This article addresses the derivation and analysis of an optimal quadrature formula for numerical integration of fractional integrals in the Hilbert space W 2 ( 2 , 1 ) ( t , 1 ) , where functions φ with prescribed properties reside. The quadrature formula is expressed as a linear combination of function values and their first-order derivative at equidistant nodes in the interval [ t,1]. The coefficients are determined by minimizing the norm of the error functional in the dual space W 2 ( 2 , 1 ) ∗ ( t , 1 ) . The error functional is defined as the difference between the integral of a function over the interval and the quadrature approximation. Key results include explicit expressions for the coefficients and the norm of the error functional. The optimization problem is formulated and solved, leading to a system of linear equations for the coefficients. Analytical solutions of the system are obtained, providing an explicit expression for the optimal coefficients. Fractional integrals of several functions are numerically calculated with the constructed optimal quadrature formula, and the convergence with the exact value of the integral is analyzed in numerical experiments.

The axially-symmetric solutions to the Navier-Stokes equations coupled with the heat conduction are considered. in a bounded cylinder Ω ⊂ R 3 . We assume that v r , v φ , ω φ vanish on the lateral part S 1 of the boundary ∂Ω and v z , ω φ , ∂ z v φ vanish on the top and bottom of the cylinder, where we used standard cylindrical coordinates and ω = rot v is the vorticity of the fluid. Moreover, vanishing of the heat flux through the boundary is imposed. Assuming existence of a sufficiently regular solution we derive a global a priori estimate in terms of data. The estimate is such that a global regular solutions can be proved. We prove the estimate because some reduction of nonlinearity are found. Moreover, we need that f ( p ) ≡ ∥ v φ ∥ L t ∞ L x p / ∥ v φ ∥ L t ∞ L x ∞ is bounded from below by a positive constant. The quantity f( p) is close to 1 for large p because f(∞)=1. Moreover, deriving the global estimate for a local solution implies a possibility of its extension in time as long as the estimate holds.

Unemployment remains a major challenge in many developing countries in the world, including Kenya. It is worthy noting that, youths unemployment rate is high and mismatch of skills in the job market is phenomenal. This paper develops a deterministic mathematical model to assess the role of post-secondary education on the unemployment problem in Kenya. We perform a thorough quantitative and qualitative analysis of the model. We compute the unemployment reproduction R u , prove that the model has a unique unemployment free equilibrium (UFE) when R u < 1 and admits a unique unemployment endemic equilibrium (UEE) whenever R u > 1 . The unique UFE and UEE are both locally and globally asymptotically stable whenever R u < 1 and R u > 1 respectively. Further, we validate the model by fitting it to real data of unemployed persons in Kenya in the year 1991−2023 as reported by the International Labour Organization (ILO). Numerical results indicate that, an increased individual’s skills match to the job market demands, significantly decreases the number of unemployed persons and increases the number of employed individuals. Moreover, it is demonstrated that, increasing the rate of employment to the unemployed persons reduces the problem of unemployment substantially. This can be achieved by setting up more sectors for job opportunities.

In this paper, a class of hematopoietic stem cell transplantation model with virus-to-cell HIV infection is proposed to characterize the competitive exclusion and coexistence between the host CD4+T cells and donor CD4+T cells. First, the positivity and boundedness of solutions as well as the basic reproduction number R are obtained. Second, criteria on the locally and globally asymptotical stability of all feasiable equilibria are established. Furthermore, bifurcation analysis is performed on the mixed chimerism infection equilibrium. Finally, the theoretical results are illustrated by numerical simulation, we find that chimerism is an important indicator of model stability, and AIDS may be cured when chimerism reaches a certain threshold.

In this paper, we propose a review of the free boundary formulation for BVPs defined on semi-infinite intervals. The main idea and theorem are illustrated, for the reader convenience, by using a class of second-order BVPs. Moreover, we are able to show the effectiveness of the proposed approach using two examples where the exact solution both for the BVPs and their free boundary formulation are available. Then, we describe the free boundary formulation for a general class of BVPs governed by an n-order differential equation. In this context, we report three problems solved using the free boundary formulation. The reported numerical results, obtained by the iterative transformation method or Keller’s second-order finite difference method, are found to be in very good agreement with those available in the literature. The last result of this research is that, in order to orient the interested reader, we provide an extensive bibliography. Of course, we may aspect further and more interesting applications of the free boundary formulation in the future.

Explicit formulas to calculate MV functions in a basis-free representation are presented for an arbitrary Clifford geometric algebra Cl p , q . The formulas are based on analysis of the roots of minimal MV polynomial and covers defective MVs, i.e. the MVs that have non-diagonalizable matrix representations. The method may be generalized straightforwardly to matrix functions and to finite dimensional linear operators. The results can find wide application in Clifford algebra analysis.

In this paper, we introduce a new family of Baskakov-Kantorovich operators that depend on a function ψ. We compare these new ψ-Baskakov-Kantorovich operators with the classical Baskakov-Kantorovich operators to evaluate their approximation results. Our analysis shows that these new operators provide better approximation results across the entire interval [0 ,∞). We demonstrate their uniform convergence in weighted spaces and determine their convergence rates using both first and second-order moduli of continuity. We also prove that these operators preserve shape preserving properties. We support our findings with graphical and numerical examples.

Abstract: In this paper we investigate the asymptotic stability of a fourth-order PDE with a fading memory forcing term and boundary conditions arising from a flexible robotics model. We carry on our study by using an abstract formulation of the problem based on the C 0 -semigroup. To achieve our objective, we first provide new results on the existence, uniqueness, continuous dependence on initial data of either mild and strong solutions for semilinear integro-differential equations in Banach spaces. Then, we also find sufficient conditions for the uniform asymptotic stability of solutions and for the existence of attactors. As an application of these abstract results, we can ensure existence, uniqueness and continuous dependence on initial data for the solutions of the boundary value problem under investigation and, finally, we prove the uniform asymptotic stability of solutions and the existence of attactors under suitable conditions on the nonlinear term.

In this paper, we study the time asymptotic behavior of solutions toward rarefaction wave for the impermeable wall problems of the one-dimensional isentropic compressible Navier-Stokes-Allen-Cahn system with density and phase field variable dependent viscosity on the half line R + = ( 0 , + ∞ ) , which models the motion of a mixture of two viscous compressible fluids. We consider the case when the viscosity coefficient ν ( ρ , χ ) = ρ α χ β and the pressure p ( ρ ) = ρ γ , where ρ= ρ( t,x) is the density of the fluid, χ= χ( t,x) is the phase field variable, and α,β,γ∈R are parameters. Under some suitable assumptions on the boundary data and spatial-asymptotic state of the velocity field, the time asymptotic profile of these problems is the 2-rarefaction wave of the resulting Euler system. By employing the elementary L 2 -energy method, it is shown that this 2-rarefaction wave is time-asymptotically stable provided that the parameters α,β,γ and the initial perturbation satisfy some conditions. Here the initial perturbation for the phase field variable is small, but the initial perturbation for the density and velocity of the fluid, and the strength of the rarefaction wave can be arbitrarily large.

In this paper, we consider the H e ̵́ non equation { − ∆ u = | x | α u 2 ∗ − 1 + ϵ , u > 0 , in Ω , u = 0 , on ∂ Ω . where Ω is a bounded smooth domain in R N containing the origin, N≥3 and ϵ is a positive constant. M. Ben Ayed et.[2] proved the equation without | x | α doesn’t have the single peaked solution. Moreover, there are few works on the H e ̵́ non equation in a general domain. Inspired by them, we proved the non-existence of the single peak solution to the H e ̵́ non problem by Pohozaev identities.

This work is the demonstration of a two-storage inventory model for deteriorating items, considering bi-level trade credit facility. The customers’ demand is dependent on selling price of the item and credit period offered by the retailer. Further, the items are deteriorating at a constant rate during stock-in situation but the deterioration rates of the items in different warehouse are different. According to the credit period provided by the supplier to retailer, three distinct scenarios may arise among which are discussed in detail. The corresponding optimization problems are solved with the help of three tournament-based hybrid algorithms (tournament differential evolution (TDE), tournament teaching learning based optimization (TTLBO) and tournament RAO-3 algorithms). Further, the feasibility of the proposed model is validated by considering a numerical example. Finally, the impacts of different system parameters on optimal policy are studied with the help of sensitivity experiment and its impacts are shown graphically.

To define the optimistic multiple-granulation(OMG) supra interior operator i n t R A and the OMG-supra closure operator c l R A associated with a set A in a multiple-granulation approximation space, we used the concepts of the OMG-lower set O L Σ m i = 1 R i ( A ) and the OMG-upper set O U Σ m i = 1 R i ( A ) . These operators establish an approximate topological space referred to as optimistic multiple-granulation supra topological spaces. Additionally, we introduce the concepts of OMG-supra-open and OMG-supra-closed approximation sets, as well as OMG-supra-continuous and OMG-irresolute approximations.

The paper investigates the dispersion of quasi-Lamb waves in a hydro-elastic system consisting of a plate of multilayer composite material and a compressible, inviscid fluid layer whose flow is confined by a rigid wall. The multilayer composite material is modeled as a transversely isotropic material whose axis of symmetry is aligned along the thickness of the plate. The motion of the plate is described by the exact equations and relations of elastodynamics for anisotropic bodies, but the fluid flow is described by the linearized Euler equations. The analytical solutions of these equations are found and according to these solutions the dispersion equations are obtained from the corresponding boundary, compatibility and impermeability conditions. The dispersion equation is solved numerically and the numerical results are obtained for the case where the plate material consists of two alternating layers. Based on these results, a parametric study of the influence of the density ratio of the plate and fluid materials and the ratio of the shear wave propagation velocity in the stiffer material layer of the plate to the sound velocity of the fluid on the dispersion curves with respect to the first four subsequent modes (denoted as A 0, S 0, A 1 and S 1 modes) of the quasi-Lamb waves is performed. The influence of the ratio of the shear moduli of the components of the multilayer material and the influence of the contact of the plate with the fluid are investigated.

In this paper we consider the following equation involving uniformly elliptic nonlocal operator in the upper half space { A 2 s u ( x )= f ( u ( x )) , x ∈ R + n , u ( x ) > 0 , x ∈ R + n , u ( x )= 0 , x ∉ R + n . We first develop a narrow region principle for antisymmetric functions in unbounded domains, in which we assume that u has polynomial growth instead of the usual decay condition u→0 at infinity, which is the improvement of the partial result of Wu-Qu-Zhang-Zhang[Math Meth Appl Sci. 2023;46:3721-3740]. Then we obtain the monotonicity of positive solutions in the upper half space by a direct method of moving planes, which extends the monotonic result of Li[Adv. Nonlinear Stud., 24 (2024), 451-462]. Based on the monotonicity results, we prove the existence of positive bounded solutions for uniformly elliptic nonlocal equation on the whole space. We believe that these methods will also be helpful for equations involving other non local operators.

We establish necessary and sufficient conditions for a weight v guaranteeing the boundedness of multilinear Riemann-Liouville operators from ∏ j = 1 m L p j to L v q .

We offer a detailed treatment of minimal, maximal, dissipative, accumulative, self‐adjoint operator realizations with exit from space of boundary–value problem for Sturm‐Liouville equation with unbounded operator coefficient having discrete spectrum andwith boundary condition dependent on Herglotz‐Nevanlinna function of eigenparameter. We also study self‐adjoint operator realizations having purely discrete spectrum or having continuous spectrum which coincide with any interval on real axis. That is done in terms of boundary conditions. In addition, in particular case we obtain asymptotics of spectrum of self‐adjoint operators with discrete spectrum.

Structure-preserving two-step linearly implicit exponential integrators are constructed for Hamiltonian partial differential equations with linear constant damping combining exponential integrators and polarization of the polynomial Hamiltonian function. We also construct an exponential version of the well-known one-step Kahan’s method by polarizing the quadratic vector field. These integrators are applied to one-dimensional damped Burger’s, Korteweg-de Vries, and nonlinear Schrödinger equations. Preservation of the dissipation rate is demonstrated for linear, quadratic conformal invariants and of the Hamiltonians by numerical experiments.

Onchocerciasis also known as River Blindness, is a vector-borne disease caused by Onchocerca volvulus and transmitted by infected female blackflies. It affects millions of people globally, with the greatest impact in sub-Saharan Africa. In this study, we develop a deterministic mathematical model that integrates multiple control measures, including Sterile Insect Technique (SIT), mechanical control, chemical control, public health education and ivermectin treatment, to manage the transmission of onchocerciasis. We employ the next-generation matrix method to calculate the blackfly offspring reproduction number N 0 and the basic reproduction number R 0 . Sensitivity analysis, conducted using the normalized forward sensitivity index, highlights the biting rate as the most positive influence on driving onchocerciasis dynamics, while the mortality rate of female blackflies has a significant negative impact on disease containment. To identify the optimal control strategy, we apply optimal control theory, considering five time-dependent controls which are public health education, treatment, mechanical control, SIT and chemical control. Using Pontryagin’s maximum principle, we derive the optimality system for controlling onchocerciasis. By implementing forward-backward Runge-Kutta method in Matlab, we identify the most optimal strategy for controlling, preventing and treating onchocerciasis in both human and blackfly populations. The results suggest that a combined strategy focusing on public health education, treatment and SIT offers the most effective approach for combating onchocerciasis.