In this scholarly analysis, we aim to address the inaccuracies and unsupported claims presented in the paper “Math. Meth. Appl. Sci. 2023;1–3,”’authored by Elsayed M. E. Zayed and Shoukry El-Ganaini. Their paper  asserts that the authors of “Math Meth Appl Sci. 2021;44:2682–2691”employed the Khater method to solve the nonlinear fractional Cahn–Allen equation. It is imperative to emphasize that this claim lacks validity. Contrarily, upon a meticulous examination of our own work , we find that Mostafa M. A. Khater, Ahmet Bekir, Dianchen Lu, and Raghda A. M. Attia employed the modified Khater method, a robust computational technique recently introduced by Prof. Mostafa M. A. Khater. This assertion is substantiated by a substantial body of Prof. Mostafa M. A. Khater’s work (as cited in references [3–6]). These findings unmistakably contradict the assertions made in “Math. Meth. Appl. Sci. 2023, ”which erroneously claim the utilization of the Khater method. It is also of utmost importance to address the implications of these inaccuracies. The claims put forth by Zayed and El-Ganaini not only misrepresent the actual methodology used but also erroneously question the authorship of the modified Khater method, implying that it does not originate from Prof. Mostafa M. A. Khater. To counter this assertion, we emphasize Prof. Khater’s extensive publication history, which indisputably demonstrates his scholarly contributions employing the modified Khater method in the years 2018 and 2019 (as thoroughly documented in references [3–6]). In summary, our analysis underscores the discrepancies and ethical concerns within the aforementioned paper, asserting that it not only misrepresents the methodology employed but also inaccurately questions the ownership of the modified Khater method. This casts doubt on the integrity of the authors’ claims, suggesting an unwarranted attempt to appropriate the contributions of another researcher.
This manuscript is devoted to a derivative-free parametric iterative step to obtain roots simultaneously for both nonlinear systems and equations. We prove that when it is added to an arbitrary scheme, it doubles the convergence order of the original procedure and defines a new algorithm that obtains several solutions simultaneously. Different numerical experiments are carried out to check the behaviour of the iterative methods by changing the value of the parameter and the initial guesses. Also, it is perform a numerical example where the dynamical planes are carried out to see and compare the basins of attraction.
The models for symmetric stochastic matrices that we consider in this study are developed using the spectral analysis of the respective mean matrices. The adjustment and validation of the models require the usage of the estimated structure vectors. The information enclosed in these matrices can be condensed into the pair consisting of the estimated structure vectors and the sum of squares of residuals. The results obtained allow for cross-sectional and longitudinal inference. For models of degree greater than one, it is also considered the possibility of truncating the model when eigenvalues are much higher than the others. A direct consequence of the adopted methodology is the application of the degree-one models to cross-product matrices and Hilbert-Schmidt scalar product matrices. In addition to these models, structured family models were also considered. The models of these families are associated with the treatments of a base design. The action of the factors considered in the base design on the structure vectors is also analyzed. In structured families with orthogonal base design, the designs are associated to partitions, and the hypotheses formulated are associated with the spaces of these partitions. We carry out ANOVA-like analysis for the action of the factors in the base design, on homolog components on estimated structure vectors, considering that the estimator’s structure vectors have, approximately, the same covariance matrix. To apply our results, we assume the factors in the base design to have fixed effects and that the base design has orthogonal structure. The action of factors in the base design is studied. An application is given, using a data set from a breeding program of durum wheat ( Triticum turgidum L., Durum Group) conducted in Portugal. The results show that our methodology is fully applicable to complete and incomplete data sets, often observed in multi-environmental trials.
Polylactic acid (PLA) is a bio-based plastic that is biodegradable under appropriate conditions of temperature, humidity and oxygen, which are achieved in the composting process. The objective of this work is to formulate a mathematical model that predicts the biodegradation of polylactic acid in composting processes. We performed a qualitative analysis of the reduced composting mass system, which is non-linear and non-autonomous. First, the reduced model was transformed into an autonomous system, showing that their solutions are positive, bounded and non-periodic. Furthermore, it was shown that the origin is locally and globally exponentially stable, the axial equilibrium is unstable and that a degenerate transcritical bifurcation exists at the origin. Simulations of the reduced system indicated that the PLA mass is completely biodegraded when the time tends to infinity, which was shown theoretically. In addition, numerical simulations of the complete composting system were performed considering three initial values of the carbon/nitrogen ratio. It was concluded that the initial carbon/nitrogen ratio of 32.5 reached 90% of PLA biodegradation in approximately 150 days. This work provides a mathematical tool applied to the field of biotechnology of biodegradable plastics.
The construction of (hierarchical) curl-conforming basis functions has been a hot topic in the last decades in the finite element community. Especially, functions applied to simplices have been quite popular after the work by Nédélec in 1980. Many mixed-order and full-order families have been provided in the last years, but sometimes it is difficult to assess if they belong to the original space proposed by Nédélec (especially when orthogonalization procedures are applied). Here, a tool to determine if a family of basis functions belongs to the Nédélec space is provided. Since affine coordinates are the most frequent choice for simplices, particularities about its use with this kind of coordinates are detailed. A detailed survey of existing families is provided, and the practical application of the tool to a representative set of these families is discussed. The tool is also available for the community in a public repository.
In this paper, we investigate the the existence and stability of non-trivial steady state solutions of a class of chemotaxis models with zero-flux boundary conditions and Dirichlet boundary conditions on one-dimensional bounded interval. By using upper-lower solution and the monotone iteration scheme method, we get the existence of the steady-state solution of the chemotaxis model. Moreover, by adopting the “inverse derivative” technique and the weighted energy method to obtain the stability of the steady-state solution of this chemotaxis model.
It is well known that investigation on exact solutions of nonlinear fractional partial differential equations (PDEs) is a very difficult work compared with integer-order nonlinear PDEs. In this paper, based on the separation method of semi-fixed variables and integral bifurcation method, a combinational method is proposed. By using this new method, a class of generalized time-fractional thin-film equations are studied. Under two kinds of definitions of fractional derivatives, exact solutions of two generalized time-fractional thin-film equations are investigated respectively. Different kinds of exact solutions are obtained and their dynamic properties are discussed. Compared to the results in the existing references, the types of solutions obtained in this paper are abundant and very different from those in the existing references. Investigation shows that the solutions of the model defined by Riemann-Liouville differential operator converge faster than those defined by Caputo differential operator. It is also found that the profiles of some solutions are very similar to solitons, but they are not true soliton solutions. In order to visually show the dynamic properties of these solutions, the profiles of some representative exact solutions are illustrated by 3D-graphs.
With the inevitable environmental perturbations and complex population movements, the analysis of troublesome influenza is harder to proceed. Studies about the epidemic mathematical models can not only forecast the development trend of influenza, but also have a beneficial influence on the protection of health and the economy. Motivated by this, a stochastic influenza model incorporating human mobility and the Ornstein-Uhlenbeck process is established in this paper. Based on the existence of the unique global positive solution, we obtain sufficient conditions for influenza extinction and persistence, which are related to the basic reproduction number in the corresponding deterministic model. Notably, the analytical expression of the probability density function of stationary distribution near the quasi-endemic equilibrium is obtained by solving the challenging Fokker--Planck equation. Finally, numerical simulations are performed to support the theoretical conclusions, and the effect of main parameters and environmental perturbations on influenza transmission are also investigated.
We present some existence and localization results for periodic solutions of impulsive first-order coupled non-linear systems of two equations, without requiring periodicity for the nonlinearities. The arguments are based on Schauder's Fixed Point Theorem together with the upper and lower solution method, where the upper and lower solutions are not necessarily well-ordered. In addition, results on equi-regulated functions are required for the impulsive analysis. An application to a Wilson-Cowan system of two strongly coupled neurons illustrates one of the main results.
This paper is concerned with quantum stochastic differential equations driven by the fermion field in noncommutative space L p ( C ) for p>2. We investigate the existence and uniqueness of L p -solution of quantum stochastic differential equations in infinite time horizon by the Burkholder-Gundy inequalities for noncommutative martingales given by Pisier and Xu. Finally, we obtain Markov property and the self-adjointness which is of great significance in the study of optimal control problems. 2020 AMS Subject Classification: 46L51, 47J25, 60H10, 81J25.
This paper is devoted to analysing a kind of fractional neutral stochastic system (FNSS). Firstly, by introducing the notion of newly defined two-parameter Mittag-Leffler matrix function, we derive the solution of the corresponding linear stochastic system. Subsequently, for the linear case, by virtue of the Grammian matrix, we give a suffcient and necessary condition to guarantee the relatively exact controllability for the addressed case. Furthermore, for the nonlinear one, the relatively exact controllability is obtained by fixed point and explore it via Banach contraction principle. Finally, two examples are provided to intensify our theoretical conclusions.
This paper is concerned with high moment and pathwise error estimates for both velocity and pressure approximations of the Euler-Maruyama scheme for time discretization and its two fully discrete mixed finite element discretizations. Optimal rates of convergence are established for all pth moment errors for p≥2 using a novel bootstrap technique. The almost optimal rates of convergence are then obtained using Kolmogorov’s theorem based on the high moment error estimates. Unlike for the velocity error estimate, the high moment and pathwise error estimates for the pressure approximation are proved in a time-averaged norm. In addition, the impact of noise types on the rates of convergence for both velocity and pressure approximations is also addressed. Finally, numerical experiments are also provided to validate the proven theoretical results and to examine the dependence/growth of the error constants on the moment order p.
In this paper we study the spectral properties of a family of discrete one-dimensional quasi-periodic Schrödinger operators (depending on a phase theta). In large disorder, under some suitable conditions on v and a diophantine rotation number, we prove using basically K.A.M theory that the spectrum of this operator is pure point for all θ∈[0 ,2 π) with exponential decaying eigenfunctions.
This paper addresses the construction of Cauchy operators and related identities from R( p,q)-deformed quantum algebras. The generating function, Mehler and Rogers formulae, and their extended identities for the homogeneous Rogers-Szegö polynomials are computed and discussed. Relevant particular identities extracted from known quantum algebras are highlighted.
In this paper, we use the generalized notions of Riemann-Liouville (fractional calculus with respect to a regular function σ) to extend the definitions of fractional integration and derivative from the functional sense to the distributional sense. First, we give some properties of fractional integral and derivative for the functions infinitely differentiable with compact support. Then, we define the weak derivative, as well as the integral and derivative of a distribution with compact support, the integral and derivative of a distribution using the convolution product. Then, we generalize those concepts from the unidimensional to the multidimensional case. Finally, we propose the definitions of some usual differential operators.
One-hidden-layer feedforward neural networks are described as functions having many real-valued parameters. The larger the number of parameters is, neural networks can approximate various functions (universal approximation property). The essential optimal order of approximation bounds is already derived in 1996. We focused on the numerical experiment that indicates the neural networks whose parameters have stochastic perturbations gain better performance than ordinary neural networks, and explored the approimation property of neural networks with stochastic perturbations. In this paper, we derived the quantitative order of variance of stochastic perturbations to achieve the essential approximation order.