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The purpose of this paper is to introduce stability analysis for the initial value problem for the fractional Schrödinger differential equation: Equation 1 in a Hilbert space H with a self‐adjoint positive definite operator A under the condition |α(s)|<M₁/s^1/2 and the first order of accuracy difference scheme for the approximate solution of this initial value problem.

The paper considers the stability estimates for the solution of this problem and the stability estimate for the approximate solution of first order of accuracy difference scheme of this problem.

The paper finds the stability for the fractional Schrödinger differential equation and the first order of accuracy difference scheme of that equation by applications to one‐dimensional fractional Schrödinger differential equation with nonlocal boundary conditions and multidimensional fractional Schrödinger differential equation with Dirichlet conditions.

The paper is a significant work on stability of fractional Schrödinger differential equation and its difference scheme presenting some numerical experiments which resulted from applying obtained theorems on several mixed fractional Schrödinger differential equations.

The purpose of this paper is to derive a dynamic equation for modelling the behaviour of smectic‐C liquid crystals under the effect of an electric field.

The model equation is solved using a finite difference approximation, method of lines and pseudo‐spectral methods. The solutions are compared for accuracy and efficiency. Comparison is made of the efficiency of finite differences, method of lines and pseudo‐spectral methods.

The Fourier pseudo‐spectral method is shown to be the most efficient approach.

This work is original; a computational comparison of numerical schemes applied to liquid crystals has not been found in the literature.

The purpose of this paper is to report the effect of radiation on flow of a magneto‐micropolar fluid past a continuously moving plate with suction and blowing.

The governing equations are transformed into dimensionless nonlinear ordinary differential equations by similarity transformation. Analytical technique, namely the homotopy perturbation method (HPM) combining with Padé approximants and finite difference method, are used to solve dimensionless non‐linear ordinary differential equations. The skin friction coefficient and local Nusselt numbers are also calculated. Beside this, the comparison of the analytical solution with numerical solution is illustrated by the graphs for different values of dimensionless pertinent parameters.

The authors have studied laminar magneto‐micropolar flow in the presence of radiation by using HPM‐Padé and finite difference methods. Results obtained by HPM‐Padé are in excellent agreement with the results of numerical solution.

The HPM‐Padé is used in a direct way without using linearization, discritization or restrictive assumption. The authors have attempted to show the capabilities and wide‐range applications of the HPM‐Padé in comparison with the finite difference solution of magneto‐micropolar flow in the presence of radiation problem.

The purpose of this paper is to build a linear time-varying discrete Verhulst model (LTDVM), to realise the convert from continuous forms to discrete forms, and to eliminate traditional grey Verhulst model's error caused by difference equations directly jumping to differential equations.

The methodology of the paper is by the light of discrete thoughts and countdown to the original data sequence.

The research of this model manifests that LTDVM is unbiased on the “s” sequential simulation.

The example analysis shows that LTDVM embodies simulation and prediction with high precision.

This paper is to realise the convert from continuous forms to discrete forms, and to eliminate traditional grey Verhulst model's error caused by difference equations directly jumping to differential equations. Meanwhile, the research of this model manifests that LTDVM is unbiased on the “s” sequential simulation.

The purpose of this study is to present a detailed quantitative and qualitative fuzzy approach to the Allee effect that permits dealing with uncertainty.

The Allee effect is related to those aspects of population dynamics that are connected with a decrease in individual fitness when the population size diminishes to very low levels. It allows to model the evolution of certain sectors or clusters which, due to their low population density, may have problems of survival. In uncertain environments, an estimate of the effect’s parameters can be performed in the form of fuzzy numbers, which means that this study is using the methodology of fuzzy arithmetic.

This study reveals that fuzziness changes the behavior of the set of solutions when the strong Allee effect is studied under uncertainty from the point of view of standard difference or generalization of the Hukuhara difference.

The value and originality of the work consists in offering a set of tools for studying the evolution of a group of firms subject to an Allee effect in uncertain environments.

The purpose of this study is the application of the following concepts to the time discrete form. Variational Calculus, potential and kinetic energies, velocity proportional Rayleigh dissipation function, the Lagrange and Hamilton formalisms, extended Hamiltonians and Poisson brackets are all defined and applied for time-continuous physical processes. Such processes are not always time-continuously observable; they are also sometimes time-discrete.

The classical approach is developed with the benefit of giving only a short table on charge and flux formulation, as they are similar to the classical case just like all other formulation types. Moreover, an electromechanical example is represented as well.

Lagrange and Hamilton formalisms together with the velocity proportional (Rayleigh) dissipation function can also be used in the discrete time case, and as a result, dissipative equations of generalized motion and dissipative canonical equations in the discrete time case are obtained. The discrete formalisms are optimal approaches especially to analyze a coupled physical system which cannot be observed continuously. In addition, the method makes it unnecessary to convert the quantities to the other. The numerical solutions of equations of dissipative generalized motion of an electromechanical (coupled) system in continuous and discrete time cases are presented.

The formalisms and the velocity proportional (Rayleigh) dissipation function aforementioned are used and applied to a coupled physical system in time-discrete case for the first time to the best of the author’s knowledge, and systems of difference equations are obtained depending on formulation type.

The Debye theory of dielectric relaxation as corrected for inertial effects has as yet been only considered in the linear approximation. There, the rise and decay transients are identical. Here a method recently developed for the treatment of a rotator in a periodic potential is applied to calculate the transient behaviour when the linear approximation is discarded. The Kramers equation for the problem is expanded in a set of orthogonal functions which lead to a set of linear differential difference equations giving the relaxation behaviour. It is shown that the Mori formalism for the problem leads to the same set of differential difference equations as the Kramers equation.

The purpose of this study is to identify a novel methodology for direct calculation of steady-state periodic solutions for electrical circuits described by nonlinear differential equations, in the time domain.

An iterative algorithm was created to determine periodic steady-state solutions for circuits with nonlinear elements in a chosen set of time instants.

This study found a novel differential operator for periodic functions and its application in the steady-state analysis.

This approach can be extended to the determination of two- or multi-periodic solutions of nonlinear dynamic systems.

The complexity of the steady-state analysis can be reduced in comparison with the frequency-domain approach.

This study identified novel difference equations for direct steady-state analysis of nonlinear electrical circuits.

The purpose of this paper is to develop an efficient numerical scheme for non-linear two-dimensional (2D) parabolic partial differential equations using modified bi-cubic B-spline functions. As a test case, method has been applied successfully to 2D Burgers equations.

The scheme is based on collocation of modified bi-cubic B-Spline functions. The authors used these functions for space variable and for its derivatives. Collocation form of the partial differential equation results into system of first-order ordinary differential equations (ODEs). The obtained system of ODEs has been solved by strong stability preserving Runge-Kutta method. The computational complexity of the method is O(p log(p)), where p denotes total number of mesh points.

Obtained numerical solutions are better than those available in literature. Ease of implementation and very small size of computational work are two major advantages of the present method. Moreover, this method provides approximate solutions not only at the grid points but also at any point in the solution domain.

First time, modified bi-cubic B-spline functions have been applied to non-linear 2D parabolic partial differential equations. Efficiency of the proposed method has been confirmed with numerical experiments. The authors conclude that the method provides convergent approximations and handles the equations very well in different cases.