A Modified Phase-Fitted Runge–Kutta Method for the Numerical Solution of the Schrödinger Equation

A modified phase-fitted Runge–Kutta method (i.e., a method with phase-lag of order infinity) for the numerical solution of periodic initial-value problems is constructed in this paper. This new modified method is based on the Runge–Kutta fifth algebraic order method of Dormand and Prince [33]. The numerical results indicate that this new method is more efficient for the numerical solution of periodic initial-value problems than the well known Runge–Kutta method of Dormand and Prince [33] with algebraic order five.     

A Family of Exponentially-fitted Runge–Kutta Methods with Exponential Order Up to Three for the Numerical Solution of the Schrödinger Equation

We have constructed three Runge–Kutta methods based on a classical method of Fehlberg with eight stages and sixth algebraic order. These methods have exponential order one, two and three. We show through the error analysis of the methods that by increasing the exponential order, the maximum power of the energy in the error expression decreases. So the higher the exponential order the smaller the local truncation error of the method compared to the corresponding classical method. The difference is higher for higher values of energy. The results confirm this, when integrating the resonance problem of the one-dimensional time-independent Schr?dinger equation.     

Wavelet-based collocation method for stiff systems in process engineering

Abrupt phenomena in modelling real-world systems such as chemical processes indicate the importance of investigating stiff systems. However, it is difficult to get the solution of a stiff system analytically or numerically. Two such types of stiff systems describing chemical reactions were modelled in this paper. A numerical method was proposed for solving these stiff systems, which have general nonlinear terms such as exponential function. The technique of dealing with the nonlinearity was based on the Wavelet-Collocation method, which converts differential equations into a set of algebraic equations. Accurate and convergent numerical solutions to the stiff systems were obtained. We also compared the new results to those obtained by the Euler method and 4th order Runge–Kutta method.     

On some discretization methods for solving a linear matrix ordinary differential equation

In this paper, some discretization methods are considered for solving a linear matrix ordinary differential equation. Discussion is focused on a family of one step methods which include Euler, backward Euler, and Crank–Nicolson schemes as special cases, as well as the Runge–Kutta methods. As an illustration, detailed convergence and error analysis are given for the family of one step methods. Some numerical examples are provided to show the good performance of the methods.     

Trigonometrically fitted Runge–Kutta methods for the numerical solution of the Schrödinger equation

In this paper we construct two trigonometrically fitted methods based on a classical Runge–Kutta method of England with fifth algebraic order. The methods will be used for the integration of the radial Schrödinger equation and have high efficiency as the results show. The efficiency is higher when using higher energy and this can be explained by the error analysis of the methods. More specifically the new methods have lower powers of the energy in the local truncation error and that keeps the error at lower values.PACS: 0.260, 95.10.EActive Member of the European Academy of Sciences and Arts     

Numerical solutions for a coupled non-linear oscillator

A second-order accurate numerical method has been proposed for the solution of a coupled non-linear oscillator featuring in chemical kinetics. Although implicit by construction, the method enables the solution of the model initial-value problem (IVP) to be computed explicitly. The second-order method is constructed by taking a linear combination of first-order methods. The stability analysis of the system suggests the existence of a Hopf bifurcation, which is confirmed by the numerical method. Both the critical point of the continuous system and the fixed point of the numerical method will be seen to have the same stability properties. The second-order method is more competitive in terms of numerical stability than some well-known standard methods (such as the Runge–Kutta methods of order two and four).     

Modeling Structural Changes in the Deposit during Galvanostatic Electrolysis and Contact Deposition of Metals

The crystallization of dendritic deposits during a galvanostatic electrolysis and a contact deposition of metals from aqueous solutions is modeled. Structural changes in a deposit are characterized by its distribution by radii of apexes of growing dendrites. The calculation is carried out by solving a set of differential equations by the Runge–Kutta method within the Mathcad Plus 6 package. The reduction of metals with different exchange currents during electrolysis and with different exchange currents of the component being reduced during cementation is considered. The calculated results qualitatively conform to available experimental data.     

Construction of Trigonometrically and Exponentially Fitted Runge–Kutta–Nyström Methods for the Numerical Solution of the Schrödinger Equation and Related Problems – a Method of 8th Algebraic Order

General conditions are presented for an m-stage Runge–Kutta–Nyström fitting to exponential and trigonometric functions. As an example an 8th order Runge–Kutta–Nyström method is constructed. Numerical results on the numerical solution of the Schrödinger equation and related problems indicate that the new method is more accurate than the classical one (we call classical Runge–Kutta–Nyström method the corresponding method with constant coefficients).     

A Runge–Kutta type four-step method with vanished phase-lag and its first and second derivatives for each level for the numerical integration of the Schrödinger equation

The investigation of the impact of the vanishing of the phase-lag and its first and second derivatives on the efficiency of a four-step Runge–Kutta type method of sixth algebraic order is presented in this paper. Based on the above mentioned investigation, a Runge–Kutta type of two level four-step method of sixth algebraic order is produced. The error and the stability of the new obtained method are also studied in the present paper. The obtained new method is applied to the resonance problem of the Schrödinger equation the efficiency of the method to be examined.     

Kinetic parameters determination in non-isothermal conditions for the crystallisation of a silica-soda-lead glass

Two integral isoconversional methods (Flynn–Wall–Ozawa and Kissinger–Akahira–Sunose) and the invariant kinetic parameters method (IKP) were used in order to examine the kinetics of the non-isothermal crystallisation of a silica-soda-lead glass. The objective of the paper is to show the usefulness of the IKP method to determine both the activation parameters and the kinetic model of the investigated process. Thismethod associated with the criterion of coincidence of kinetic parameters for all heating rates and some procedures of the evaluation of the parameter from Johnson–Mehl–Avrami–Erofeev–Kolmogorov (JMAEK) equation led us to the following kinetic triplet: activation energy, E=170.5±2.5 kJ mol^–1 , pre-exponential factor, A=1.178±0.350·10 10 min^–1 and JMAEK model (A _m) m=1.5.     

Absorption spectra of carbohydrates and related compounds in H2SO4

The possibility of modifying the Dreywood anthrone method for spectrophotometric determination of carbohydrates in order to increase the accuracy and reproducibility of the results was examined. The classical method and a developed modification were compared. The relative uncertainty in a determination by the modified method was less than 4%. __________ Translated from Khimiya Prirodnykh Soedinenii, No. 3, pp. 218–220, May–June, 2006.     

Heat capacity increases with pressure

Approximation polynomial of temperature for the emf of a thermocouple is of high order, with low accuracy and many digits in the polynomial coefficients. These disadvantages are shown clearly in comparison with the approximation of high-temperature heat capacity. The fitting problems result from the fundamental reason, namely, the particular analytical expression for the emf as the function of temperature. In the approximation theory, this disadvantage is known as Runge’s phenomenon. In this report, it is shown to be typical of the functions with a negative power of the variable, where the derivation produces a factorial.     

Capillary electrophoresis–electrospray mass spectrometry and HR–ICP–MS for the detection and quantification of 10B-boronophenylalanine (10B–BPA) used in boron neutron capture therapy

Boron neutron capture therapy (BNCT) is a bimodal radiotherapeutic treatment based on the irradiation of neoplastic tissues with neutrons after the tissues have selectively accumulated molecules loaded with nuclides with large neutron capture cross-sections (such boron-10). Boron-10 carriers have been tested to a limited extent, and clinical trials have been conducted on sulfhydryl borane (¹⁰B-BSH) and boronophenylalanine (¹⁰B-BPA). However, precise and accurate measurements of boron-10 concentrations (0.1–100 μg/g) in specimens and samples of limited size (μg scale) are needed in order to be able to biologically characterise new compounds in predictive tissue dosimetry, toxicology and pharmacology studies as well as in clinical investigations. A new approach based on fast separation and detection of ¹⁰B-BPA performed by coupling capillary electrophoresis to electrospray mass spectrometry is reported. This method allows the quantitative analysis and characterisation of ¹⁰B-BPA in a short time with a high separation efficiency. Detection limits of 3 μM for ¹⁰B-BPA and 30 ng/mL for ¹⁰B were obtained with CE–ESI–MS. A quantification limit of 10 μM for ¹⁰B-BPA (100 ng/mL for ¹⁰B) was attained. The total boron-10 concentration was determined by high-resolution inductively coupled mass spectrometry in order to validate the method. Boron-10 isotope measurements were carried out by HR–ICP–MS at medium resolution (R=4000) due to the presence of an isobaric interference at mass 10. Good agreement was obtained between the values from CE–ESI–MS and those from HR–ICP–MS. The method has been successfully used to determine the ¹⁰B-BPA in two lines of cultured cells.     

Phase Equilibria in the Ni–Al–W System at 900°C

Summary. Multicomponent Ni-base alloys exhibit good mechanical properties even at elevated temperatures and they are widely used for industrial production of exertion-resistive parts of engines. These properties are mainly determined by the coexistence of a disordered γ matrix with a face centred cubic lattice and cuboidal domains of its ordered γ′ structure. Therefore it is useful to study phase equilibria in Ni-base systems, namely in the regions involving both mentioned phases. One of the conclusions of our recent work on Ni–Al–Cr–W system was a necessity of modification of selected thermodynamic parameters of the ternary Ni–Al–W subsystem in order to achieve a better agreement of our experimental observations with theoretical modelling. This involves new measurements of the microstructure of selected samples of the Ni–Al–W system at 900°C and the comparison of the results with existing literature data in order to confirm our conclusions on higher order system investigated before. It is a first step on the way to an assessment of the Ni–Al–W system, which has not been done before.     

Phase Equilibria in the Ni–Al–W System at 900°C

Multicomponent Ni-base alloys exhibit good mechanical properties even at elevated temperatures and they are widely used for industrial production of exertion-resistive parts of engines. These properties are mainly determined by the coexistence of a disordered γ matrix with a face centred cubic lattice and cuboidal domains of its ordered γ′ structure. Therefore it is useful to study phase equilibria in Ni-base systems, namely in the regions involving both mentioned phases. One of the conclusions of our recent work on Ni–Al–Cr–W system was a necessity of modification of selected thermodynamic parameters of the ternary Ni–Al–W subsystem in order to achieve a better agreement of our experimental observations with theoretical modelling. This involves new measurements of the microstructure of selected samples of the Ni–Al–W system at 900°C and the comparison of the results with existing literature data in order to confirm our conclusions on higher order system investigated before. It is a first step on the way to an assessment of the Ni–Al–W system, which has not been done before.     

Determination of nicotinamide (NA) using its polarographic catalytic wave in the presence of KIO3

Nicotinamide (NA) yields a polarographic catalytic wave with a peak potential –1.38 V (vs. SCE) in 0.1 mol/L HAc-NaAc (pH 4.7)/4 × 10^–3 mol/L KIO₃ buffer solution. The sensitivity of the catalytic wave increased in one order of magnitude as compared to that of the responding reduction wave without KIO₃. Based on this observation, a new method for the determination of NA was recommended. The second order derivative peak current was proportional to the NA concentration in the range of 5 × 10^–8 – 6 × 10^–7 mol/L. 0.11-fold vitamin B₁, 0.13-fold B₂, 0.14-fold B₆ and 8-fold nicotinic acid amounts do not interfere the determination of 1 × 10^–6 mol/L NA. The proposed method was used to determine the NA content in multivitamin tablets, with good agreement to the declared amount. Received: 25 October 1999 / Revised: 28 March 2000 / /Accepted: 31 March 2000     

Multivariate optimisation of the microwave-assisted extraction of oleuropein and related biophenols from olive leaves

Microwave assistance is proposed for the first time in order to accelerate the extraction of biophenols from olive leaves. Under optimal working conditions, obtained using a multivariate methodology, complete extraction of the target analytes was achieved in 8 min. The extracts required no clean-up nor concentration prior to injection into a chromatograph–photodiode array detector assembly for individual separation–quantification. The optimal extractant (an 80:20 ethanol–water mixture) was also used in the development of a stirring-based extraction method which required around 24 h for complete extraction of the target compounds. These mixtures can be used as replacements for toxic extractants, with a view to exploiting olive leaves in order to obtain biophenols for human use.     

The numerical solution of the radial Schrödinger equation via a trigonometrically fitted family of seventh algebraic order Predictor–Corrector methods

In this paper, we develop new seventh order trigonometrically fitted Adams–Bashforth–Moulton predictor–corrector (P–C) algorithms. Our predictor is based on the sixth algebraic order Adams–Bashforth scheme and our corrector on the seventh algebraic order Adams–Moulton scheme. In order to assess the efficiency of our new methods, we contacted appropriate comparisons of our schemes against well known methods and the numerical experimentations demonstrated that our schemes behave more efficiently.T.E. Simos: Active Member of the European Academy of Sciences and Arts; Fellow of the Royal Society of Chemistry (FRSC).