Ibraheem Alolyan

A family of high-order multistep methods with vanished phase-lag and its derivatives for the numerical solution of the Schrödinger equation

Many simulation algorithms (chemical reaction systems, differential systems arising from the modelling of transient behaviour in the process industries etc.) contain the numerical solution of systems of differential equations. For the efficient solution of the above mentioned problems, linear multistep methods or Runge-Kutta single-step methods are used. For the simulation of chemical procedures the radial Schrodinger equation is used frequently. In the present paper we will study a class of linear multistep methods. More specifically, the purpose of this paper is to develop an efficient algorithm for the approximate solution of the radial Schrodinger equation and related problems. This algorithm belongs in the category of the multistep methods. In order to produce an efficient multistep method the phase-lag property and its derivatives are used. Hence the main result of this paper is the development of an efficient multistep method for the numerical solution of systems of ordinary differential equations with oscillating or periodical solutions. The reason of their efficiency, as the analysis proved, is that the phase-lag and its derivatives are eliminated. Another reason of the efficiency of the new obtained methods is that they have high algebraic order

A new family of symmetric linear four-step methods for the efficient integration of the Schrödinger equation and related oscillatory problems

Abstract In this article we develop a family of three explicit symmetric linear four-step methods. The new methods, with nullified phase-lag, are optimized for the efficient solution of the Schrodinger equation and related oscillatory problems. We perform an analysis of the local truncation error of the methods for the general case and for the special case of the Schrodinger equation, where we show the decrease of the maximum power of the energy in relation to the corresponding classical methods. We also perform a periodicity analysis, where we find that there is a direct relationship between the periodicity intervals of the methods and their local truncation errors. In addition we determine their periodicity regions. We finally compare the new methods to the corresponding classical ones and other known methods from the literature, where we show the high efficiency of the new methods.

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.

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

In this paper, we study the effects of the vanishing of the phase-lag and its first, second and third derivatives on the effectiveness of a four-step hybrid type method of sixth algebraic order. As a result of the above described study, a Hybrid type of three level four-step method of sixth algebraic order is obtained. We investigate the new produced method theoretically and computationally. The theoretical investigation of the new hybrid method consists of:The development of the new method.The computation of the Local Truncation Error.The Comparison of the Local Truncation Error analysis with other known methods of the same form.The Stability Analysis. The computational investigation consists of the application of the new obtained hybrid method to the numerical solution of the resonance problem of the radial time independent Schrödinger equation.

Efficient low computational cost hybrid explicit four-step method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical integration of the Schrödinger equation

Based on an optimized explicit four-step method, a new hybrid high algebraic order four-step method is introduced in this paper. For this new hybrid method, we investigate the procedure of vanishing of the phase-lag and its first, second, third and fourth derivatives. More specifically, we investigate: (1) the construction of the new method, i.e. the computation of the coefficients of the method in order its phase-lag and first, second, third and fourth derivatives of the phase-lag to be eliminated, (2) the definition of the local truncation error, (3) the analysis of the local truncation error, (4) the stability (interval of periodicity) analysis (using scalar test equation with frequency different than the frequency of the scalar test equation for the phase-lag analysis). Finally, we investigate computationally the new obtained method by applying it to the numerical solution of the resonance problem of the radial Schrödinger equation. The efficiency of the new developed method is tested comparing this method with well known methods of the literature but also using very recently developed methods.

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

A family of explicit linear sixth algebraic order six-step methods with vanished phase-lag and its first derivative is obtained in this paper. The investigation of the above family of methods contains:theoretical study of the new family of methods andcomputational study of the new family of methods. The theoretical study of the above mentioned family of methods contains:1.the development of the method,2.the computation of the local truncation error,3.the comparative local truncation error analysis. The comparison is taken place between the new family of methods with the corresponding method with constant coefficients and4.the stability analysis of the new family of methods. The stability analysis is taken place using test equation with different frequency than the frequency of the test equation used for the phase-lag analysis of the methods. The application of the new family of linear six-step sixth algebraic order methods to the resonance problem of the one-dimensional time independent Schrödinger equation is used for the computational study of the new family of methods. The result of the above mentioned theoretical and computational investigation is that the new proposed family of linear explicit schemes are computationally and theoretically more effective than other well known methods for the approximate solution of the radial Schrödinger equation and related initial or boundary value problems with periodic and/or oscillating solutions.

A high algebraic order predictor–corrector explicit method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical solution of the Schrödinger equation and related problems

In this paper an eighth algebraic order predictor–corrector explicit four-step method is studied. The main scope of this paper is to study the consequences of (1) the vanishing of the phase-lag and its first, second, third and fourth derivatives and (2) the high algebraic order on the efficiency of the new developed method. A theoretical and computational study of the obtained method is also presented. More specifically, the theoretical study of the new predictor–corrector method consists of:The development of the new predictor–corrector method, i.e. the definition of the coefficients of the method in order its phase-lag and phase-lag’s first, second, third and fourth derivatives to be vanishedThe computation of the local truncation errorThe comparative local truncation error analysisThe stability (interval of periodicity) analysis, using scalar test equation with frequency different than the frequency of the scalar test equation for the phase-lag analysis. Finally, the computational study of the new predictor–corrector method consists of the application of the new produced predictor–corrector explicit four-step method to the numerical solution of the resonance problem of the radial time independent Schrödinger equation.

A predictor–corrector explicit four-step method with vanished phase-lag and its first, second and third derivatives for the numerical integration of the Schrödinger equation

A predictor–corrector explicit four-step method of sixth algebraic order is investigated in this paper. More specifically, we investigate the results of the elimination of the phase-lag and its first, second and third derivatives on the efficiency of the proposed method. The resultant method is studied theoretically and computationally. The theoretical investigation of the new hybrid method consists of: (1) the construction of the new method, (2) the definition (calculation) of the local truncation error, (3) the comparative local truncation error analysis (with other known methods of the same form), (4) the stability analysis using scalar test equation with frequency different than the frequency of the phase-lag analysis. Finally, we will study computationally the new obtained method. This study is based on the application of the new produced predictor–corrector explicit four-step method to the approximate solution of the resonance problem of the radial time independent Schrödinger equation.

A high algebraic order multistage explicit four-step method with vanished phase-lag and its first, second, third, fourth and fifth derivatives for the numerical solution of the Schrödinger equation

A new Multistage high algebraic order four-step method is obtained in this paper. It is the first time in the literature that a method of this category is developed and has vanishing of the phase-lag and its first, second, third, fourth and fifth derivatives. We study this new method by investigating: (1) the development of the new method, i.e. the calculation of the coefficients of the method in order the phase-lag and its first, second, third, fourth and fifth derivatives of the phase-lag to be vanished, (2) the determination of the formula of the Local Truncation Error, (3) the comparative analysis of the Local Truncation Error (with this we mean the application of the new method and similar methods on a test problem and the analysis of their behavior), (4) the stability of the new method, by applying the new obtained method to a scalar test equation with frequency different than the frequency of the scalar test equation for the phase-lag analysis and by studying the results of this application i.e. by investigating the interval of periodicity of the new obtained method. We finally study the computational behavior the new developed method by using the application of the new method to the approximate solution of the resonance problem of the radial Schrödinger equation. We prove the effectiveness of the new obtained method by comparing it with (1) well known methods of the literature and (2) very recently obtained methods.

A new four-step hybrid type method with vanished phase-lag and its first derivatives for each level for the approximate integration of the Schrödinger equation

In this paper we present a new methodology for the development of four-step hybrid type methods of sixth algebraic order with vanished phase-lag and its derivatives. The methodology is based on the vanishing of the phase-lag and its derivatives on its level of the hybrid method. We present a comparative error and stability analysis for the produced new method. The efficiency of the new obtained methods is examined by application to the resonance problem of the Schrödinger equation.