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This paper investigates global dynamics in fractional-order dual inertial neural networks with time lags. Firstly, according to some crucial features of Mittag-Leffler functions and Banach contracting mapping principle, the existence and uniqueness of $ S $-asymptotically $ \omega $-periodic oscillation of the model are gained. Secondly, by using the comparison principle and the stability criteria of delayed Caputo fractional-order differential equations, global asymptotical stability of the model is studied. In the end, the feasibility and effectiveness of the obtained conclusions are supported by two numerical examples. There are few papers focus on $ S $-asymptotically $ \omega $-periodic dynamics in fractional-order dual inertial neural networks with time-varying lags, apparently, the works in this paper fill some of the gaps.

Using the approach of so-called c-sequences introduced by the fifth author in his recent work, we give much simpler and shorter proofs of multivalued Perov's type results with respect to the ones presented in the recently published paper by M. Abbas et al. Our proofs improve, complement, unify and enrich the ones from the recent papers. Further, in the last section of this paper, we correct and generalize the well-known Perov's fixed point result. We show that this result is in fact equivalent to Banach's contraction principle.

There is a close analogy and similarity between topology and rough set theory. As, the leading idea of this theory is depended on two approximations, namely lower and upper approximations, which correspond to the interior and closure operators in topology, respectively. So, the joined study of this theory and topology becomes fundamental. This theory mainly propose to enlarge the lower approximations by adding new elements to it, which is an equivalent goal for canceling elements from the upper approximations. For this intention, one of the primary motivation of this paper is the desire of improving the accuracy measure and reducing the boundary region. This aim can be achieved easily by utilizing ideal in the construction of the approximations as it plays an important role in removing the vagueness of concept. The emergence of ideal in this theory leads to increase the lower approximations and decrease the upper approximations. Consequently, it minimizes the boundary and makes the accuracy higher than the previous. Therefore, this work expresses the set of approximations by using new topological notions relies on ideals namely $ \mathcal{I} $-$ {\delta_{\beta}}_{J} $-open sets and $ \mathcal{I} $-$ {\bigwedge_{\beta}}_{J} $-sets. Moreover, these notions are also utilized to extend the definitions of the rough membership relations and functions. The essential properties of the suggested approximations, relations and functions are studied. Comparisons between the current and previous studies are presented and turned out to be more precise and general. The brilliant idea of these results is increased in importance by applying it in the chemical field as it is shown in the end of this paper. Additionally, a practical example induced from an information system is introduced to elucidate that the current rough membership functions is better than the former ones in the other studies.

In a recent paper [4], Xu studied some alternating multiple zeta values. In particular, he gave two recurrence formulas of alternating multiple zeta values $ \zeta^\star({\bar 1}, \{1\}_{m-1}, {\bar 1}) $ and $ \zeta^\star(2, \{1\}_{m-1}, {\bar 1}) $. In this paper, we will give the closed forms representations of $ \zeta^\star({\bar 1}, \{1\}_{m-1}, {\bar 1}) $ and $ \zeta^\star(2, \{1\}_{m-1}, {\bar 1}) $ in terms of single zeta values and polylogarithms.

In this paper, we prove some new Simpson's type inequalities for partial $ (p, q) $-differentiable convex functions of two variables in the context of $ (p, q) $-calculus. We also show that the findings in this paper are generalizations of comparable findings in the literature.

In this paper, we derive the explicit analytical solution of incommensurate fractional differential equation systems with fractional order $ 1 < \alpha, \beta < 2 $. The derivation is extended from a recently published paper by Huseynov et al. in [1], which is limited for incommensurate fractional order $ 0 < \alpha, \beta < 1 $. The incommensurate fractional differential equation systems were first converted to Volterra integral equations. Then, the Mittag-Leffler function and Picard's successive approximations were used to obtain the analytical solution of incommensurate fractional order systems with $ 1 < \alpha, \beta < 2 $. The solution will be simplified via some combinatorial concepts and bivariate Mittag-Leffler function. Some special cases will be discussed, while some examples will be given at the end of this paper.

In this paper, we investigate the Hyers-Ulam stability of additive and Fréchet functional equations on restricted domains. We improve the bounds and thus the results obtained by S. M. Jung and J. M. Rassias. As a consequence, we obtain asymptotic behaviors of functional equations of different types. One of the objectives of this paper is to bring out the involvement of functional equations in various characterizations of inner product spaces.

In this paper, it is explicitly determined the r-dynamic chromatic number of the direct product of any given path with either a complete graph or a wheel graph. Illustrative examples are shown for each one of the cases that are studied throughout the paper. Junta de Andalucía FQM-016

In this paper, variational discretization directed against the optimal control problem governed by nonlinear parabolic equations with control constraints is studied. It is known that the a priori error estimates is $|||u-u_h|||_{L^\infty(J; L^2(\Omega))} = O(h+k)$ using backward Euler method for standard finite element. In this paper, the better result $|||u-u_h|||_{L^\infty(J; L^2(\Omega))} = O(h^2+k)$ is gained. Beyond that, we get a posteriori error estimates of residual type.

This paper establishes a model of continuous-state branching processes with time inhomogeneous competition in Levy random environments. Some results on extinction are presented, including the distribution of the extinction time, the limiting distribution conditioned on large extinction times and the asymptotic behavior near extinction. This paper also provides a new time-space transformation which can be used for further exploration in similar models. The results are applied to an epidemic model to describe the dynamics of infectious population and a virus model to describe the dynamics of viral load.

Fractal is a common feature of many deterministic complex networks. The complex networks with fractal features have interesting structure and good performance. The network based on hypergraph is named hypernetwork. In this paper, we construct a hypernetwork model with fractal properties, and obtain its topological properties. Moreover, according to the exact controllability theory, we obtain the node controllability and the hyperedge controllability of the fractal hypernetwork. The simulation results show that the measure of hyperedge controllability is smaller than that of node in the fractal hypernetwork. In addition, We compare the controllability of three types of hypernetwork, which are easier to control by their hyperedges. It is shown the fractal hypernetwork constructed in this paper has the best controllability. Because of the good controllability of our fractal hypernetwork model, it is suitable for the topology structure of many real systems.

The mean curvature-based image deblurring model is widely used to enhance the quality of the deblurred images. However, the discretization of the associated Euler-Lagrange equations produce a nonlinear ill-conditioned system which affect the convergence of the numerical algorithms like Krylov subspace methods. To overcome this difficulty, in this paper, we present two new symmetric positive definite (SPD) preconditioners. An efficient algorithm is presented for the mean curvature-based image deblurring problem which combines a fixed point iteration (FPI) with new preconditioned matrices to handle the nonlinearity and ill-conditioned nature of the large system. The eigenvalues analysis is also presented in the paper. Fast convergence has shown in the numerical results by using the proposed new preconditioners.

The main objective of this work is to explore and characterize the idea of $ s $-type preinvex function and related inequalities. Some interesting algebraic properties and logical examples are given to support the newly introduced idea. In addition, we attain the novel version of Hermite-Hadamard type inequality utilizing the introduced preinvexity. Furthermore, we establish two new identities, and employing these, we present some refinements of Hermite-Hadamard-type inequality. Some special cases of the presented results for different preinvex functions are deduced as well. Finally, as applications, some new inequalities for the arithmetic, geometric and harmonic means are established. Results obtained in this paper can be viewed as a significant improvement of previously known results. The awe-inspiring concepts and formidable tools of this paper may invigorate and revitalize for additional research in this worthy and absorbing field.

In this paper, we discuss a novel numerical algorithm for solving boundary value problems. We introduce an orthonormal basis generated from compressed Legendre polynomials. This basis can avoid Runge phenomenon caused by high-order polynomial approximation. Based on the new basis, a numerical algorithm of two-point boundary value problems is established. The convergence and stability of the method are proved. The whole analysis is also applicable to higher order equations or equations with more complex boundary conditions. Four numerical examples are tested to illustrate the accuracy and efficiency of the algorithm. The results show that our algorithm have higher accuracy for solving linear and nonlinear problems.

Aiming at the problem of solving nonlinear ordinary differential equations with variable coefficients, this paper introduces the elastic transformation method into the process of solving ordinary differential equations for the first time. A class of first-order and a class of third-order ordinary differential equations with variable coefficients can be transformed into the Laguerre equation through elastic transformation. With the help of the general solution of the Laguerre equation, the general solution of these two classes of ordinary differential equations can be obtained, and then the curves of the general solution can be drawn. This method not only expands the solvable classes of ordinary differential equations, but also provides a new idea for solving ordinary differential equations with variable coefficients.

In the philosophy of rough set theory, the methodologies of rough soft sets and rough fuzzy sets have been being examined to be efficient mathematical tools to deal with unpredictability. The basic of approximations in rough set theory is based on equivalence relations. In the aftermath, such theory is extended by arbitrary binary relations and fuzzy relations for more wide approximation spaces. In recent years, the notion of picture hesitant fuzzy relations by Mathew et al. can be considered as a novel extension of fuzzy relations. Then this paper proposes extended approximations into rough soft sets and rough fuzzy sets from the viewpoint of its. We give corresponding examples to illustrate the correctness of such approximations. The relationships between the set-valued picture hesitant fuzzy relations with the upper (resp., lower) rough approximations of soft sets and fuzzy sets are investigated. Especially, it is shown that every non-rough soft set and non-rough fuzzy set can be induced by set-valued picture hesitant fuzzy reflexive relations and set-valued picture hesitant fuzzy antisymmetric relations. By processing the approximations and advantages in the new existing tools, some terms and products have been applied to semigroups. Then, we provide attractive results of upper (resp., lower) rough approximations of prime idealistic soft semigroups over semigroups and fuzzy prime ideals of semigroups induced by set-valued picture hesitant fuzzy relations on semigroups.

In this paper, we introduce new type of generalized Kantorovich variant of $ \alpha $-Bernstein operators and study their approximation properties. We obtain estimates of rate of convergence involving first and second order modulus of continuity and Lipschitz function are studied for these operators. Furthermore, we establish Voronovskaya type theorem of these operators. The last section is devoted to bivariate new type $ \alpha $-Bernstein-Kantorovich operators and their approximation behaviors. Also, some graphical illustrations and numerical results are provided.

In this paper, a flexible probability mass function is proposed for modeling count data, especially, asymmetric, and over-dispersed observations. Some of its distributional properties are investigated. It is found that all its statistical properties can be expressed in explicit forms which makes the proposed model useful in time series and regression analysis. Different estimation approaches including maximum likelihood, moments, least squares, Andersonӳ-Darling, Cramer von-Mises, and maximum product of spacing estimator, are derived to get the best estimator for the real data. The estimation performance of these estimation techniques is assessed via a comprehensive simulation study. The flexibility of the new discrete distribution is assessed using four distinctive real data sets ԣoronavirus-flood peaks-forest fire-Leukemia? Finally, the new probabilistic model can serve as an alternative distribution to other competitive distributions available in the literature for modeling count data.

In this paper, we introduce a new class of bi-univalent functions defined in the open unit disc and connected with a $ q $-convolution. We find estimates for the general Taylor-Maclaurin coefficients of the functions in this class by using Faber polynomial expansions, and we obtain an estimation for Fekete-Szegö problem for this class.

This research paper deals with two novel varieties of boundary value problems for nonlinear hybrid fractional differential equations involving generalized fractional derivatives known as the $ \Psi $-Caputo fractional operators. Such operators are generated by iterating a local integral of a function with respect to another increasing positive function $ \Psi $. The existence results to the proposed systems are obtained by using Dhage's fixed point theorem. Two pertinent examples are provided to confirm the feasibility of the obtained results. Our presented results generate many special cases with respect to different values of a $ \Psi $ function.

The main purpose of this research is to establish a new generalized $ \xi^{\ast } $-Kannan type double controlled contraction on a sequence and obtain fixed point results for a pair of multivalued mappings in left $ K $-sequentially complete double controlled dislocated quasi metric type spaces. New results in different setting of generalized metric spaces and ordered spaces and also new results for graphic contractions can be obtained as corollaries of our results. An example is presented to show the novelty of results. In this paper, we unify and extend some recent results in the existing literature.

In this paper, we propose a novel fractional-order delayed financial crises contagions model. The stability, Hopf bifurcation and its control of the established fractional-order delayed financial crises contagions model are studied. A delay-independent sufficient condition ensuring the stability and the occurrence of Hopf bifurcation for the fractional-order delayed financial crises contagions model is obtained. By applying time delay feedback controller, a novel delay-independent sufficient criterion guaranteeing the the stability and the occurrence of Hopf bifurcation for the fractional-order controlled financial crises contagions model with delays is set up.

This paper is devoted to studying the transcendental directions of entire solutions of $ f^{(n)}+A_{n-1}f^{(n-1)}+...+A_0f = 0 $, where $ n(\geq 2) $ is an integer and $ A_i(z)(i = 0, 1, ..., n-1) $ are entire functions of finite lower order. With some additional conditions, the set of common transcendental directions of non-trivial solutions, their derivatives and their primitives must have a definite range of measure. Moreover, we obtain the lower bound of the measure of the set defined by the common transcendental directions of Jackson difference operator of non-trivial solutions.

This paper applies two computational techniques for constructing novel solitary wave solutions of the ill-posed Boussinesq dynamic wave (IPB) equation. Jacques Hadamard has formulated this model for studying the dynamic behavior of waves in shallow water under gravity. Extended simple equation (ESE) method and novel Riccati expansion (NRE) method have been applied to the investigated model's converted nonlinear ordinary differential equation through the wave transformation. As a result of this research, many solitary wave solutions have been obtained and represented in different figures in two-dimensional, three-dimensional, and density plots. The explanation of the methods used shows their dynamics and effectiveness in dealing with certain nonlinear evolution equations.

In this paper, we study the existence of a positive ground state solution for a class of generalized quasilinear Schrödinger equations with asymptotically periodic potential. By the variational method, a positive ground state solution is obtained. Compared with the existing results, our results improve and generalize some existing related results.

In this paper, we introduce the notion of pseudo-semi-normed linear spaces, following the concept of pseudo-norm which was presented by Schaefer and Wolff, and illustrate their relationship. On the other hand, we introduce the concept of fuzzy pseudo-semi-norm, which is weaker than the notion of fuzzy pseudo-norm initiated by N$ \tilde{\rm{a}} $d$ \tilde{\rm{a}} $ban. Moreover, we give some examples which are according to the commonly used $ t $-norms. Finally, we establish norm structures of fuzzy pseudo-semi-normed spaces and provide (fuzzy) topological spaces induced by (fuzzy) pseudo-semi-norms, and prove that the (fuzzy) topological spaces are (fuzzy) Hausdorff.

The main purpose of this paper is using the properties of the classical Gauss sums and the analytic methods to study the computational problem of one kind of character sums analogous to high dimensional Kloosterman sums, and give some interesting identities for it.

In this paper, we consider a two-times nonlinear fractional differential inequality involving both Hadamard-Caputo and Caputo fractional derivatives of different orders, with a singular potential term. We obtain sufficient criteria depending on the parameters of the problem, for which a global solution does not exist. Some examples are provided to support our main results.

In this work, we construct the fractional order model for chemical kinetics issues utilizing novel fractal operators such as fractal fractional by using generalized Mittag-Leffler Kernel. To overcome the constraints of the traditional Riemann-Liouville and Caputo fractional derivatives, a novel notion of fractional differentiation with non-local and non-singular kernels was recently presented. Many scientific conclusions are presented in the study, and these results are supported by effective numerical results. These findings are critical for solving the nonlinear models in chemical kinetics. These concepts are very important to use for real life problems like brine tank cascade, recycled brine tank cascade, pond pollution, home heating and biomass transfer problem. Many scientific results are presented in the paper also prove these results by effective numerical results. These results are very important for solving the nonlinear model in chemistry kinetics which will be helpful to understand the chemical reactions and its actual behavior; also the observation can be developed for future kinematic chemical reactions with the help of these results.

The theory of convex function has a lot of applications in the field of applied mathematics and engineering. The Caputo-Fabrizio non-singular operator is the most significant operator of fractional theory which permits to generalize the classical theory of differentiation. This study consider the well known Hermite-Hadamard type and associated inequalities to generalize further. To full fill this mileage, we use the exponential convexity and fractional-order differential operator and also apply some existing inequalities like Holder, power mean, and Holder-Iscan type inequalities for further extension. The generalized exponential type fractional integral Hermite-Hadamard type inequalities establish involving the global integral. The applications of the developed results are displayed to verify the applicability. The establish results of this paper can be considered an extension and generalization of the existing results of convex function and inequality in literature and we hope that will be more helpful for the researcher in future work.

In this paper, we investigate the spectral element approximation for the optimal control problem of parabolic equation, and present a hp spectral element approximation scheme for the parabolic optimal control problem. For improve the accuracy of the algorithm and construct an adaptive finite element approximation. Under the Scott-Zhang type quasi-interpolation operator, a $ L^2(H^1)-L^2(L^2) $ posteriori error estimates of the hp spectral element approximated solutions for both the state variables and the control variable are obtained. Adopting two auxiliary equations and stability results, a $ L^2(L^2)-L^2(L^2) $ posteriori error estimates are derived for the hp spectral element approximation of optimal parabolic control problem.

The basic objective of this paper is to utilize the factorization technique method to derive several properties such as, shift operators, recurrence relation, differential, integro-differential, partial differential expressions for Gould-Hopper-Frobenius-Genocchi polynomials, which can be utilized to tackle some new issues in different areas of science and innovation.

In this paper, we first study continuous linear functionals on a fuzzy quasi-normed space, obtain a characterization of continuous linear functionals, and point out that the set of all continuous linear functionals forms a convex cone and can be equipped with a weak fuzzy quasi-norm. Next, we prove a theorem of Hahn-Banach type and two separation theorems for convex subsets of fuzzy quasinormed spaces.

In this paper, we introduce a generalized fractional operator in the setting of Hilfer fractional derivatives, the $ \psi $-Hilfer generalized proportional fractional derivative of a function with respect to another function. The proposed operator can be viewed as an interpolator between the Riemann-Liouville and Caputo generalized proportional fractional operators. The properties of the proposed operator are established under some classical and standard assumptions. As an application, we formulate a nonlinear fractional differential equation with a nonlocal initial condition and investigate its equivalence with Volterra integral equations, existence, and uniqueness of solutions. Finally, illustrative examples are given to demonstrate the theoretical results.

In this paper, a dynamic model of Middle East Respiratory Syndrome Coronavirus (MERS-CoV) with periodic coefficients is proposed and studied. By using the continuation theorem of the coincidence degree theory, we obtain some sufficient conditions for the existence of positive periodic solutions of the model. The periodic model degenerates to an autonomous case, and our conditions can be degenerated to the basic reproductive number $ R_0 > 1 $. Finally, we give some numerical simulations to illustrate our main theoretical results.

In this paper, a general HIV model incorporating intracellular time delay is investigated. Taking the latent virus infection, both virus-to-cell and cell-to-cell transmissions into consideration, the model exhibits threshold dynamics with respect to the basic reproduction number $ \mathfrak{R}_0 $. If $ \mathfrak{R}_0 < 1 $, then there exists a unique infection-free equilibrium $ E_0 $, which is globally asymptotically stable. If $ \mathfrak{R}_0 > 1 $, then there exists $ E_0 $ and a globally asymptotically stable infected equilibrium $ E^* $. When $ \mathfrak{R}_0 = 1 $, $ E_0 $ is linearly neutrally stable and a forward bifurcation takes place without time delay around $ \mathfrak{R}_0 = 1 $. The theoretical results and corresponding numerical simulations show that the existence of latently infected cells and the intracellular time delay have vital effect on the global dynamics of the general virus model.

In this paper, we present new Lyapunov-type inequalities for Hilfer-Katugampola fractional differential equations. We first give some unique properties of the Hilfer-Katugampola fractional derivative, and then by using these new properties we convert the multi-point boundary value problems of Hilfer-Katugampola fractional differential equations into the equivalent integral equations with corresponding Green's functions, respectively. Finally, we make use of the Banach's contraction principle to derive the desired results, and give a series of corollaries to show that the current results extend and enrich the previous results in the literature.

A warrant is a financial agreement that gives the right but not the responsibility, to buy or sell a security at a specific price prior to expiration. Many researchers inadvertently utilize call option pricing models to price equity warrants, such as the Black Scholes model which had been found to hold many shortcomings. This paper investigates the pricing of equity warrants under a hybrid model of Heston stochastic volatility together with stochastic interest rates from Cox-Ingersoll-Ross model. This work contributes to exploration of the combined effects of stochastic volatility and stochastic interest rates on pricing equity warrants which fills the gap in the current literature. Analytical pricing formulas for hybrid equity warrants are firstly derived using partial differential equation approaches. Further, to implement the pricing formula to realistic contexts, a calibration procedure is performed using local optimization method to estimate all parameters involved. We then conducted an empirical application of our pricing formula, the Black Scholes model, and the Noreen Wolfson model against the real market data. The comparison between these models is presented along with the investigation of the models' accuracy using statistical error measurements. The outcomes revealed that our proposed model gives the best performance which highlights the crucial elements of both stochastic volatility and stochastic interest rates in valuation of equity warrants. We also examine the warrants' moneyness and found that 96.875% of the warrants are in-the-money which gives positive returns to investors. Thus, it is beneficial for warrant holders concerned in purchasing warrants to elect the best warrant with the most profitable and more benefits at a future date.

In this paper, we have established some new Hermite–Hadamard–Mercer type of inequalities by using $ {\kappa} $–Riemann–Liouville fractional integrals. Moreover, we have derived two new integral identities as auxiliary results. From the applied identities as auxiliary results, we have obtained some new variants of Hermite–Hadamard–Mercer type via $ {\kappa} $–Riemann–Liouville fractional integrals. Several special cases are deduced in detail and some know results are recaptured as well. In order to illustrate the efficiency of our main results, some applications regarding special means of positive real numbers and error estimations for the trapezoidal quadrature formula are provided as well.

Let $ G $ be a connected standard simple algebraic group of type $ C $ or $ D $ over an algebraically closed field $ \Bbbk $ of positive characteristic $ p > 0 $, and $ \mathfrak{g}: = \mathrm{Lie}(G) $ be the Lie algebra of $ G $. Motivated by the variety of $ \mathbb{E}(r, \mathfrak{g}) $ of $ r $-dimensional elementary subalgebras of a restricted Lie algebra $ \mathfrak{g} $, in this paper we characterize the irreducible components of $ \mathbb{E}(\mathrm{rk}_{p}(\mathfrak{g})-1, \mathfrak{g}) $ where the $ p $-rank $ \mathrm{rk}_{p}(\mathfrak{g}) $ is defined to be the maximal dimension of an elementary subalgebra of $ \mathfrak{g} $.

SARS-COV-2 (Coronavirus) viral growth kinetics within-host become a key fact to understand the COVID-19 disease progression and disease severity since the year 2020. Quantitative analysis of the viral dynamics has not yet been able to provide sufficient information on the disease severity in the host. The SARS-CoV-2 dynamics are therefore important to study in the context of immune surveillance by developing a mathematical model. This paper aims to develop such a mathematical model to analyse the interaction between the immune system and SARS-CoV-2 within the host. The model is developed to explore the viral load dynamics within the host by considering the role of natural killer cells and T-cell. Through analytical simplifications, the model is found well-posed and asymptotically stable at disease-free equilibrium. The numerical results demonstrate that the influx of external natural killer (NK) cells alone or integrating with anti-viral therapy plays a vital role in suppressing the SARS-CoV-2 growth within-host. Also, within the host, the virus can not grow if the virus replication rate is below a threshold limit. The developed model will contribute to understanding the disease dynamics and help to establish various potential treatment strategies against COVID-19.

In this paper, we consider the common Re-nonnegative definite (Re-nnd) and Re-positive definite (Re-pd) solutions to a pair of linear matrix equations $ A_1XA_1^\ast = C_1, \ A_2XA_2^\ast = C_2 $ and present some necessary and sufficient conditions for their solvability as well as the explicit expressions for the general common Re-nnd and Re-pd solutions when the consistent conditions are satisfied.

This paper investigates the event-triggered state estimation problem for a class of complex networks (CNs) suffered by hybrid cyber-attacks. It is assumed that a wireless network exists between sensors and remote estimators, and that data packets may be modified or blocked by malicious attackers. Adaptive event-triggered scheme (AETS) is introduced to alleviate the network congestion problem. With the help of two sets of Bernoulli distribution variables (BDVs) and an arbitrary function related to the system state, a mathematical model of the hybrid cyber-attacks is developed to portray randomly occurring denial-of-service (DoS) attacks and deception attacks. CNs, AETS, hybrid cyber-attacks, and state estimators are then incorporated into a unified architecture. The system state is cascaded with state errors as an augmented system. Furthermore, based on Lyapunov stability theory and linear matrix inequalities (LMIs), sufficient conditions to ensure the asymptotic stability of the augmented system are derived, and the corresponding state estimator is designed. Finally, the effectiveness of the theoretical method is demonstrated by numerical examples and simulations.

Physical constraints must be taken into account in solving partial differential equations (PDEs) in modeling physical phenomenon time evolution of chemical or biological species. In other words, numerical schemes ought to be devised in a way that numerical results may have the same qualitative properties as those of the theoretical results. Methods with monotonicity preserving property possess a qualitative feature that renders them practically proper for solving hyperbolic systems. The need for monotonicity signifies the essential boundedness properties necessary for the numerical methods. That said, for many linear multistep methods (LMMs), the monotonicity demands are violated. Therefore, it cannot be concluded that the total variations of those methods are bounded. This paper investigates monotonicity, especially emphasizing the stepsize restrictions for boundedness of A-BDF methods as a subclass of LMMs. A-stable methods can often be effectively used for stiff ODEs, but may prove inefficient in hyperbolic equations with stiff source terms. Numerical experiments show that if we apply the A-BDF method to Sod's problem, the numerical solution for the density is sharp without spurious oscillations. Also, application of the A-BDF method to the discontinuous diffusion problem is free of temporal oscillations and negative values near the discontinuous points while the SSP RK2 method does not have such properties.

Throughout the paper, we introduce a new subclass $ \mathcal{H}_{\alpha, \mu, \rho, m, \beta }^{n, q, \lambda, l}\ f(z)$ by using the Bazilevič functions with the idea of bounded boundary rotation and $ q $-analogue Cătaş operator. Also we find the estimate of the coefficients for functions in this class. Finally, in the concluding section, we have chosen to reiterate the well-demonstrated fact that any attempt to produce the rather straightforward $ (p, q) $-variations of the results, which we have presented in this article, will be a rather trivial and inconsequential exercise, simply because the additional parameter $ p $ is obviously redundant.

In this paper, we study a boundary value problem consisting of Hahn integro-difference equation supplemented with four-point fractional Hahn integral boundary conditions. The novelty of this problem lies in the fact that it contains two fractional Hahn difference operators and three fractional Hahn integrals with different quantum numbers and orders. Firstly, we convert the given nonlinear problem into a fixed point problem, by considering a linear variant of the problem at hand. Once the fixed point operator is available, we make use the classical Banach's and Schauder's fixed point theorems to establish existence and uniqueness results. An example is also constructed to illustrate the main results. Several properties of fractional Hahn integral that will be used in our study are also discussed.

The novel corona virus (COVID-19) has badly affected many countries (more than 180 countries including China) in the world. More than 90% of the global COVID-19 cases are currently outside China. The large, unanticipated number of COVID-19 cases has interrupted the healthcare system in many countries and created shortages for bed space in hospitals. Consequently, better estimation of COVID-19 infected people in Sri Lanka is vital for government to take suitable action. This paper investigates predictions on both the number of the first and the second waves of COVID-19 cases in Sri Lanka. First, to estimate the number of first wave of future COVID-19 cases, we develop a stochastic forecasting model and present a solution technique for the model. Then, another solution method is proposed to the two existing models (SIR model and Logistic growth model) for the prediction on the second wave of COVID-19 cases. Finally, the proposed model and solution approaches are validated by secondary data obtained from the Epidemiology Unit, Ministry of Health, Sri Lanka. A comparative assessment on actual values of COVID-19 cases shows promising performance of our developed stochastic model and proposed solution techniques. So, our new finding would definitely be benefited to practitioners, academics and decision makers, especially the government of Sri Lanka that deals with such type of decision making.

In this paper, we study the summability of solutions to the following semilinear elliptic equations involving mixed local and nonlocal operators \begin{document}$ \left\{ \begin{matrix} - \Delta u(x)+{{(-\Delta )}^{s}}u(x)=f(x), & x\in \Omega , \\ u(x)\ge 0,~~~~~ & x\in \Omega , \\ u(x)=0,~~~~~ & x\in {{\mathbb{R}}^{N}}\setminus \Omega , \\ \end{matrix} \right. $\end{document} where $ 0 < s < 1 $, $ \Omega\subset \mathbb{R}^N $ is a smooth bounded domain, $ (-\Delta)^s $ is the fractional Laplace operator, $ f $ is a measurable function.

The $ r $-Lah numbers generalize the Lah numbers to the $ r $-Stirling numbers in the same sense. The Stirling numbers and the central factorial numbers are one of the important tools in enumerative combinatorics. The $ r $-Lah number counts the number of partitions of a set with $ n+r $ elements into $ k+r $ ordered blocks such that $ r $ distinguished elements have to be in distinct ordered blocks. In this paper, the $ r $-central Lah numbers and the $ r $-central Lah-Bell numbers ($ r\in \mathbb{N} $) are introduced parallel to the $ r $-extended central factorial numbers of the second kind and $ r $-extended central Bell polynomials. In addition, some identities related to these numbers including the generating functions, explicit formulas, binomial convolutions are derived. Moreover, the $ r $-central Lah numbers and the $ r $-central Lah-Bell numbers are shown to be represented by Riemann integral, respectively.

The aim of this paper is to combine the innovative concept of single valued neutrosophic sets and ordered semirings. It studies ordered semirings by the properties of their single valued neutrosphic subsets. In this regard, we define single valued neutrosophic $ (m, n) $-ideals (SVN-$ (m, n) $-ideals) of ordered semirings. First, we illustrate our new definition by non-trivial examples. Second, we study these SVN-$ (m, n) $-ideals under different operations of SVNS. Finally, we find a relationship between the $ (m, n) $-ideals of ordered semirings and level sets by finding a necessary and sufficient condition for an SVNS of an ordered semiring $ R $ to be an SVN-$ (m, n) $-ideal of $ R $.