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Abstract: Abstract We investigate a generalized class of fractional hemivariational inequalities involving the time-fractional aspect. The existence result is established by employing the Rothe method in conjunction with the surjectivity of multivalued pseudomonotone operators and the properties of the Clarke generalized gradient. We are also exploring a numerical approach to address the problem, utilizing both spatially semi-discrete and fully discrete finite elements, along with a discrete approximation of the fractional derivative. All these results are applied to the analysis and numerical approximations of a frictional contact model that describes the quasi-static contact between a viscoelastic body and a solid foundation. The constitutive relation is modeled using the fractional Kelvin-Voigt law. The contact and friction are described by the subdifferential boundary conditions. The variational formulation of this problem leads to a fractional hemivariational inequality. The error estimates for this problem are derived. Finally, here’s a second concrete example to illustrate the application. We propose a frictional contact model that incorporates normal compliance and Coulomb friction to describe the quasi-static contact between a viscoelastic body and a solid foundation. PubDate: 2024-07-20

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Abstract: Abstract We are concerned with a simplified quantum energy-transport model for bipolar semiconductors, which consists of nonlinear parabolic fourth-order equations for the electron and hole density; degenerate elliptic heat equations for the electron and hole temperature; and Poisson equation for the electric potential. For the periodic boundary value problem in the torus \(\mathbb{T}^{d}\) , the global existence of weak solutions is proved, based on a time-discretization, an entropy-type estimate, and a fixed-point argument. Furthermore, the semiclassical limit is obtained by using a priori estimates independent of the scaled Planck constant. PubDate: 2024-07-15

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Abstract: Abstract We present a new diagonal quasi-Newton method for solving unconstrained optimization problems based on the weak secant equation. To control the diagonal elements, the new method uses new criteria to generate the Hessian approximation. We establish the global convergence of the proposed method with the Armijo line search. Numerical results on a collection of standard test problems demonstrate the superiority of the proposed method over several existing diagonal methods. PubDate: 2024-07-15

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Abstract: Abstract We present a unified approach to studying the superconvergence property of the spectral volume (SV) method for high-order time-dependent partial differential equations using the local discontinuous Galerkin formulation. We choose the diffusion and third-order wave equations as our models to illustrate approach and the main idea. The SV scheme is designed with control volumes constructed using the Gauss points or Radau points in subintervals of the underlying meshes, which leads to two SV schemes referred to as GSV and RSV schemes, respectively. With a careful choice of numerical fluxes, we demonstrate that the schemes are stable and exhibit optimal error estimates. Furthermore, we establish superconvergence of the GSV and RSV for the solution itself and the auxiliary variables. To be more precise, we prove that the errors of numerical fluxes at nodes and for the cell averages are superconvergent with orders of \(\cal{O}(h^{2k+1})\) and \(\cal{O}(h^{2k})\) for RSV and GSV, respectively. Superconvergence for the function value and derivative value approximations is also studied and the superconvergence points are identified at Gauss points and Radau points. Numerical experiments are presented to illustrate theoretical findings. PubDate: 2024-06-27

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Abstract: Abstract The Fletcher-Reeves (FR) method is widely recognized for its drawbacks, such as generating unfavorable directions and taking small steps, which can lead to subsequent poor directions and steps. To address this issue, we propose a modification to the FR method, and then we develop it into the three-term conjugate gradient method in this paper. The suggested methods, named “HZF” and “THZF”, preserve the descent property of the FR method while mitigating the drawbacks. The algorithms incorporate strong Wolfe line search conditions to ensure effective convergence. Through numerical comparisons with other conjugate gradient algorithms, our modified approach demonstrates superior performance. The results highlight the improved efficacy of the HZF algorithm compared to the FR and three-term FR conjugate gradient methods. The new algorithm was applied to the problem of image restoration and proved to be highly effective in image restoration compared to other algorithms. PubDate: 2024-06-07 DOI: 10.21136/AM.2024.0009-24

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Abstract: Abstract We multiply both sides of the complex symmetric linear system Ax = b by 1 − iω to obtain a new equivalent linear system, then a dual-parameter double-step splitting (DDSS) method is established for solving the new linear system. In addition, we present an upper bound for the spectral radius of iteration matrix of the DDSS method and obtain its quasi-optimal parameter. Theoretical analyses demonstrate that the new method is convergent when some conditions are satisfied. Some tested examples are given to illustrate the effectiveness of the proposed method. PubDate: 2024-06-01 DOI: 10.21136/AM.2024.0133-23

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Abstract: Abstract We consider the inverse nodal problem for Sturm-Liouville (S-L) equation with frozen argument. Asymptotic behaviours of eigenfunctions, nodal parameters are represented in two cases and numerical algorithms are produced to solve the given problems. Subsequently, solution of inverse nodal problem is calculated by the second Chebyshev wavelet method (SCW), accuracy and effectiveness of the method are shown in some numerical examples. PubDate: 2024-06-01 DOI: 10.21136/AM.2024.0038-21

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Abstract: Abstract We study an n-dimensional system of ordinary differential equations with a constant matrix, a relay-type nonlinearity, and an external disturbance in the right-hand side. We consider a nonideal relay characteristic. The external disturbance is described by the product of an exponential function and a sine function with an initial phase as a parameter. We assume the matrix of the linear part and the vector at the relay characteristic such that, by a nonsingular transformation, the system is reduced to the form with the diagonal matrix and the vector being opposite to the unit vector. We establish a necessary and sufficient condition for the existence of two-point oscillatory solutions, i.e., the solutions with two fixed points on the hyperplanes of the relay switching in phase space. Also, we give the sufficient conditions under which such solutions do not exist. We provide a supporting example, which demonstrates how to apply the obtained results. PubDate: 2024-06-01 DOI: 10.21136/AM.2024.0152-22

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Abstract: Abstract Initial value problem for three dimensional (3D) elastodynamic system in two dimensional (2D) inhomogeneous quasicrystals is considered. An analytical method is studied for the solution of this problem. The system is written in terms of Fourier images of displacements with respect to lateral variables. The resulting problem is reduced to integral equations of the Volterra type. Finally, using Paley Wiener theorem it is shown that the solution of the initial value problem can be found by the inverse Fourier transform. A numerical example is considered for the comparison of the exact solution with the computed solution obtained by using the method. PubDate: 2024-06-01 DOI: 10.21136/AM.2024.0045-23

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Abstract: Abstract The Sturm-Liouville eigenvalue problem is symmetric if the coefficients are even functions and the boundary conditions are symmetric. The eigenfunction is expressed in terms of orthonormal bases, which are constructed in a linear space of trial functions by using the Gram-Schmidt orthonormalization technique. Then an n-dimensional matrix eigenvalue problem is derived with a special matrix A:= [aij], that is, aij = 0 if i + j is odd. Based on the product formula, an integration method with a fictitious time, namely the fictitious time integration method (FTIM), is developed to obtain the higher-index eigenvalues. Also, we recover the symmetric potential function q(x) in the Sturm-Liouville operator by specifying a few lower-index eigenvalues, based on the product formula and the Newton iterative method. PubDate: 2024-06-01 DOI: 10.21136/AM.2024.0005-21

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Abstract: Abstract To have accuracy in the extracted information is the goal of the reliability theory investigation. In information theory, varentropy has recently been proposed to describe and measure the degree of information dispersion around entropy. Theoretical investigation on varentropy of past life has been initiated, however details on its stochastic properties are yet to be discovered. In this paper, we propose a novel stochastic order and introduce new classes of life distributions based on past varentropy. Further, we illustrate some of its applications in reliability modeling and in the diversity measure of Boltzmann distribution. PubDate: 2024-06-01 DOI: 10.21136/AM.2024.0163-23

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Abstract: Abstract We investigate the Cauchy problem of the one dimensional Maxwell-Schrödinger (MS) system under the Lorenz gauge condition. Different from the classical case, we consider the electromagnetic and electrostatic potentials which are growing at space infinity. More precisely, the electrostatic potential is allowed to grow linearly, while for the electromagnetic potential the growth is sublinear. Based on the energy estimates and the gauge transformation, we prove the global existence and the uniqueness of the weak solutions to this system. PubDate: 2024-05-31 DOI: 10.21136/AM.2024.0180-23

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Abstract: Abstract The aim of this article is to investigate the well-posedness, stability of solutions to the time-dependent Maxwell’s equations for electric field in conductive media in continuous and discrete settings, and study convergence analysis of the employed numerical scheme. The situation we consider would represent a physical problem where a subdomain is emerged in a homogeneous medium, characterized by constant dielectric permittivity and conductivity functions. It is well known that in these homogeneous regions the solution to the Maxwell’s equations also solves the wave equation, which makes computations very efficient. In this way our problem can be considered as a coupling problem, for which we derive stability and convergence analysis. A number of numerical examples validate theoretical convergence rates of the proposed stabilized explicit finite element scheme. PubDate: 2024-05-27 DOI: 10.21136/AM.2024.0248-23

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Abstract: The product q · q in Section 4 and Appendix should be replaced by qq. PubDate: 2024-04-01 DOI: 10.21136/AM.2024.0056-24

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Abstract: Abstract The evolutions of small and large compressive pulses are studied in a two-phase flow of gas and dust particles with a variable azimuthal velocity. The method of relatively undistorted waves is used to study the mechanical pulses of different types in a rotational, axisymmetric dusty gas. The results obtained are compared with that of nonrotating medium. Asymptotic expansion procedure is used to discuss the nonlinear theory of geometrical acoustics. The influence of the solid particles and the rotational effect of the medium on the distortion are investigated. In a rotational flow it is observed that with the increase in the value of rotational parameter, the steepening of the pulses also increases. The presence of dust in the rotational flow delays the onset of shock formation thereby increasing the distance where the shock is formed first. The rotational and the dust parameters are observed to have the same effect on the shock strength. PubDate: 2024-04-01 DOI: 10.21136/AM.2024.0107-23

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Abstract: Abstract In this work, we consider an inverse backward problem for a nonlinear parabolic equation of the Burgers’ type with a memory term from final data. To this aim, we first establish the well-posedness of the direct problem. On the basis of the optimal control framework, the existence and necessary condition of the minimizer for the cost functional are established. The global uniqueness and stability of the minimizer are deduced from the necessary condition. Numerical experiments demonstrate the effectiveness of this approach. PubDate: 2024-04-01 DOI: 10.21136/AM.2024.0049-23

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Abstract: Abstract We analyse a finite-element discretisation of a differential equation describing an axisymmetrically loaded thin shell. The problem is singularly perturbed when the thickness of the shell becomes small. We prove robust convergence of the method in a balanced norm that captures the layers present in the solution. Numerical results confirm our findings. PubDate: 2024-04-01 DOI: 10.21136/AM.2024.0134-23

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Abstract: Abstract The existence of synchronization is an important issue in complex dynamical networks. In this paper, we study the synchronization of impulsive coupled oscillator networks with the aid of rotating periodic solutions of impulsive system. The type of synchronization is closely related to the rotating matrix, which gives an insight for finding various types of synchronization in a united way. We transform the synchronization of impulsive coupled oscillators into the existence of rotating periodic solutions in a relevant impulsive system. Some existence theorems about rotating periodic solutions for a non-homogeneous linear impulsive system and a nonlinear perturbation system are established by topology degree theory. Finally, we give two examples to show synchronization behaviors in impulsive coupled oscillator networks. PubDate: 2024-04-01 DOI: 10.21136/AM.2024.0183-23

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Abstract: Abstract Post-training rounding, also known as quantization, of estimated parameters stands as a widely adopted technique for mitigating energy consumption and latency in machine learning models. This theoretical endeavor delves into the examination of the impact of rounding estimated parameters in key regression methods within the realms of statistics and machine learning. The proposed approach allows for the perturbation of parameters through an additive error with values within a specified interval. This method is elucidated through its application to linear regression and is subsequently extended to encompass radial basis function networks, multilayer perceptrons, regularization networks, and logistic regression, maintaining a consistent approach throughout. PubDate: 2024-04-01 DOI: 10.21136/AM.2024.0090-23

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Abstract: Abstract In this paper, we deal with the construction of symmetric matrix whose corresponding graph is connected and unicyclic using some pre-assigned spectral data. Spectral data for the problem consist of the smallest and the largest eigenvalues of each leading principal submatrices. Inverse eigenvalue problem (IEP) with this set of spectral data is generally known as the extremal IEP. We use a standard scheme of labeling the vertices of the graph, which helps in getting a simple relation between the characteristic polynomials of each leading principal submatrix. Sufficient condition for the existence of the solution is obtained. The proof is constructive, hence provides an algorithmic procedure for finding the required matrix. Furthermore, we provide the condition under which the same problem is solvable when two particular entries of the required matrix satisfy a linear relation. PubDate: 2024-04-01 DOI: 10.21136/AM.2024.0084-23