Computational and Mathematical Methods最新文献

In the present article, the periodic solutions of the N-body with quasi-homogeneous potential in the Sitnikov sense by applying the multiple methods of scale (MMS) and Lindstedt–Poincaré (LP) technique are obtained. However, these methods are used to find the approximate periodic solutions in the closed form by eliminating the secular terms. In addition of the Newtonian potential and forces, we consider that the big bodies create quasi-homogeneous potentials. We add the inverse cubic corrective term to the inverse square Newtonian law of gravitation, in order to approximate the various phenomena due to the shape of the bodies or the radiation emitting from them. We study the Sitnikov motion in the N-bodies under this consideration. We, further, analyzed the obtain approximate periodic solutions of the Sitnikov motion, for $��\nu �=�2�,�7�$ by using the MMS and LP-method, in closed form. The numerical comparisons are presented in the first and second approximated solutions obtained by using MMS and numerical solutions obtained by LP-method are illustrated graphically. The effect of initial conditions on the solutions of the Sitnikov motion is illustrated graphically obtained by both the techniques. It is observed that the choice of initial conditions plays a crucial role in the numerical and approximate solutions.

This study aims at constructing a robust numerical scheme for solving singularly perturbed parabolic delay differential equations arising in the modeling of neuronal variability. Taylor's series expansion is applied to approximate the shift terms. The obtained result is approximated by using the implicit Euler method in the temporal discretization on a uniform step size with the hybrid numerical scheme consisting of the midpoint upwind method in the outer layer region and the cubic spline in tension method in the inner layer region on a piecewise uniform Shishkin mesh in the spatial discretization. The constructed scheme is shown to be an $��\epsilon �$-uniformly convergent accuracy of order $��O�\left(�\Lambda �t�+�{�N�}^{�-�2�}�{�\ln �}^{�3�}�N�\right)�$. Two model examples are given to testify the theoretical findings.

In this work, we investigate an effective linearized spectral collocation method for two-dimensional (2D) Riesz space fractional nonlinear reaction–diffusion equations with homogeneous boundary conditions. The proposed method is based on the Jacobi–Gauss–Lobatto spectral collocation method for spatial discretization and the finite difference method for temporal discretization. The full implementation of the method is demonstrated in detail. The stability of the numerical scheme is rigorously discussed and the errors with benchmark solutions that show second-order convergence in time and spectral convergence in space are numerically analyzed. Finally, numerical simulations for 2D Riesz space fractional Allen–Cahn and FitzHugh–Nagumo models are carried out to illustrate the effectiveness of the developed method and its ability for long-time simulations.

We propose two positive parameters based on the choice of Birgin and Martínez search direction. Using the two classical choices of the Barzilai-Borwein parameters, two positive parameters were derived by minimizing the distance between the relative matrix corresponding to the propose search direction and the scaled memory-less Broyden–Fletcher–Goldfarb-Shanno (BFGS) matrix in the Frobenius norm. Moreover, the resulting direction is descent independent of any line search condition. We established the global convergence of the proposed algorithm under some appropriate assumptions. In addition, numerical experiments on some benchmark test problems are reported in order to show the efficiency of the proposed algorithm.

In this article we give some integral inequalities for tgs-convex functions which provide refinements of well-known integral inequalities for unified integral operators. Consequently, various results for convex functions are compared. Furthermore, applications are studied by considering the particular functions to get results for fractional integral operators.

The article introduces the scale-deviating operator. The scale-deviating differential operator comprises the parameters to designate the operator order and the parameters to define the dimension of space. The operator order depends on the characteristic length κ. There are two types of linear scale-deviating operators. For the distances r, which are much less than κ, the scale-deviating operator of the first type $��A�$ reduces to the common operators. For the distances, which exceed the length κ, this operator reduces to the fractional Riesz operator. The second type of the scale-deviating operator $��B�$ behaves oppositely. For the distances r, which are much higher than κ, the scale-deviating operator of the second type reduces to the common operators. Finally, for the distances, which below the length κ, this operator lessens to the fractional Riesz operator. These linear, isotropic operators possess the order, less than two. The solutions of new scale-deviating equations and the shell theorem for these operators are provided closed form.

In Statistics of Extremes, the estimation of the extreme value index is an essential requirement for further tail inference. In this work, we deal with the estimation of a strictly positive extreme value index from a model with a Pareto-type right tail. Under this framework, we propose a new class of weighted Hill estimators, parameterized with a tuning parameter a. We derive their non-degenerate asymptotic behavior and analyze the influence of the tuning parameter in such result. Their finite sample performance is analyzed through a Monte Carlo simulation study. A comparison with other important extreme value index estimators from the literature is also provided.

This article proposes a two-step hybrid block method (TSHBM) with a fourth derivative for solving third-order boundary value problems in ordinary differential equations. The mathematical formulation of the proposed approach depends on interpolation and collocation techniques. The order of convergence of the TSHBM is showed to be seventh-order convergent and zero-stable. A few numerical examples are given to evaluate its performance. Numerical outcomes show that the TSHBM scheme is more efficient than some existing numerical techniques.

In this work, we consider control problems represented by a linear differential equation assuming that all the coefficients are random variables and with an additive control that is a stochastic process. Specifically, we will work with controllable problems in which the initial condition and the final target are random variables. The probability density function of the solution and the control has been calculated. The theoretical results have been applied to study, from a probabilistic standpoint, a damped oscillator.

Many problems of modern laser physics are governed by equations or sets of equations in an unbounded domain. For solving these problems using the computer simulation, it is necessary to introduce the bounded domain, which size should be extended significantly to avoid the spurious wave reflection from the domain boundaries. Alternatively, the artificial (non-reflective or transparent) boundary conditions should be stated. This approach is also effective for enhancing computation performance at the numerical solution of the nonlinear partial differential equations (PDEs). In the current paper, we investigate the laser pulse propagation in a semiconductor, governed by the Schrödinger equation, under the appearance of spatio-temporal contrast structures of semiconductor characteristics. Their evolution is described by a set of PDEs. The optical pulse is partly reflected from the boundaries of these structures. Consequently, even a little reflection of the optical pulse from the artificial boundaries can essentially distort the numerical solution. Thus, these artificial boundary conditions must possess a high quality to minimize their reflection coefficients. With this aim, we propose the method for constructing adaptive artificial boundary conditions and discuss their advantages.