The numerical solution of the generalized nonlinear Schrödinger equation by simple splitting methods can be disturbed by so-called spurious instabilities. We analyze these numerical instabilities for an arbitrary splitting method and apply our results to several well-known higher-order splittings. We find that the spurious instabilities can be suppressed to a large extent. However, they never disappear completely if one keeps the integration step above a certain limit and applies what is considered to be a more accurate higher-order method. The latter can be used to make calculations more accurate with the same numerically stable step, but not to make calculations faster with a much larger step.

The fractional exponential function is considered. General expansions in fractional powers are used to solve fractional population dynamics problems. Laguerretype exponentials are also considered, and an application to Laguerre-type fractional logistic equation is shown.

The Benjamin-Bona-Mahony-Burgers equation (BBMBE) plays a fundemental role in many application scenarios. In this paper, we study a spectral method for the BBMBE with homogeneous boundary conditions. We propose a spectral scheme using the transformed generalized Jacobi polynomial in combination of the explicit fourth-order Runge-Kutta method in time. The boundedness, the generalized stability and the convergence of the proposed scheme are proved. The extensive numerical examples show the efficiency of the new proposed scheme and coincide well with the theoretical analysis. The advantages of our new approach are as follows: (i) the use of the transformed generalized Jacobi polynomial simplifies the theoretical analysis and brings a sparse discrete system; (ii) the numerical solution is spectral accuracy in space.

This research examines the propagation of waves in a semi-infinite, isotropic magneto-thermoelastic solid, and a semi-infinite thermoelastic solid with welded contact. The study investigates the influence of a magnetic field on amplitude coefficients for the incidence of thermal, SV, and P waves in the magnetothermoelastic solid in a semi-infinite space. The incidence of these waves results in a total of six waves, including both refracted and reflected waves. The fluctuation of amplitude coefficients for various magnetic pressure values is explored for copper and aluminum as numerical constants. The study observes that the amplitude coefficients of seismic waves, occurring during the incidence of thermal, SV, and P waves in the magneto-thermoelastic solid semi-infinite space, are dependent on the incident angle, magnetic field, and material constants. Notably, the amplitude coefficients for the incidence of SV waves exhibit only a minor influence from the magnetic field. The implications of this research extend to applications in ocean acoustics, geophysics, acoustic devices, composite materials, and non-destructive testing.

The objective of this work is to establish the existence of entropy solutions to degenerate nonlinear elliptic problems for $L^1$-data $f$ with a Hardy potential, in weighted Sobolev spaces with variable exponent, which are represented as follows \begin{gather*} -\text{div}\big(\Phi(z,v,\nabla v)\big)+g(z,v,\nabla v)=f+\rho\frac{\vert v \vert^{p(z)-2}v}{|v|^{p(z)}}, \end{gather*} where $-\text{div}(\Phi(z,v,\nabla v))$ is a Leray-Lions operator from $W_{0}^{1,p(z)}(\Omega,\omega)$ into its dual, $g(z,v,\nabla v)$ is a non-linearity term that only meets the growth condition, and $\rho>0$ is a constant.

We consider a mathematical model of two-dimensional electrically driven laminar free shear flows in a straight duct under action of an applied uniform homogeneous magnetic field. The mathematical approach is based on studies by J.C.R. Hunt and W.E. Williams [10], Yu. Kolesnikov and H. Kalis [22,23]. We solve the system of stationary partial differential equations (PDEs) with two unknown functions of velocity U and induced magnetic field H. The flows are generated as a result of the interaction of injected electric current in liquid and the applied field using one or two couples of linear electrodes located on duct walls: three cases are considered. In dependence on direction of current injection and uniform magnetic field, the flows between the end walls are realized. Distributions of velocities and induced magnetic fields, electric current density in dependence on the Hartmann number Ha are studied. The solution of this problem is obtained analytically and numerically, using the Fourier series method and Matlab.

In this work, we introduce a numerical approach that utilizes spline quasi-interpolation operators over a bounded interval. This method is designed to provide a numerical solution for a class of Fredholm integro-differential equations with weakly singular kernels. We outline the computational components involved in determining the approximate solution and provide theoretical findings regarding the convergence rate. This convergence rate is analyzed in relation to both the degree of the quasi-interpolant and the grading exponent of the graded grid partition. Finally, we present numerical experiments that validate the theoretical findings.

We consider the discretization method for solving three-dimensional variable-order (3D-VO) time-fractional partial differential equations. The proposed method is developed based on discrete shifted Hahn polynomials (DSHPs) and their operational matrices. In the process of method implementation, the modified operational matrix (MOM) and complement vector (CV) of integration and pseudooperational matrix (POM) of VO fractional derivative plays an important role in the accuracy of the method. Further, we discuss the error of the approximate solution. At last, the methodology is validated by well test examples in two types of space domains. In order to evaluate the accuracy and applicability of the approach, the results are compared with other methods.

For Volterra integro-differential equations (VIDEs) with weakly singular kernels, their solutions are singular at the initial time. This property brings a great challenge to traditional numerical methods. Here, we investigate the numerical approximation for the solution of the nonlinear model with weakly singular kernels. Due to its characteristic, we split the interval and focus on the first one to save operation. We employ the corresponding singular functions as basis functions in the first interval to simulate its singular behavior, and take the Legendre polynomials as basis functions in the other one. Then the corresponding hp-version spectral method is proposed, the existence and uniqueness of solution to the numerical scheme are proved, the hp-version optimal convergence is derived. Numerical experiments verify the effectiveness of the proposed method.

In the paper, we consider simultaneous approximation of a pair of analytic functions by discrete shifts $\zeta_{u_N}(s+ikh_1; \ga)$ and $\zeta_{u_N}(s+ikh_2, \alpha; \gb)$ of the absolutely convergent Dirichlet series connected to the periodic zeta-function with multiplicative sequence $\ga$, and the periodic Hurwitz zeta-function, respectively. We suppose that $u_N\to\infty$ and $u_N\ll N^2$ as $N\to\infty$, and the set $\{(h_1\log p:\! p\in\! \PP), (h_2\log(m+\alpha): m\in \NN_0), 2\pi\}$ is linearly independent over $\QQ$.