On the efficiency of high-order difference schemes for the Schro¨dinger equation

Исследуется ряд двух- и трехслойных разностных схем, построенных на расширенных шаблонах, до восьмого порядка точности для уравнения Шрёдингера. Наряду с многоточечными схемами рассматривается метод коррекции Ричардсона в приложении к схеме четвертого порядка аппроксимации, повышающий порядок точности путем построения линейных комбинаций приближенных решений, полученных на различных вложенных сетках. Проведено сравнение методов по устойчивости, сложности реализации алгоритмов и объему вычислений, необходимых для достижения заданной точности. На основе теоретического анализа и численных экспериментов выявлены методы, наиболее эффективные для практического применения The efficiency of difference methods for solving problems of nonlinear wave optics is largely determined by the order of accuracy. Schemes up to the fourth order of accuracy have the traditional architecture of three-point stencils and standard conditions for the application of algorithms. However, a further increase in the order in the general case is associated with the need to expand the stencils using multipoint difference approximations of the derivatives. The use of such schemes forces formulating additional boundary conditions, which are not present in the differential problem, and leads to the need to invert the matrices of the strip structure, which are different from the traditional tridiagonal ones. An exception is the Richardson correction method, which is aimed at increasing the order of accuracy by constructing special linear combinations of approximate solutions obtained on various nested grids according to traditional structure schemes. This method does not require the formulation of additional boundary conditions and inversion of strip matrices. In this paper, we consider several explicit and implicit multipoint difference schemes up to the eighth order of accuracy for the Schr¨odinger equation. In addition, a simple and double Richardson correction method is also investigated in relation to the classical fourth-order scheme. A simple correction raises the order to sixth and a double correction to eighth. This large collection of schemes is theoretically compared in terms of their properties such as the order of approximation, stability, the complexity of the implementation of a numerical algorithm, and the amount of arithmetic operations required to achieve a given accuracy. The theoretical analysis is supplemented by numerical experiments on the selected test problem. The main conclusion drawn from the research results is that of all the considered schemes, the Richardson-corrected scheme is the most preferable in terms of the investigated properties

Generalized Ulam–Hyers–Rassias Stability Results of Solution for Nonlinear Fractional Differential Problem with Boundary Conditions

The problem of existence and generalized Ulam–Hyers–Rassias stability results for fractional differential equation with boundary conditions on unbounded interval is considered. Based on Schauder’s fixed point theorem, the existence and generalized Ulam–Hyers–Rassias stability results are proved, and then some examples are given to illustrate our main results.

Uniqueness of Solutions, Stability and Simulations for a Differential Problem Involving Convergent Series and Time Variable Singularities

We focus on a new type of nonlinear integro-differential equations with nonlocal integral conditions. The considered problem has one nonlinearity with time variable singularity. It involves also some convergent series combined to Riemann-Liouville integrals. We prove a uniqueness of solutions for the proposed problem, then, we provide some examples to illustrate this result. Also, we discuss the Ulam-Hyers stability for the problem. Some numerical simulations, using Rung Kutta method, are discussed too. At the end, a conclusion follows.

Nonlinear Differential Problem with p-Laplacian and via Phi-Hilfer Approach: Solvability and Stability Analysis

This paper we consider a study of a general class of nonlinear singular fractional DEs with p-Laplacian for the existence and uniqueness solution and the Hyers-Ulam (HU) stability. result via ϕ−Hilfer derivative is studied. Then, an existence of one solution is investigated. Some illustrative examples are discussed at the end.

Remarks on Parabolicity in a One-Dimensional Interdiffusion Model with the Vegard Rule

Until 1948 the interdiffusion theory was based on the Onsager phenomenology, namely thermodynamics of irreversible processes, and a drift was not included. Its main limitation is practical impossibility of the experimental as well as theoretical determination of mobilities (diffusivities) in multicomponent systems ($$r > 2$$ r > 2 ). After experimental discovery of the drift by Smigelskas and Kirkendall (Trans AIME 171:130–142, 1947), Darken (Trans AIME 175:184–201, 1948) formulated his famous model for the binary system. Consequently, the bi-velocity approach dominates interdiffusion studies (e.g. in more than 500 papers in 2020). In this paper, we consider the diffusional transport in a one-dimensional r-component solid solution. The model is expressed by the nonlinear system of strongly coupled evolution differential equations with initial and nonlinear coupled boundary conditions. We present a non-trivial proof of a theorem called the criterion of parabolicity, which implies the generalized parabolicity condition formulated without a proof in our previous works. This condition is a key in the proofs of our previous theorems on existence, uniqueness and properties of global weak solutions of the differential problem studied. The criterion of parabolicity works if diffusion coefficients are not too dispersed, and it is true in many physical systems. The numerical simulations consistent with real experiments for which our criterion works are given.

Existence of solution for Hilfer fractional differential problem with nonlocal boundary condition in Banach spaces

"This paper is devoted to study the existence of a solution to Hilfer fractional differential equation with nonlocal boundary condition in Banach spaces. We use the equivalent integral equation to study the considered Hilfer differential problem with nonlocal boundary condition. The Monch type fixed point theorem and the measure of the noncompactness technique are the main tools in this study. We demonstrate the existence of a solution with a suitable illustrative example."

On the Approximated Solution of a Special Type of Nonlinear Third-Order Matrix Ordinary Differential Problem

Matrix differential equations are at the heart of many science and engineering problems. In this paper, a procedure based on higher-order matrix splines is proposed to provide the approximated numerical solution of special nonlinear third-order matrix differential equations, having the form Y(3)(x)=f(x,Y(x)). Some numerical test problems are also included, whose solutions are computed by our method.

Numerical Solutions of Fractional Differential Equations by Using Laplace Transformation Method and Quadrature Rule

This paper introduces an efficient numerical scheme for solving a significant class of fractional differential equations. The major contributions made in this paper apply a direct approach based on a combination of time discretization and the Laplace transform method to transcribe the fractional differential problem under study into a dynamic linear equations system. The resulting problem is then solved by employing the numerical method of the quadrature rule, which is also a well-developed numerical method. The present numerical scheme, which is based on the numerical inversion of Laplace transform and equal-width quadrature rule is robust and efficient. Some numerical experiments are carried out to evaluate the performance and effectiveness of the suggested framework.

A Theoretical Model for Debonding Prediction in the RC Beams Externally Strengthened with Steel Strip and Inorganic Matrix

This paper shows a theoretical model for predicting the moment–curvature/load–deflection relationships and debonding failure of reinforced concrete (RC) beams externally strengthened with steel reinforced geopolymeric matrix (SRGM) or steel reinforced grout (SRG) systems. Force equilibrium and strain compatibility equations for a beam section divided into several segments are numerically solved using non-linear behaviour of concrete and internal steel bars. The deflection is then obtained from the flexural stiffness at a mid-span section. Considering the appropriate SRGM-concrete bond–slip law, calibrated on single-lap shear bond tests, both end and intermediate debonding failures are analysed. To predict the end debonding, an anchorage strength model is adopted. To predict intermediate debonding, at each pair of flexural cracks a shear stress limitation is placed at concrete–matrix interface and the differential problem is solved at steel strip–matrix interface. Based on the theoretical predictions, the comparisons with experimental data show that the proposed model can accurately predict the structural response of SRGM/SRG strengthened RC beams. It can be a useful tool for evaluating the behaviour of externally strengthened RC beams, avoiding experimental tests.