New five-stages finite difference pair with optimized phase properties

A new five-stages symmetric two-step finite difference pair is developed, for the first time in the literature, in this paper. This new finite difference pair has optimal phase and stability properties The main characteristics of the new finite difference pair are:

1. 1.
   it is of symmetric type,
    
2. 2.
   it is of two-step algorithm,
    
3. 3.
   it is of five-stages,
    
4. 4.
   it is of twelfth-algebraic order,
    
5. 5.
   the new nonlinear finite difference pair is produced using the following approximations:

   • An approximation developed on the first layer on the point \(x_{n-1}\),
   • An approximation developed on the second layer on the point \(x_{n-1}\),
   • An approximation developed on the third layer on the point \(x_{n-1}\),
   • An approximation developed on the fourth layer on the point \(x_{n}\) and finally,
   • An approximation developed on the fifth (final) layer on the point \(x_{n+1}\),
    
6. 6.
   it has vanished the phase-lag and its first derivative,
    
7. 7.
   it has optimized stability properties for the general problems,
    
8. 8.
   it is a P-stable method since it has an interval of periodicity equal to \(\left( 0, \infty \right) \).
    

A full numerical analysis (error and stability analysis) is given for the new finite difference pair.

The effectiveness of the new finite difference pair is evaluated by applying it on the approximate solution of systems of coupled differential equations of the Schrödinger form.     

A four stages numerical pair with optimal phase and stability properties

In this paper we obtain, for the first time in the literature a new numerical pair. The properties of the new pair are:

1. 1.
   symmetric two-step algorithm,
    
2. 2.
   four-stages scheme,
    
3. 3.
   scheme of tenth-algebraic order,
    
4. 4.
   the approximations of the layers are taken place as follows: first and second layer on the point \(x_{n-1}\), third layer on the point \(x_{n}\), fourth layer on the point \(x_{n+1}\),
    
5. 5.
   the method has eliminated the phase-lag and its first derivative,
    
6. 6.
   excellent stability and interval of periodicity properties (i.e. interval of periodicity equal to \(\left( 0, \infty \right) \).
    

For this new numerical pair we present a detailed theoretical analysis. Finally, we prove the efficiency of the new scheme by applying it to the solution of systems of coupled Schrödinger equations.

     

New finite difference pair with optimized phase and stability properties

In this paper and for the first time in the literature, a new four-stages symmetric two-step finite difference pair with optimized phase and stability properties is introduced. The new scheme has the following characteristics:

1. 1.
   is of symmetric type,
    
2. 2.
   is of two-step algorithm,
    
3. 3.
   is of four-stages,
    
4. 4.
   is of tenth-algebraic order,
    
5. 5.
   the approximations which produces the new finite difference pair are the following:

   • An approximation developed on the first layer on the point \(x_{n-1}\),
   • An approximation developed on the second layer on the point \(x_{n-1}\),
   • An approximation developed on the third layer on the point \(x_{n}\) and finally,
   • An approximation developed on the fourth (final) layer on the point \(x_{n+1}\),
    
6. 6.
   it has eliminated the phase-lag and its first, second, third and fourth derivatives,
    
7. 7.
   it has optimized stability properties,
    
8. 8.
   is a P-stable methods since it has an interval of periodicity equal to \(\left( 0, \infty \right) \).
    

For the new finite difference scheme we describe an error and stability analysis. The examination of the efficiency of the new obtained finite difference pair is based on its application on 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 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.

     

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 Runge–Kutta type symmetric two-step method with optimized characteristics

In this paper and for the first time in the literature, we build a new hybrid symmetric two-step method with the following properties: (1) the new scheme is of symmetric type, (2) the new scheme is of two-step, (3) the new scheme is of five-stages, (4) the new scheme is of twelfth-algebraic order, (5) the new scheme has eliminated the phase-lag and its first, second, third, fourth and fifth derivatives, (6) the new scheme has improved stability characteristics for the general problems, (7) the new scheme is P-stable [with interval of periodicity equal to \(\left( 0, \infty \right) \)] and (8) the new scheme builded based on the following approximations:

• the first stage is approximation on the point \(x_{n-1}\),
• the second stage is approximation on the point \(x_{n-1}\),
• the third stage is approximation on the point \(x_{n-1}\),
• the fourth stage is approximation on the point \(x_{n}\) and finally,
• the fifth stage is approximation on the point \(x_{n+1}\),

For the new builded scheme we give a full numerical analysis (local truncation error and stability analysis). The efficiency of the new builded scheme is examined with the numerical solution of systems of coupled differential equations of the Schrödinger type.

     

A hybrid finite difference pair with maximum phase and stability properties

We develop, in the present paper and for the first time in the literature, a new hybrid finite difference pair of symmetric two-step. The basic properties of the new pair are:

• is of symmetric form,
• is of two-step,
• is of four-stages,
• is of tenth-algebraic order,
• the development of the hybrid symmetric two-step pair is of the following form:

  • first layer is an approximation on the point \(x_{n-1}\),
  • second layer is an approximation on the point \(x_{n-1}\),
  • third layer is an approximation on the point \(x_{n}\) and finally,
  • fourth layer is an approximation on the point \(x_{n+1}\),
• it has vanished the phase-lag and its first, second and third derivatives,
• it has excellent stability properties,
• it has an interval of periodicity equal to \(\left( 0, \infty \right) \).

For the new symmetric finite difference pair a full analysis is presented. We evaluate the efficiency of the new obtained symmetric finite difference pair by applying it on systems of coupled differential equations of the Schrödinger form.

     

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.

     

New hybrid symmetric two step scheme with optimized characteristics for second order problems

In the present paper, for the first time in the literature, we build a new three-stages symmetric two-step finite difference pair with optimized properties. In more details the new method: (1) is of symmetric type, (2) is of two-step algorithm, (3) is of three-stages—i.e. hybrid or Runge–Kutta type, (4) it is of tenth-algebraic order, (5) it has vanished the phase-lag and its first and second derivatives, (6) it has optimized stability properties for the general problems, (7) it is a P-stable finite difference scheme since it has an interval of periodicity equal to \(\left( 0, \infty \right) \). The new Runge–Kutta type algorithm is builded based on the following approximations:

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

A full theoretical analysis (local truncation error analysis, comparative error analysis and stability and interval of periodicity analysis) is given for the new builded finite difference pair. The effectiveness of the new builded hybrid scheme is evaluated on the numerical solution of systems of coupled differential equations of the Schrödinger type.

     

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.

     

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 five–stages symmetric method with improved phase properties

In this paper, for the first time in the literature, we introduce a new five–stages symmetric two–step method with improved phase and stability properties. The characteristics of the new non linear two–step method are: (1) it is of symmetric type, (2) it is of two–step finite difference pair, (3) it is of five–stages, (4) it is of twelfth–algebraic order, (5) it has vanished the phase–lag and its first and second derivatives, (6) it has improved stability properties for the general problems, (7) it is a P-stable method since it has an interval of periodicity equal to \(\left( 0, \infty \right) \), (8) for the development of the new scheme, the following approximations are used: (a) an approximation developed on the first layer on the point \(x_{n-1}\), (b) an approximation developed on the second layer on the point \(x_{n-1}\), (c) an approximation developed on the third layer on the point \(x_{n-1}\), (d) an approximation developed on the fourth layer on the point \(x_{n}\) and finally, (e) an approximation developed on the fifth (final) layer on the point \(x_{n+1}\). For the new developed method we give a full theoretical analysis (error and stability analysis). The efficiency of the new five–stages symmetric two–step method is examined by applying it on the numerical solution of systems of coupled differential equations arising from the Schrödinger equation.     

A finite difference pair with improved phase and stability properties

A new finite difference pair is produced in this paper, for the first time in the literature. The characteristics of the new finite diffence pair are: (1) is of symmetric two-step, (2) is four-stages, (3) is of tenth-algebraic order, (4) the production of the pair is based on the following approximations for the layers: first and second layer are approximated on the point \(x_{n-1}\), third layer is approximated on the point \(x_{n}\) and finally fourth layer is approximated on the point \(x_{n+1}\), (5) has vanished the phase-lag and its first and second derivatives, (6) has excellent stability properties for all type of problems, (7) has an interval of periodicity equal to \(\left( 0, \infty \right) \). We present for the new obtained finite difference pair a full theoretical analysis. The effectiveness of the new developed finite difference pair is proved by its application on systems of coupled differential equations arising from the Schrödinger equation.     

New three–stages symmetric two step method with improved properties for second order initial/boundary value problems

In this paper and for the first time in the literature we develop a new three–stages symmetric two–step method with improved properties. More specifically the new scheme: (1) is of symmetric type, (2) is of two–step algorithm, (3) is of three–stages, (4) it is of tenth–algebraic order, (5) it has eliminated the phase–lag and its first derivative, (6) it has improved stability properties for the general problems, (7) it is a P–stable method since it has an interval of periodicity equal to \(\left( 0, \infty \right) \). In order to develop the new hybrid algorithm we use the following approximations: (i) An approximation developed on the first layer on the point \(x_{n-1}\), (ii) An approximation developed on the second layer on the point \(x_{n}\) and finally, (iii) An approximation developed on the third (final) layer on the point \(x_{n+1}\). For the new proposed method we give a full theoretical analysis (local truncation error analysis and stability and interval of periodicity analysis). The efficiency of the new proposed method is examined on the numerical solution of coupled differential equations arising from the Schrödinger equation.     

Structural,Energetic, and Magnetic Properties of Ag$$_{}$$Rh$$_{m}$$m and Ag$$_{m}$$mRh$$_{}$$ Clusters with $$n \le 20$$n≤20 and $$m=0,1$$m=0,1

At first, a genetic algorithm in combination with either the parametrized density-functional tight-binding method or a Gupta-potential is used to determine the putative global minimum energy structures of mixed Ag\(_{n-m}\)Rh\(_{m}\) and Ag\(_{m}\)Rh\(_{n-m}\) clusters with \(n\le 20\) and \(m=0,1\). Subsequently, the resulting structures are re-optimized with a first-principles method. The results demonstrate that the exchange of a single silver atom by rhodium leads to compact core-shell-like structures with structural motifs well known from the Lennard-Jones system. For the systems of the present study, AgRh\(_{n-1}\) clusters retain their cube-based structural motif and the silver atoms typically avoid the corner positions within a cube if possible. Population analysis of both cluster systems shows that the total magnetic moment is mainly due to unpaired electrons on the rhodium atoms with a small ferro-magnetic contribution of the silver host in Ag\(_{n-1}\)Rh and virtually no contribution to the total magnetic moment from the single silver atom in AgRh\(_{n-1}\) clusters.     

An economical eighth-order method for the approximation of the solution of the Schrödinger equation

In this paper we introduce, for the first time in the literature, a three-stages two-step method. The new algorithm has the following characteristics: (1) it is a two-step algorithm, (2) it is a symmetric method, (3) it is an eight-algebraic order method (i.e of high algebraic order), (4) it is a three-stages method, (5) the approximation of its first layer is done on the point \(x_{n-1}\) and not on the usual point \(x_{n}\), (6) it has eliminated the phase–lag and its derivatives up to order two, (7) it has good stability properties (i.e. interval of periodicity equal to \(\left( 0, 22 \right) \). For this method we present a detailed analysis : development, errorand stability analysis. The new proposed algorithm is applied to systems of differential equations of the Schrödinger type in order to examine its efficiency.     

Solubility Thermodynamics of CyMe$$_{}$$-BTBP in Various Diluents Mixed with TBP

The two organic ligands 6,6′-bis(5,5,8,8-tetramethyl-5,6,7,8-tetrahydrobenzo[1,2,4]triazin-3-yl)[2,2′]bipyridine (CyMe\(_{4}\)-BTBP) and tri-butyl phosphate (TBP) have previously been investigated in different diluents for use within recycling of used nuclear fuel through solvent extraction. The thermodynamic parameters, \(K_{\mathrm{S}}\), \(\Delta C_{p}\), \(\Delta H^{0}\) and \(\Delta S^{0}\), of the CyMe\(_{4}\)-BTBP solubility in three diluents (cyclohexanone, octanol and phenyl trifluoromethyl sulfone) mixed with TBP have been studied at 288, 298 and 308 K, both as pristine solutions and pre-equilibrated with 4 mol\(\cdot \)L\(^{-1}\) nitric acid. In addition, the amount of acid in the organic phase and density change after pre-equilibration have been measured. The solubility of CyMe\(_{4}\)-BTBP increases with an increased temperature in all systems, especially after acid pre-equilibration. This increased CyMe\(_{4}\)-BTBP solubility after pre-equilibration could be explained by acid dissolution into the solvent. Comparing the \(\Delta H^{0}\) and \(\Delta S^{0}\) calculated using \(\Delta C_{p}\) with the same parameters derived from a linear fit indicates temperature independence of all three thermodynamic systems. The change in enthalpy is positive in all solutions.     

An efficient and economical high order method for the numerical approximation of the Schrödinger equation

A new scheme is developed in this paper, for the first time in the literature. The new scheme: (1) is a symmetric two-step method, (2) is of three-stages scheme, (3) is a high order method (i.e of eight-algebraic order), (4) the approximations of the layers are taken place as follows: first layer on the point \(x_{n-1}\), second layer on the point \(x_{n}\), third layer on the point \(x_{n+1}\), (5) has vanished the phase-lag and its derivatives up to order four, (6) has good interval of periodicity properties [i.e. interval of periodicity equal to (0, 9.8)]. A detailed theoretical analysis is also presented. More specifically we present: (1) the development of the new method, (2) comparative error analysis (3) stability analysis. The effectiveness of the new scheme is tested via the solution of systems of coupled differential equations of the Schrödinger type.     

Magnetic Silicon Fullerenes: Experimental Exploration and Theoretical Insight

The present article summarizes progress in research on silicon clusters with encapsulated metal atoms, and specifically focuses on the recent identification of magnetic silicon fullerenes. Considering that C\(_{20}\) forms the smallest known fullerene, the Si\(_{20}\) cluster is of particular interest in this context. While the pure hollow Si\(_{20}\) cage is unstable due to the lack of \(sp^2\) hybridization, endohedral doping with a range of metal atoms has been considered to be an effective way to stabilize the cage structure. In order to seek out suitable embedded atoms for stabilizing Si\(_{20}\), a broad search has been made across elements with relatively large atomic radius. The rare earth elements have been found to be able to stabilize the Si\(_{20}\) cage in the neutral state by forming R@Si\(_{20}\) fullerene cages. Among these atoms, Eu@Si\(_{20}\) has been reported to yield a stable magnetic silicon fullerene. The central europium atom has a large magnetic moment of nearly 7.0 Bohr magnetons. In addition, based on a stable Eu\(_2\)Si\(_{30}\) tube, a magnetic silicon nanotube has been constructed and discussed. These magnetic silicon fullerenes and nanotubes may have potential applications in the fields of spintronics and high-density magnetic storage.