eliminated phase-lag and its first derivative is also given. Finally, we study the accuracy and computational effectiveness of the new developed algorithm for the on the approximation of the solution of the Schrödinger equation. The above analysis which is described in this paper, leads to the conclusion that the new algorithm is more efficient than other known or recently obtained schemes of the literature.

     

A new multistep finite difference pair for the Schrödinger equation and related problems

In this paper and for the first time in the literature, we introduce a new three-stages symmetric six-step finite difference pair with optimal phase and stability properties. The basic characteristics of the new finite difference pair are:

1. 1.
   Is a symmetric hybrid six-step method,
    
2. 2.
   Is of three stages
    
3. 3.
   Is of twelfth algebraic order,
    
4. 4.
   Has vanished the phase-lag,
    
5. 5.
   Has vanished the derivatives of the phase-lag up to order four.
    

For this new finite difference pair we present a detailed analysis which consists of the following:

1. 1.
   The development of the new three-stages symmetric six-step finite difference pair
    
2. 2.
   The presentation of the local truncation error of the new finite difference pair
    
3. 3.
   A comparative error analysis of the new finite difference pair with other finite difference pairs of the same family: the the classical finite difference pair of the family (i.e. the finite difference pair with constant coefficients), the recently developed finite difference pairs of the same family with vanished phase-lag and its first derivative, the recently developed scheme of the same family with vanished phase-lag and its first and second derivatives and finally with the recently developed finite difference algorithm of the same family with vanished phase-lag and its first, second and third derivatives .
    
4. 4.
   A stability and an interval of periodicity analysis and
    
5. 5.
   Finally, the evaluation of the accuracy and computational efficiency of the new three-stages symmetric six-step finite difference pair for the solution of the Schrödinger equation.
    

The theoretical and numerical analysis of the produced new three-stages symmetric six-step finite difference pair, which are presented in this paper, show the effectiveness of the new scheme compared with other known or recently developed algorithms of the literature.

     

New four-stages symmetric six-step method with improved phase properties for second order problems with periodical and/or oscillating solutions

In this paper, we build, for the first time in the literature, a new four-stages symmetric six-step finite difference pair with optimized properties. The method:

1. 1.
   is a symmetric non-linear six-step method,
    
2. 2.
   is of four stages
    
3. 3.
   is of fourteenth algebraic order,
    
4. 4.
   has eliminated the phase-lag,
    
5. 5.
   has eliminated the first and second derivatives of the phase-lag.
    

An analysis of the new proposed method is given in details in this paper. More specifically, we present:

1. 1.
   the building of the new four-stages symmetric six-step method,
    
2. 2.
   the computation of the local truncation error of the new proposed method,
    
3. 3.
   the comparative local truncation error analysis of the new proposed method with other finite difference pairs of the same family.
    
4. 4.
   the stability and the interval of periodicity analysis and
    
5. 5.
   finally, the investigation and evaluation of the computational efficiency of the new proposed scheme for the approximate solution of the Schrödinger equation.
    

The theoretical, computational and numerical results for the new proposed method show its effectiveness compared with other known or recently obtained finite difference pairs in the literature.

     

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.     

High order computationally economical six-step method with vanished phase-lag and its derivatives for the numerical solution of the Schrödinger equation

A computationally economical symmetric six-step algorithm with high algebraic and phase-lag order is obtained in this paper, for the first time in the literature. Some characteristics of the new algorithm are: (1) algebraic order ten tenth, (2) eliminated phase-lag and its first, second, third, fourth and fifth derivatives, (3) the first layer is an approximation on the point \(x_{n+3}\) and no at the usual point \(x_{n}\). A detailed analysis is also presented. In order to evaluate the efficiency of the new algorithm, we compare it with other well known and recently developed algorithms on three stages of evaluation: (1) evaluation based on local truncation error. (2) Evaluation based on stability analysis. (3) Evaluation based on accuracy and computational efficiency of the numerical approximation of the Schrödinger equation. Based on the above analysis, we arrive to the conclusion that the new developed method is more effective than other well known or recently produced methods of the literature.     

Two stages six-step method with eliminated phase-lag and its first,second, third and fourth derivatives for the approximation of the Schrödinger equation

For the first time in the literature, a new two-stages symmetric six-step algorithm is developed and analyzed. The new algorithm has:

• tenth algebraic order (which is the highest possible order),
• vanished phase-lag and its first, second, third and fourth derivatives.
• good stability properties i.e. an interval of periodicity, equal to \(\left( 0, 133.36 \right) \),
• the approximation of the first stage of the algorithm is done on the point \(x_{n+3}\) and no at the usual point \(x_{n}\).

We also present a full analysis of the new algorithm (i.e. error, stability and interval of periodicity analysis). Finally, we also examine the effectiveness of the new obtained algorithm by comparing it with well known algorithms and very recently produced algorithms in the literature. Three stages of comparison for the efficiency of the algorithm are used:

• Comparison on local truncation error analysis,
• Comparison on stability analysis,
• Comparison on accuracy and computational effectiveness of the solution of the Schrödinger equation.

The theoretical and numerical achievements lead to the conclusion that the new algorithm is more efficient than other well known or recently obtained algorithms.

     

A new family of two stage symmetric two-step methods with vanished phase-lag and its derivatives for the numerical integration of the Schrödinger equation

A family of two stage low computational cost symmetric two-step methods with vanished phase-lag and its derivatives is developed in this paper. More specifically we produce:

• a two-stage symmetric two-step eighth algebraic order method which has the phase-lag and its first, second and third derivatives vanished and
• a two-stage symmetric two-step sixth algebraic order method, which is P-stable and has the phase-lag and its first and second derivatives vanished.

The local truncation error, the interval of periodicity and the effect of the vanishing of the phase-lag and its derivatives on the efficiency of the obtained method are also studied in this paper.

     

Three stages symmetric six-step method with eliminated phase-lag and its derivatives for the solution of the Schrödinger equation

For the first time in the literature we develop in this paper a three stages symmetric six-step scheme with twelfth algebraic and eliminated phase-lag and its first derivative. An additional characteristic of the new scheme is that the first and the second layer denote the approximation of the function on the point \(x_{n+3}\). We also present a local truncation error analysis and a stability and interval of periodicity analysis and we compared the new scheme with the classical scheme (i.e. scheme with constant coefficients). Additionally, we examine in details the accuracy and computational efficiency of the new developed scheme on the numerical solution of the Schrödinger equation. The study and investigation which are presented in this paper, lead to the conclusion that the new obtained scheme is more effective than other known or recently developed methods of the literature.     

A new three-stages six-step finite difference pair with optimal phase properties for second order initial and/or boundary value problems with periodical and/or oscillating solutions

A new symmetric six-step method with improved phase properties is introduced, for the first time in the literature, in this paper. The proposed method: (1) is a symmetric nonlinear six-step method, (2) is of three-stages, (3) is of twelfth algebraic order, (4) has vanished the phase-lag and (5) has vanished the derivatives of the phase-lag up to order five. For the new symmetric six-step method we present a detailed theoretical, numerical and computational analysis which consists of: (a) the development of the new three-stages symmetric six-step method, (b) the computation of the local truncation error of the new scheme, (c) the comparative error analysis of the new algorithm with other schemes of the same family: (i) the classical algorithm of the family (i.e. the method with constant coefficients), (ii) the recently proposed method of the same family with vanished phase-lag and its first derivative, (iii) the recently proposed method of the same family with vanished phase-lag and its first and second derivatives, (iv) the recently proposed method of the same family with vanished phase-lag and its first, second and third derivatives and finally (v) the recently proposed method of the same family with vanished phase-lag and its first, second, third and fourth derivatives, (d) the stability and the interval of periodicity analysis for the new developed method and finally (e) the study of the accuracy and computational effectiveness of the new proposed method for the solution of the Schrödinger equation. The theoretical, numerical and computational analysis of the new obtained three-stages symmetric six-step method show the efficiency of the new algorithm compared with other known or recently developed schemes of the literature.     

Direct determination of pair natural orbitals

A method is proposed to solve the two-electron Schrödinger equation by a rapidly converging iterative procedure. The wavefunction is obtained in terms of its NO's. The special features of the present method are:

1. Each iteration requires only the computational equivalent of a conventional Hartree-Fock iteration.
2. Within each iteration we improve simultaneously the NO's, the CI expansion coefficients and the total energy.
3. The construction of a CI matrix is never required.

We further propose simplified NO-equations the solution of which requires a small fraction of computertime only. As examples of the efficiency of these methods we report applications to the 1¹ Sstate of He, the 1¹∑ _g ⁺ , 1³∑ _u ⁺ states of H₂, and IEPA,PNO-CI, and CEPA type computations on CH₄.     

A new six-step algorithm with improved properties for the numerical solution of second order initial and/or boundary value problems

A new four-stages symmetric six-step finite difference pair with improved properties is developed in this paper and for the first time in the literature. The new finite difference pair: (1) is a symmetric non-linear six-step method, (2) is of four stages, (3) is of fourteenth algebraic order, (4) has eliminated the phase-lag, (5) has eliminated the first derivative of the phase-lag. A numerical analysis of the new developed finite difference pair is presented. The analysis of the scheme consists of: (1) the production of the new four-stages symmetric six-step finite difference pair, (2) the calculation of the local truncation error of the new finite difference pair, (3) the comparative error analysis of the new method with other scheme of the same family which is the classical finite difference pair of the family (i.e. the finite difference pair with constant coefficients). (4) The stability and an interval of periodicity analysis and (5) finally, the study of the numerical and computational efficiency of the new method for the solution of the Schrödinger equation. The theoretical and numerical achievements for the new obtained four-stages symmetric six-step finite difference pair show its efficiency compared with other known or recently obtained schemes of the literature.     

New Runge–Kutta type symmetric two step finite difference pair with improved properties for second order initial and/or boundary value problems

A new three-stages symmetric two-step method with improved properties is developed in this paper and for the first time in the literature. The properties of the new proposed algorithm are:

• is a symmetric finite difference pair,
• is a scheme of two-step,
• is an algorithm of three-stages—i.e. hybrid or Runge–Kutta type,
• is of tenth-algebraic order,
• it has vanished the phase-lag and its first, second and third derivatives,
• it has improved stability properties for the general problems,
• it is a P-stable method since it has an interval of periodicity equal to \(\left( 0, \infty \right) \).

The new proposed scheme is constructed based on the following layers:

• An approximation denoted on the first layer on the point \(x_{n-1}\),
• An approximation denoted on the second layer on the point \(x_{n}\) and finally,
• An approximation denoted on the third (final) layer on the point \(x_{n+1}\),

For the new proposed method we give a full theoretical analysis which consists of: (1) local truncation error analysis, (2) comparative local truncation error analysis, (3) stability analysis and (4) interval of periodicity analysis. The efficiency of the new proposed algorithm is tested on the approximate solution of systems of coupled differential equations arising from the Schrödinger equation.

     

A new two-step finite difference pair with optimal properties

A new four-stages symmetric two-step finite difference pair with optimal error, phase-lag and general stability properties is obtained, for the first time in the literature, in this paper.

The new scheme has the following properties:

• is of symmetric form,
• is a two-step finite difference pair,
• is of four-stages finite difference pair,
• is of tenth-algebraic order,
• the approximations which are obtained at each level of the new finite difference scheme are the following:

  1. 1.
     An approximation obtained on the first level on the point \(x_{n-1}\),
      
  2. 2.
     An approximation obtained on the second level on the point \(x_{n-1}\),
      
  3. 3.
     An approximation obtained on the third level on the point \(x_{n}\) and finally,
      
  4. 4.
     An approximation obtained on the fourth (final) level on the point \(x_{n+1}\),
      
• it has vanished the phase-lag and its first, second, third, fourth and fifth derivatives,
• it has optimized stability properties,
• has efficient stability properties since it has an interval of periodicity equal to \(\left( 0, 9,2 \right) \).

For the new four-stages symmetric two-step finite difference pair we present a full theoretical analysis (error and stability analysis).

The evaluation of the efficiency of the new developed four-stages symmetric two-step finite difference pair is based on its application on systems of coupled differential equations of the Schrödinger form.     

New hybrid two-step method with optimized phase and stability characteristics

In this paper and for the first time in the literature, we develop a new Runge–Kutta type symmetric two-step finite difference pair with the following characteristics:

• the new algorithm is of symmetric type,
• the new algorithm is of two-step,
• the new algorithm is of five-stages,
• the new algorithm is of twelfth-algebraic order,
• the new algorithm is based on the following approximations:

  1. 1.
     the first layer on the point \(x_{n-1}\),
      
  2. 2.
     the second layer on the point \(x_{n-1}\),
      
  3. 3.
     the third layer on the point \(x_{n-1}\),
      
  4. 4.
     the fourth layer on the point \(x_{n}\) and finally,
      
  5. 5.
     the fifth (final) layer on the point \(x_{n+1}\),
      
• the new algorithm has vanished the phase-lag and its first, second, third and fourth derivatives,
• the new algorithm has improved stability characteristics for the general problems,
• the new algorithm is of P-stable type since it has an interval of periodicity equal to \(\left( 0, \infty \right) \).

For the new developed algorithm we present a detailed numerical analysis (local truncation error and stability analysis). The effectiveness of the new developed algorithm is evaluated with the approximate solution of coupled differential equations arising from the Schrödinger type.

     

Theoretical discussion of methods of elimination of matrix effects in non-dispersive X-ray analysis

Theoretical consideration concerning some possibilities for the elimination of matrix effects in non-dispersive X-ray fluorescence and absorption analysis are discussed. The theoretical treatment is concerned with the following methods:

1. double-channel absorption edge analysis,
2. concentration increase and dilution method in fluorescence analysis,
3. fluorescence-absorption method,
4. emission-transmission method,
5. fluorescence-Compton scattering method,
6. method of multicomponent analysis.

On the basis of the derived formulas, nomographic methods of interpretation of the data are given. Using these methods it is possible to determine unambiguously the concentration of the relevant element. The formulas are also convenient for numerical interpretation. The introduction of the concept of “generalized sensitivity” allows the comparison of various radiometric methods.     

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.     

New 8-step symmetric embedded predictor–corrector (EPCM) method with vanished phase-lag and its first derivative for the numerical integration of the Schrödinger equation

In the present work, a new embedded predictor–corrector phase-fitted method with vanished phase-lag is developed for the first time in literature. It is about a multistep symmetric method built on the multistep method of Quinlan and Tremaine (Astron J 100(5):1694–1700, 1990) with eight steps. It can be used to solve numerically IVPs with oscillatory solutions, orbital problems and the Schrödinger equation. Initially we present a pair of embedded predictor–corrector method upon which the new method is built. We test our method and the numerical results indicate that this method is much more accurate than other well known methods including the radial Schrödinger equation.