Mathematical Modelling and Analysis

An efficient spectral method for nonlinear Volterra integro-differential equations with weakly singular kernels

2024-05-14T18:25:42+03:00

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.

An accurate numerical scheme for three-dimensional variable-order time-fractional partial differential equations in two types of space domains

2024-05-14T18:25:42+03:00

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.

Mathematical modelling electrically driven free shear flows in a duct under uniform magnetic field

2024-05-21T18:25:49+03:00

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.

Spline quasi-interpolation numerical methods for integro-differential equations with weakly singular kernels

2024-05-21T18:25:49+03:00

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.

Existence results in weighted Sobolev space for quasilinear degenerate p(z)−elliptic problems with a Hardy potential

2024-05-21T18:25:50+03:00

The objective of this work is to establish the existence of entropy solutions to degenerate nonlinear elliptic problems for L¹-data f with a Hardy potential, in weighted Sobolev spaces with variable exponent, which are represented as follows where is a Leray-Lions operator from into its dual, is a non-linearity term that only meets the growth condition, and ρ > 0 is a constant.

A note on fractional-type models of population dynamics

2024-06-12T18:26:14+03:00

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.

Plane waves at an interface of thermoelastic and magneto-thermoelastic media

2024-06-12T18:26:15+03:00

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.

Spectral method for one dimensional Benjamin-Bona-Mahony-Burgers equation using the transformed generalized Jacobi polynomial

2024-06-12T18:26:14+03:00

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.

Effectivity of the vaccination strategy for a fractional-order discrete-time SIC epidemic model

2024-06-27T18:26:32+03:00

Indirect disease transmission is modeled via a fractional-order discretetime Susceptible-Infected-Contaminant (SIC) model vaccination as a control strategy. Two control actions are considered, giving rise to two different models: the vaccine efficacy model and the vaccination impact model. In the first model, the effectiveness of the vaccine is analyzed by introducing a new parameter, while in the second model, the impact of the vaccine is studied incorporating a new variable into the model. Both models are studied giving population thresholds to ensure the eradication of the disease. In addition, a sensitivity analysis of the Basic Reproduction Number has been carried out with respect to the effectiveness of the vaccine, the fractional order, the vaccinated population rate and the exposure rate. This analysis has been undertaken to study its effect on the dynamics of the models. Finally, the obtained results are illustrated and discussed with a simulation example related to the evolution of the disease in a pig farm.

Compensation problem in linear fractional order disturbed systems

2024-06-27T18:26:33+03:00

In this paper, we study fractional-order linear, finite-dimensional disturbed systems. The fundamental objective of this work is to study the remediability or compensation problem in linear fractional-order time-invariant perturbed systems. The remediability was introduced with the aim of finding an appropriate control that steers the output of the perturbed system towards normal observation at the final moment. We begin first by giving some characterizations of compensation, and then we prove that a rank condition is sufficient to assure the remediability of our system. The relationship between controllability and compensation is also given, and we provide some examples to illustrate our results.

Stability of the higher-order splitting methods for the nonlinear Schrödinger equation with an arbitrary dispersion operator

2024-06-27T18:26:31+03:00

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.

Solving a spectral problem for large-area photonic crystal surface-emitting lasers

2024-06-27T18:26:32+03:00

We present algorithms for constructing and resolving spectral problems for novel photonic crystal surface-emitting lasers with large emission areas, given by first-order PDEs with two spatial dimensions. These algorithms include methods to overcome computer-arithmetic-related challenges when dealing with huge and small numbers. We show that the finite difference schemes constructed using relatively coarse numerical meshes enable accurate estimation of several major optical modes, which are essential in practical applications.