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The explanation of Einstein’s special theory of relativity in his original 1905 paper is examined. His analysis is confusing, as terms x, y, z, t, etc., have different meanings at various points, and he presents equations based on different and inconsistent assumptions. Adding subscripts clarifies these issues but exposes errors in his reasoning. To calculate his transformation equations, he selects a combination of equations which gives results matching the Lorentz transformation but he ignores other possible valid solutions. Also his calculations contain serious errors. Therefore, he fails to prove that his theory leads to the Lorentz equations as a unique solution. Einstein’s analysis includes “moving” clocks which show “stationary” time t, so the idea that a moving clock should run slower than a stationary clock is incompatible with his theory. Also, his calculation of time dilation contains serious errors. As a result, he fails to provide a theoretical justification for his famous “clocks paradox.” [ABSTRACT FROM AUTHOR]

Here we study the physical problem in which a thin bar, such as a strip of paper, is bent under several types of loads and boundary conditions. Initially, we consider lengthwise compression of the strip due to concentrated forces applied on its ends. We present this problem for the three boundary conditions: doubly-clamped, hinged–clamped, and doubly-hinged ends constrained on a flat surface. We also consider both analytically and experimentally the problems of distributed loading such as the heavy cantilever and the post-buckling heavy column. We find the equations that govern the shapes of the bent strips via a minimization flow of the bending energy. In all cases, we find that photographically-captured profiles agree very well with the theoretical predictions. We also validate the theory with measurements of the deflection of the middle or the tip of the strip as its length is varied. [ABSTRACT FROM AUTHOR]

We discuss the recent paper by A. Darós and L.K. Arruda (2019) [6]. Our intention is to correct some imperfections left by the authors and present the orbital stability of periodic snoidal waves in the energy space. [ABSTRACT FROM AUTHOR]

I point out a conceptually simple way of establishing one of the main lemmas in the paper "Solvability of equations in elementary functions" by A. Kanel-Belov, A. Malistov and R. Zaytsev. [ABSTRACT FROM AUTHOR]

The study of the dynamics of the size of a population via mathematical modelling is a problem of interest and widely studied. Traditionally, continuous deterministic methods based on differential equations have been used to deal with this problem. However, discrete versions of some models are also available and sometimes more adequate. In this paper, we randomize the Pielou logistic equation in order to include the inherent uncertainty in modelling. Taking advantage of the method of transformation of random variables, we provide a full probabilistic description to the randomized Pielou logistic model via the computation of the probability density functions of the solution stochastic process, the steady state, and the time until a certain level of population is reached. The theoretical results are illustrated by means of two examples: The first one consists of a numerical experiment and the second one shows an application to study the diffusion of a technology using real data. [ABSTRACT FROM AUTHOR]

We describe a method to construct smooth and compactly supported solutions of 3D incompressible Euler equations and related models. The method is based on localizable Grad–Shafranov equations and is inspired by the recent result (Gavrilov in A steady Euler flow with compact support. Geom Funct Anal 29(1):90–197, [Gav19]). [ABSTRACT FROM AUTHOR]

This work is concerned with the consistency conditions for the equations in max plus algebra. The three classes of the max plus linear equations presented in the paper Shinzawa [Ultra discrete permanent and the consistency of max plus linear equations, Linear Algebra Appl. 506 (2016) 445–477] can be equivalently converted into the corresponding system of inequalities. The equivalence relation between ultra discrete permanent and maximum cycle mean is suggested. Thus, the necessary and sufficient conditions for solvability of the equations are obtained using the maximum cycle mean in this note. [ABSTRACT FROM AUTHOR]

Paper materials are used as the insulation of power transformers. Over time these materials slowly degrade, until they reach a point when they no longer function effectively as transformer insulation, and the transformer has then reached its end of life. The aging rate of paper is affected by temperature, water, oxygen and acids. Investigations have been performed previously on Kraft and Kraft thermally upgraded types of paper. However, a different type of paper used in transformer insulation, Grade 3 presspaper (which contains cotton), has not been extensively tested and compared to Kraft paper. In these experiments we studied the aging rate of Grade 3 presspaper and compared it to our previous studies of Kraft paper. Traditionally, the aging rate of paper has been studied in sealed vessels. The problem with this approach is that the chemical environment within the vessel will change during aging, and so the aging rate will be affected. In our experiment setup we controlled the water and oxygen content to more accurately determine the aging rate. Similarly to Kraft paper, the aging rate of Grade 3 presspaper with the same water content increased with oxygen content in the oil. Life curves were developed based on the water content of the paper and the oxygen content of the oil. [ABSTRACT FROM AUTHOR]

We consider a two-dimensional model of three species in rock-paper-scissors competition and study the self-organisation of the population into fascinating spiraling patterns. Within our individual-based metapopulation formulation, the population composition changes due to cyclic dominance (dominance-removal and dominance-replacement), mutations, and pair-exchange of neighboring individuals. Here, we study the influence of mobility on the emerging patterns and investigate when the pair-exchange rate is responsible for spiral waves to become elusive in stochastic lattice simulations. In particular, we show that the spiral waves predicted by the system's deterministic partial equations are found in lattice simulations only within a finite range of the mobility rate. We also report that in the absence of mutations and dominance-replacement, the resulting spiraling patterns are subject to convective instability and far-field breakup at low mobility rate. Possible applications of these resolution and far-field breakup phenomena are discussed. [ABSTRACT FROM AUTHOR]

The rapid advancements in deep learning techniques have significantly impacted various fields, including mathematics and natural language processing. Mathematical equation identification and solving is a challenging task that has been studied by researchers for many years. Traditional approaches to this task typically rely on rule-based systems, hand-crafted features or manual work. However, these approaches are often limited in their ability to handle complex equations and there are also chances of error in it. The survey begins by discussing the challenges associated with mathematical equation identification and solving, such as the complexity of mathematical notation, variable representation, and equation structure. The paper provides an overview of the existing methodologies, algorithms, and frameworks employed in the development of systems capable of identifying and solving mathematical equations using deep learning techniques. [ABSTRACT FROM AUTHOR]

We assessed 6122 environmental modelling papers published since 2005 to determine whether the number of citations each paper had received by September 2014 could be predicted with no knowledge of the paper's quality. A random forest was applied, using a range of easily quantified or classified variables as predictors. The 511 papers published in two key journals in 2008 were further analysed to consider additional variables. Papers with no differential equations received more citations. The topic of the paper, number of authors and publication venue were also significant. Ten other factors, some of which have been found significant in other studies, were also considered, but most added little to the predictive power of the models. Collectively, all factors predicted 16–29% of the variation in citation counts, with the remaining variance (the majority) presumably attributable to important subjective factors such as paper quality, clarity and timeliness. [ABSTRACT FROM AUTHOR]

Samples of a dried composite oil-paper system were thermally aged in the laboratory under N2 in a sealed container and analysed using their dielectric response. The contributing loss processes were separated using an equivalent circuit model containing constant high frequency capacitances, dc-conductances, a dispersive capacitance, and a low frequency quasi-dc (q-dc) dispersion process as circuit elements. This low frequency dispersion in oil-paper samples is a bulk property of the system that is often mistaken for a dc conduction process when only the imaginary capacitance or ac-conductance is measured. The Dissado-Hill response function is proposed for the frequency dependence of the dispersive capacitance in the mid-frequency range rather than the Debye function because its cluster concepts correlate with the sample morphology of a matrix of oil-filled cavities. The quasi-dc dispersion is assigned to the transport of mobile charges between charge-clusters associated with locally connected oil filled cavities, which is a process that can be represented on a global scale by the form of hierarchical circuit system that leads to constant phase angle responses, and is here modelled by the Dissado-Hill q-dc function. It was found that the two dispersions moved together to higher frequency with thermal ageing of the composite material. The characteristic frequency common to the dispersions had an activation energy that decreased with thermal ageing, whereas their amplitude increased. This behavior is discussed in terms of possible changes to the oil-paper composite caused by the ageing process. [ABSTRACT FROM AUTHOR]

In this paper the accelerated aging experiments of oil-paper insulation are carried out to simulate the oil-paper aging process under normal thermal aging process. On the basis of the results of accelerated aging experiments, how the deviation degree of aging status change under the same aging factors is studied, and a method to calculate the failure probability due to the loss of mechanical strength of the insulation is put forward. This paper presents an approach for economic assessment on insulation diagnosis through accelerated aging experiments. The foundation of this approach for economic assessment on insulation diagnosis is to establish condition simulation method of triple relationships between condition status, aging time and failure probability according to data from accelerated aging experiments. The conclusions are made that the economic and technical factors can affect the optimum life cycle cost of each maintenance strategy, which consequently affects the economic feasibility of condition-based maintenance strategies. [ABSTRACT FROM AUTHOR]

This paper focusses on the nonlinear fractional Kadomtsev-Petviashvili (FKP) equation in space-time, employing the conformable fractional derivative (CFD) approach. The main objective of this paper is to examine the application of the (G'/G)-expansion method in order to find analytical solutions to the FKP equation. The (G'/G)-expansion method is a powerful tool for constructing traveling wave solutions of nonlinear evolution equations. However, its application to the FKP equation remains relatively unexplored. By employing traveling wave transformation, the FKP equation was transformed into an ordinary differential equation (ODE) to acquire exact wave solutions. A range of exact analytical solutions for the FKP equation is obtained. Graphical illustrations are included to elucidate the physical characteristics of the acquired solutions. To demonstrate the impact of the fractional operator on results, the acquired solutions are exhibited for different values of the fractional order α, with a comparison to their corresponding exact solutions when taking the conventional scenario where α equals 1. The results indicate that the (G'/G)-expansion method serves as an efficient method and dependable in solving the nonlinear FKP equation. [ABSTRACT FROM AUTHOR]

This document discusses a research paper on the discharge coefficient of Type-A piano key weirs. The discussers appreciate the authors' efforts but suggest further analysis for one equation in the original paper. They conducted simulations and analysis, providing graphs and tables to support their findings. The document also includes a sensitivity analysis of input parameters affecting the discharge coefficient, showing the varying effects of different parameters. The study provides insights into estimating the discharge coefficient and was supported by funding from the Water Resources Department of Jiangxi Province and the Natural Science Foundation of Jiangxi Province in China. [Extracted from the article]

We correct the proof of Theorem 2 in the mentioned paper concerning finite-difference equilibrium solutions. [ABSTRACT FROM AUTHOR]

• The convergence analysis and the techniques used in Lemma 3.3 by X. Wang et al. in [1] are based essentially on Sobolev's inequality in one dimension and they cannot be extended to two dimensions. • All results based on Lemma 3.3 are false, in particular Theorem 3.3, Theorem 4.1, Theorem 5.1 and Theorem 5.2. • Regarding the conservation of mass and energy, the initial terms Q 0 and E 0 depends on U 0 and on U 1 and the authors consider only Eq. (4) without taking into account Eq. (7). Therfore, there is an error in Theorem 3.1 and Theorem 3.2. • In Theorem 4.1, the authors did not show that u1 is uniquely determined by Eq. (7). In the above article, a second-order convergence in the maximum norm of the two-dimensional regularized long-wave equation is proved. However, due to a wrong inequality in Lemma 3.3 used in X. Wang et al.[1], there are some fundamental errors in this paper, in particular the proof of Theorem 3.3 and the convergence Theorem are no longer correct. The present brief paper contains clarifying comments. [ABSTRACT FROM AUTHOR]

We have studied initial value problems for Caputo fractional differential equations with singular nonlinearities involving the p-Laplacian operator. We have given a precise mathematical analysis of the equivalence of the fractional differential equations and Volterra integral equations studied in this paper. A theorem for the global existence of the solution was proven. In addition, an example was given at the end of the article as an application of the results found in this paper. [ABSTRACT FROM AUTHOR]

This research paper introduces novel findings regarding a specific class of Kirchhoff problems. Building upon the theoretical groundwork established in a previous work by Ricceri [15], the paper centers its investigation on the Sierpiński gasket (V, |•|) situated in (Rn-1,- |•|), where n≥ 2. The intrinsic boundary of the Sierpiński gasket, denoted as Vo, comprises n corners. Under certain conditions, we prove that a Kirchhoff problem has two classical solutions in the space H0¹(V). [ABSTRACT FROM AUTHOR]

In this paper we study the controllability for the 1D-Heat equation with a Dirichlet boundary condition, in the presence of a scale-invariant parameter. First, we construct the scale-invariant solutions for the one-dimensional heat equation. Then we present our problem statement. We finally prove the Dirichlet boundary controllability. [ABSTRACT FROM AUTHOR]

This letter presents comments on the paper “Critical review on the applicability of Avrami fractional kinetic equation in adsorption-based water treatment studies” that was recently published in Desalination and Water Treatment by N.A. Oladoja. This author explained the misuse of Avrami-type kinetic equation in adsorption process at solid/solution interfaces and also mentioned few of the papers published in the literature that used the Avrami kinetic equation. We, therefore, decided to write this letter to make it clearer to the reader that there are no wrong or unfounded concepts in the papers cited in the review of N.A. Oladoja. We do believe that the mechanism of adsorption could only be established by several analytical techniques and a good sense of chemistry associated with the equilibrium and kinetic data. The mechanism cannot be directly assigned based on just simple kinetic experiments. In addition, the author was very selective in choosing some specific papers, which used this valuable kinetic adsorption equation, for his criticism. The kinetic equation presented at the end of Oladoja’s paper is worthless and may not be a feasible equation that can be applied to adsorption from solutions. [ABSTRACT FROM AUTHOR]

The objective work of this paper is to transform the nonlinear space-time fraction Whitham–Broer–Kaup equation into ordinary differential equation by using the conformal fractional derivative, and find the exact solutions through the complete polynomial discriminant system. At the same time, we build the appropriate solution for the identified parameters to show the existence of the solution. In addition, we provide the 3D and 2D graphics to show that the solutions are real and effective. [ABSTRACT FROM AUTHOR]

This research paper investigates the behavior of optical fiber pulses by studying the solitary wave solutions of the perturbed Chen–Lee–Liu (CLL) equation. The perturbed CLL equation is derived from the perturbed Schrödinger equation. Two analytical approaches for obtaining the innovative soliton-wave solutions are presented in this paper, and their reliability is examined by using a well-established numerical method. The accuracy of pulse wave analysis in an optical cable is demonstrated through a series of graphical representations. Our study introduces scientific novelty, as evidenced by the comparative analysis of our data with those of previous research papers. This research contributes to enhancing the understanding of the kinetics and physical behavior of pulses in optical fibers, which holds implications for the advancement of optical communication technologies. [ABSTRACT FROM AUTHOR]

In this paper, we propose a discrete Hamilton–Jacobi theory for (discrete) Hamiltonian dynamics defined on a (discrete) contact manifold. To this end, we first provide a novel geometric Hamilton–Jacobi theory for continuous contact Hamiltonian dynamics. Then, rooting on the discrete contact Lagrangian formulation, we obtain the discrete equations for Hamiltonian dynamics by the discrete Legendre transformation. Based on the discrete contact Hamilton equation, we construct a discrete Hamilton–Jacobi equation for contact Hamiltonian dynamics. We show how the discrete Hamilton–Jacobi equation is related to the continuous Hamilton–Jacobi theory presented in this work. Then, we propose geometric foundations of the discrete Hamilton–Jacobi equations on contact manifolds in terms of discrete contact flows. At the end of the paper, we provide a numerical example to test the theory. [ABSTRACT FROM AUTHOR]

In this paper, we study the following Kirchhoff-type equation:... where M ∈ C(R+,R+) is a general function, V ≥ 0 and its zero set may have several disjoint connected components, μ > 0 is a parameter, K is permitted to be unbounded above, and f has exponential critical growth. By using the truncation technique and developing some approaches to deal with Kirchhofftype equations with critical growth in the whole space, we get the existence and concentration behavior of solutions. The results are new even for the case M ≡ 1. [ABSTRACT FROM AUTHOR]

The Functionalized Cahn Hilliard (FCH) equation was used to model micro-phase separation in mixtures of amphiphilic molecules in solvent. In this paper, we proposed a Tri-Harmonic Modified (THM) numerical approach for efficiently solving the FCH equation with symmetric double well potential by extending the ideas of the Bi-harmonic Modified (BHM) method. THM formulation allowed for the nonlinear terms in the FCH equation to be computed explicitly, leading to fast evaluations at every time step. We investigated the convergence properties of the new approach by using benchmark problems for phase-field models, and we directly compared the performance of the THM method with the recently developed scalar auxiliary variable (SAV) schemes for the FCH equation. The THM modified scheme was able to produce smaller errors than those obtained from the SAV formulation. In addition to this direct comparison with the SAV schemes, we tested the adaptability of our scheme by using an extrapolation technique which allows for errors to be reduced for longer simulation runs. We also investigated the adaptability of the THM method to other 6th order partial differential equations (PDEs) by considering a more complex form of the FCH equation with nonsymmetric double well potential. Finally, we also couple the THM scheme with a higher order timestepping method, (implicit-explicit) IMEX schemes, to demonstrate the robustness and adaptability of the new scheme. Numerical experiments are presented to investigate the performance of the new approach. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic behavior of transmission system for coupled Kirchhoff plates, where one equation is conserved and the other has dissipative property, and the dissipation mechanism is given by fractional damping (- Δ) 2 θ v t with θ ∈ [ 1 2 , 1 ] . By using the semigroup method and the multiplier technique, we obtain the exact polynomial decay rates, and find that the polynomial decay rate of the system is determined by the inertia/elasticity ratios and the fractional damping order. Specifically, when the inertia/elasticity ratios are not equal and θ ∈ [ 1 2 , 3 4 ] , the polynomial decay rate of the system is t - 1 / (10 - 4 θ) . When the inertia/elasticity ratios are not equal and θ ∈ [ 3 4 , 1 ] , the polynomial decay rate of the system is t - 1 / (4 + 4 θ) . When the inertia/elasticity ratios are equal, the polynomial decay rate of the system is t - 1 / (4 + 4 θ) . Furthermore it has been proven that the obtained decay rates are all optimal. The obtained results extend the results of Oquendo and Suárez (Z Angew Math Phys 70(3):88, 2019) for the case of fractional damping exponent 2 θ from [0, 1] to [1, 2]. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to consider a continuous data assimilation algorithm for the three-dimensional planetary geostrophic model in the case that the observable measurements, obtained continuously in time, may be contaminated by systematic errors. In this paper, we will provide some suitable conditions on the nudging parameter and the spatial resolution, which are sufficient to show that the approximation solution of the proposed continuous data assimilation algorithm converges to the unique exact unknown reference solution of the original system at an exponential rate, asymptotically in time, under the assumption that the observed data is free of error. [ABSTRACT FROM AUTHOR]

In this paper, we make the first attempt to figure out the differences on Hölder regularity in time of solutions and conserved physical quantities between the ideal electron magnetohydrodynamic equations concerning Hall term and the incompressible Euler equations involving convection term. It is shown that the regularity in time of magnetic field B is C_{t}^{\frac {\alpha }2} provided it belongs to L_{t}^{\infty } C_{x}^{\alpha } for any \alpha >0, its energy is C_{t}^{\frac {2\alpha }{2-\alpha }} as long as B belongs to L_{t}^{\infty } \dot {B}^{\alpha }_{3,\infty } for any 0<\alpha <1 and its magnetic helicity is C_{t}^{\frac {2\alpha +1}{2-\alpha }} supposing B belongs to L_{t}^{\infty } \dot {B}^{\alpha }_{3,\infty } for any 0<\alpha <\frac 12, which are quite different from the classical incompressible Euler equations. [ABSTRACT FROM AUTHOR]

In this paper, we study the spectral properties of a class of degenerate third-order partial differential operators with variable coefficients presented in a rectangle. Conditions are found to ensure the existence and compactness of the inverse operator. A theorem on estimates of approximation numbers is proven. Here, we note that finding estimates of approximation numbers, as well as extremal subspaces, for a set of solutions to the equation is a task that is certainly important from both a theoretical and a practical point of view. The paper also obtained an upper bound for the eigenvalues. Note that, in this paper, estimates of eigenvalues and approximation numbers for the degenerate third-order partial differential operators are obtained for the first time. [ABSTRACT FROM AUTHOR]

The widely used discrete-element particle flow software PFC's (PFC 7.0 and previous versions) algorithm for calculating porosity is not sufficiently accurate. Because of this, when the particles are densely packed, the solution to the equation produces an algorithm exception for odd calculations of porosity, which results in the inability to calculate the results. This paper, based on a Darcy seepage model of fluid flow through a granular bed, analyzed the shortcomings of the two porosity calculation methods of PFC and the function analysis method. Combining this analysis with the theory of computer graphics, a new and efficient porosity calculation algorithm was proposed. The result showed that the new proposed porosity calculation algorithm calculated a more accurate and reasonable porosity field and made the iterative solution of the CFD equation more stable. This method makes porosity-related models of PFC more accurate. The algorithm can be not only used to calculate porosity, but also applied to other fields. [ABSTRACT FROM AUTHOR]

A 30-year (1980–2010) climatology of the major variables and terms of the transformed Eulerian-mean (TEM) momentum and thermodynamic equations is constructed by using four global atmospheric reanalyses: the Modern-Era Retrospective analysis for Research and Applications, Version 2 (MERRA-2); the Japanese 55-year Reanalysis (JRA-55); the European Centre for Medium-Range Weather Forecasts (ECMWF) interim reanalysis (ERA-Interim); and the Climate Forecast System Reanalysis (CFSR). Both the reanalysis ensemble mean (REM) and the differences in each reanalysis from the REM are investigated in the latitude–pressure domain for December–January–February and for June–July–August. For the REM investigation, two residual vertical velocities (the original one and one evaluated from residual meridional velocity) and two mass streamfunctions (from meridional and vertical velocities) are compared. Longwave (LW) radiative heating and shortwave (SW) radiative heating are also shown and discussed. For the TEM equations, the residual terms are also calculated and investigated for their potential usefulness, as the residual term for the momentum equation should include the effects of parameterized processes such as gravity waves, while that for the thermodynamic equation should indicate the analysis increment. Inter-reanalysis differences are investigated for the mass streamfunction, LW and SW heating, the two major terms of the TEM momentum equation (the Coriolis term and the Eliassen–Palm flux divergence term), and the two major terms of the TEM thermodynamic equation (the vertical temperature advection term and the total diabatic heating term). The spread among reanalysis TEM momentum balance terms is around 10 % in Northern Hemisphere winter and up to 50 % in Southern Hemisphere winter. The largest uncertainties in the thermodynamic equation (about 50 %) are found in the vertical advection, for which the structure is inconsistent with the differences in heating. The results shown in this paper provide basic information on the degree of agreement among recent reanalyses in the stratosphere and upper troposphere in the TEM framework. [ABSTRACT FROM AUTHOR]

In this paper, we consider k-curvature equations \sigma _k(\kappa [M_u])=f(x,u,\nabla u) subject to (k+1)-convex Dirichlet boundary data instead of affine Dirichlet data of Sheng, Urbas, and Wang [Duke Math. J. 123 (2004), pp. 235–264]. By using the crucial concavity inequality for Hessian operator of Lu [Calc. Var. Partial Differential Equations 62 (2023), p.23], we derive Pogorelov estimates of semi-convex admissible solutions for these k-curvature equations. [ABSTRACT FROM AUTHOR]

This paper is devoted to some inverse problems of recovering the nonlinearity for the Jordan–Moore–Gibson–Thompson equation, which is a third order nonlinear acoustic equation. This equation arises, for example, from the wave propagation in viscous thermally relaxing fluids. The well-posedness of the nonlinear equation is obtained with the small initial and boundary data. By the second order linearization to the nonlinear equation, and construction of complex geometric optics solutions for the linearized equation, the uniqueness of recovering the nonlinearity is derived. [ABSTRACT FROM AUTHOR]

Purpose: The main aim of the current paper is to find a numerical plan for hydraulic fracturing problem with application in extracting natural gases and oil. Design/methodology/approach: First, time discretization is accomplished via Crank-Nicolson and semi-implicit techniques. At the second step, a high-order finite element method using quadratic triangular elements is proposed to derive the spatial discretization. The efficiency and time consuming of both obtained schemes will be investigated. In addition to the popular uniform mesh refinement strategy, an adaptive mesh refinement strategy will be employed to reduce computational costs. Findings: Numerical results show a good agreement between the two schemes as well as the efficiency of the employed techniques to capture acceptable patterns of the model. In central single-crack mode, the experimental results demonstrate that maximal values of displacements in x- and y- directions are 0.1 and 0.08, respectively. They occur around both ends of the line and sides directly next to the line where pressure takes impact. Moreover, the pressure of injected fluid almost gained its initial value, i.e. 3,000 inside and close to the notch. Further, the results for non-central single-crack mode and bifurcated crack mode are depicted. In central single-crack mode and square computational area with a uniform mesh, computational times corresponding to the numerical schemes based on the high order finite element method for spatial discretization and Crank-Nicolson as well as semi-implicit techniques for temporal discretizations are 207.19s and 97.47s, respectively, with 2,048 elements, final time T = 0.2 and time step size τ = 0.01. Also, the simulations effectively illustrate a further decrease in computational time when the method is equipped with an adaptive mesh refinement strategy. The computational cost is reduced to 4.23s when the governed model is solved with the numerical scheme based on the adaptive high order finite element method and semi-implicit technique for spatial and temporal discretizations, respectively. Similarly, in other samples, the reduction of computational cost has been shown. Originality/value: This is the first time that the high-order finite element method is employed to solve the model investigated in the current paper. [ABSTRACT FROM AUTHOR]

In this paper, we study the quenching phenomenon related to the ω-diffusion equation on graphs with a potential and a singular source ut(x, t) = Δωu(x, t) + b(x)(1 - u(x, t))-p, where Δω is called the discrete weighted Laplacian operator. Under some appropriate hypotheses, we prove the existence and uniqueness of the local solution via Banach fixed point theorem. We also show that the solution of the problem quenches in a finite time and that the time-derivative blows up at the quenching time. Moreover, we estimate the quenching time and the quenching rate. Finally, we verify our results through some numerical examples. [ABSTRACT FROM AUTHOR]

Purpose: This work presents a new two-step iterative technique for solving absolute value equations. The developed technique is valuable and effective for solving the absolute value equation. Various examples are given to demonstrate the accuracy and efficacy of the suggested technique. Design/methodology/approach: In this paper, we present a new two-step iterative technique for solving absolute value equations. This technique is very straightforward, and due to the simplicity of this approach, it can be used to solve large systems with great effectiveness. Moreover, under certain assumptions, we examine the convergence of the proposed method using various theorems. Numerical outcomes are conducted to present the feasibility of the proposed technique. Findings: This paper gives numerical experiments on how to solve a system of absolute value equations. Originality/value: Nowadays, two-step approaches are very popular for solving equations (1). For solving AVEs, Liu in Shams (2021), Ning and Zhou (2015) demonstrated two-step iterative approaches. Moosaei et al. (2015) introduced a novel approach that utilizes a generalized Newton's approach and Simpson's rule to solve AVEs. Zainali and Lotfi (2018) presented a two-step Newton technique for AVEs that converges linearly. Feng and Liu (2016) have proposed minimization approaches for AVEs and presented their convergence under specific circumstances. Khan et al. (2023), suggested a nonlinear CSCS-like technique and a Picard-CSCS approach. Based on the benefits and drawbacks of the previously mentioned methods, we will provide a two-step iterative approach to efficiently solve equation (1). The numerical results show that our proposed technique converges rapidly and provides a more accurate solution. [ABSTRACT FROM AUTHOR]

P- and S-wave separation plays an important role in elastic reverse-time migration. It can reduce the artifacts caused by crosstalk between different modes and improve image quality. In addition, P- and Swave separation can also be used to better understand and distinguish wave types in complex media. At present, the methods for separating wave modes in anisotropic media mainly include spatial nonstationary filtering, low-rank approximation, and vector Poisson equation. Most of these methods require multiple Fourier transforms or the calculation of large matrices, which require high computational costs for problems with large scale. In this paper, an efficient method is proposed to separate the wave mode for anisotropic media by using a scalar anisotropic Poisson operator in the spatial domain. For 2D problems, the computational complexity required by this method is 1/2 of the methods based on solving a vector Poisson equation. Therefore, compared with existing methods based on pseudo-Helmholtz decomposition operators, this method can significantly reduce the computational cost. Numerical examples also show that the P and S waves decomposed by this method not only have the correct amplitude and phase relative to the input wavefield but also can reduce the computational complexity significantly. [ABSTRACT FROM AUTHOR]

The paper presents a method for determining the dynamic response of systems containing viscoelastic damping elements with uncertain design parameters. A viscoelastic material is characterized using classical and fractional rheological models. The assumption is made that the lower and upper bounds of the uncertain parameters are known and represented as interval values, which are then subjected to interval arithmetic operations. The equations of motion are transformed into the frequency domain using Laplace transformation. To evaluate the uncertain dynamic response, the frequency response function is determined by transforming the equations of motion into a system of linear interval equations. Nevertheless, direct interval arithmetic often leads to significant overestimation. To address this issue, this paper employs the element-by-element technique along with a specific transformation to minimize redundancy. The system of interval equations obtained is solved iteratively using the fixed-point iteration method. As demonstrated in the examples, this method, which combines the iterative solving of interval equations with the proposed technique of equation formulation, enables a solution to be found rapidly and significantly reduces overestimation. Notably, this approach has been applied to systems containing viscoelastic elements for the first time. Additionally, the proposed notation accommodates both parallel and series configurations of damping elements and springs within rheological models. [ABSTRACT FROM AUTHOR]

In this paper, we investigate several Liouville-type theorems related to k-Hessian equations with non-linear gradient terms. More specifically, we study non-negative solutions to S_k[D^2u]\ge h(u,|Du|) in \mathbb {R}^n. The results depend on some qualified growth conditions of h at infinity. A Liouville-type result to subsolutions of a prototype equation S_k[D^2u]=f(u)+g(u)\varpi (|Du|) is investigated. A necessary and sufficient condition for the existence of a non-trivial non-negative entire solution to S_k[D^2u]=f(u)+g(u)|Du|^q for 0\le q

This paper investigates the inhomogeneous version of the pantograph equation. The current model includes the exponential function as the inhomogeneous part of the pantograph equation. The Maclaurin series expansion (MSE) is a well-known standard method for solving initial value problems; it may be easier than any other approaches. Moreover, the MSE can be used in a straightforward manner in contrast to the other analytical methods. Thus, the MSE is extended in this paper to treat the inhomogeneous pantograph equation. The solution is obtained in a closed series form with an explicit formula for the series coefficients and the convergence of the series is proved. Also, the analytic solutions of some models in the literature are recovered as special cases of the present work. The accuracy of the results is examined through several comparisons with the available exact solutions of some classes in the relevant literature. Finally, the residuals are calculated and then used to validate the accuracy of the present approximations for some classes which have no exact solutions. [ABSTRACT FROM AUTHOR]

We highlight some problems in this paper. They lead to question the model itself. [ABSTRACT FROM AUTHOR]

In the above paper some equations are wrong. [ABSTRACT FROM AUTHOR]

In the above paper, some basic equations are wrong. [ABSTRACT FROM AUTHOR]