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.     

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.

     

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.

     

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 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 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 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.

     

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.     

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.     

On the structure of discrete spectrum of a non-selfadjoint system of differential equations with integral boundary condition

In this paper, we investigated the spectrum of the operator \(L(\lambda )\) generated in Hilbert Space of vector-valued functions \(L_{2}(\mathbb {R}_{+},C_{2})\) by the system

$$\begin{aligned} iy_{1}^{'}(x,\lambda )+q_{1}(x)y_{2}(x,\lambda )&=\lambda y_{1}(x,\lambda )\\ -iy_{2}^{'}(x,\lambda )+q_{2}(x)y_{1}(x,\lambda )&=\lambda y_{2}(x,\lambda ),x\in \mathbb {R}_{+}:=(0,\infty ), \end{aligned}$$

and the integral boundary condition of the type

$$\begin{aligned} \int _{0}^{\infty }K(x,t)y(t,\lambda ){\mathrm {dt}}+\alpha y_{2}(0,\lambda )-\beta y_{1}(0,\lambda )=0 \end{aligned}$$

where \(\lambda \) is a complex parameter, \(q_{i},\,i=1,2\) are complex-valued functions and \(\alpha ,\beta \in \mathbb {C}\). K(x, t) is vector fuction such that \(K(x,t)=(K_{1}(x,t),K_{2}(x,t)),\,K_{i}(x,t)\in L_{1}(0,\infty )\cap L_{2}(0,\lambda ),\,i=1,2\). Under the condition

$$\begin{aligned} \left| q_{i}(x)\right| \le ce^{-\varepsilon \sqrt{x}},c>0,\varepsilon >0,i=1,2 \end{aligned}$$

we proved that \(L(\lambda )\) has a finite number of eigenvalues and spectral singularities with finite multiplicities.

     

Intrinsic ratios of glucose,fructose, glycerol and ethanol <Superscript>13</Superscript>C/<Superscript>12</Superscript>C isotopic ratio determined by HPLC-<Emphasis Type="Italic">co</Emphasis>-IRMS: toward determining constants for wine authentication

High-performance liquid chromatography linked to isotope ratio mass spectrometry (HPLC-co-IRMS) via a Liquiface© interface has been used to simultaneously determine ¹³C isotope ratios of glucose (G), fructose (F), glycerol (Gly) and ethanol (Eth) in sweet and semi-sweet wines. The data has been used the study of wine authenticity. For this purpose, 20 authentic wines from various French production areas and various vintages have been analyzed after dilution in pure water from 20 to 200 times according to sugar content. If the ¹³C isotope ratios vary according to the production area and the vintage, it appears that internal ratios of ¹³C isotope ratios \(\left( {R_{^{13} C} } \right)\) of the four compounds studied can be considered as a constant. Thus, ratios of isotope ratios are found to be 1.00?±?0.04 and 1.02?±?0.08 for \(R_{^{13} C_{G/F} }\) and \(R_{^{13} C_{Gly/Eth} }\), respectively. Moreover, \(R_{^{13} C_{Eth/Sugar} }\) is found to be 1.15?±?0.10 and 1.16?±?0.08 for \(R_{^{13} C_{Gly/Sugar} }\). Additions of glucose, fructose and glycerol to a reference wine show a variation of the \(R_{^{13} C}\) value for a single product addition as low as 2.5 g/L^?1. Eighteen commercial wines and 17 concentrated musts have been analyzed. Three wine samples are suspicious as the \(R_{^{13} C}\) values are out of range indicating a sweetening treatment. Moreover, concentrated must analysis shows that ¹³C isotope ratio can be also used directly to determine the authenticity of the matrix.

Open image in new window

Figure HPLC-co-IRMS chromatogram of a diluted sweet wine.

     

Deposition Morphology and Magnetism of Co,Pt Adatoms and Small CoPt Adclusters on Ni(100) Substrate

Theoretical calculations of Co\(_{n-x}\)Pt\(_x\) (n = 1–3; \(x \le n\)) clusters on Ni(100) surface for their spin and orbital magnetic moments, as well as the magnetic anisotropy energy (MAE), are performed by using the density-functional theory (DFT) method including a self-consistent treatment of spin–orbit coupling (SOC). The results reveal that the ferromagnetic Co atoms in intra Co\(_{n-x}\)Pt\(_x\) adclusters couple ferromagnetically to their underlayer Ni atoms. The predominant inter-interactions between Co adatoms and Ni surface with the partly filled 3d band, together with the secondary intra-interactions between Co adatoms and Pt adatoms with fully filled 5d band, lead to a strongly quenched orbital moment (\(\mu _{\mathrm{{orb}}}^{\mathrm{{Co}}}\) = 0.18–0.14 \(\mu _B\); \(\mu _{\mathrm{{orb}}}^{\mathrm{{Pt}}} \approx \) 0.24–0.19 \(\mu _B\)) but a less quenched spin moment (\(\mu _{\mathrm{{spin}}}^{\mathrm{{Co}}} \approx \) 2.0 \(\mu _B\); \(\mu _{\mathrm{{spin}}}^{\mathrm{{Pt}}} \approx \) 0.35 \( \mu _B\)). The MAEs of CoPt adclusters exhibit a strong dependence on alloying effect rather than size effect, which is direly proportional to SOC strength and orbital moment anisotropy. The oxidations of CoPt clusters always reduce orbital magnetic moments and consequently decrease the corresponding MAEs.     

Densities,Ultrasonic Velocities,Excess Properties and IR Spectra of Binary Liquid Mixtures of Organic Esters (Ethyl Lactate,Some Organic Carbonates)

Organic esters of carbonic acid {dimethyl carbonate (DMC)/diethyl carbonate (DEC)/propylene carbonate (PC)}, in combination with a lactate ester {ethyl lactate (EL)}, with green chemistry characteristics were chosen for the present study of molecular interactions in binary liquid mixtures. Densities (ρ) and ultrasonic velocities (U) of the pure solvents and liquid mixtures were measured experimentally over the entire composition range at temperatures (303.15, 308.15, 313.15 and 318.15) K and atmospheric pressure. The experimental data was used to calculate thermodynamic and acoustic parameters \( V_{\text{m}}^{\text{E}} \), \( \kappa_{S}^{\text{E}} \), \( L_{\text{f}}^{\text{E}} \), \( \bar{V}_{\text{m,1}}^{{}} \), \( \bar{V}_{\text{m,2}}^{{}} \), \( \bar{V}_{\text{m,1}}^{\text{E}} \), \( \bar{V}_{\text{m,2}}^{\text{E}} \), \( \bar{V}_{ 1}^{\text{E,0}} \) and \( \bar{V}_{ 2}^{\text{E,0}} \) and the excess functions were fitted with the Redlich–Kister polynomial equation to obtain the binary solution coefficients and the standard deviations. It was observed that the values of \( V_{\text{m}}^{\text{E}} \), \( \kappa_{S}^{\text{E}} \) and \( L_{\text{f}}^{\text{E}} \) are positive for the mixtures of (EL + DMC/DEC) and negative for those of (EL + PC) over the entire range of composition and temperature. The positive values of \( V_{\text{m}}^{\text{E}} \), \( \kappa_{S}^{\text{E}} \) and \( L_{\text{f}}^{\text{E}} \) indicate the action of dispersion forces between the component molecules of (EL + DMC/DEC) mixtures whereas negative values for the mixture (EL + PC) suggest the existence of strong specific interactions between the component molecules, probably resulting from chemical and structural contributions. The excess properties have also been analyzed by using the reduced (\( Y^{\text{E}} /x_{1} x_{2} \)) excess function approach and the results are found to be in agreement with those from the corresponding \( Y^{\text{E}} \)(= \( V_{\text{m}}^{\text{E}} \), \( \kappa_{S}^{\text{E}} \) and \( L_{\text{f}}^{\text{E}} \)) values. This is further supported by FTIR spectral analysis.     

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.     

Transition-metal dichalcogenide heterostructure solar cells: a numerical study

We evaluate the tunneling short-circuit current density \(J_{TU}\) in a p–i–n solar cell in which the transition metal dichalcogenide heterostructure (\(\hbox {MoS}_2/\hbox {WS}_2\) superlattice) is embedded in the intrinsic i region. The effects of varying well and barrier widths, Fermi energy levels and number of quantum wells in the i region on \(J_{TU}\) are examined. A similar analysis is performed for the thermionic current \(J_{TH}\) that arises due to the escape and recapture of charge carriers between adjacent potential wells in the i-region. The interplay between \(J_{TU}\) and \(J_{TH}\) in the temperature range (300–330 K) is examined. The thermionic current is seen to exceed the tunneling current considerably at temperatures beyond 310 K, a desirable attribute in heterostructure solar cells. This work demonstrates the versatility of monolayer transition metal dichalcogenides when utilized as fabrication materials for van der Waals heterostructure solar cells.