This paper investigates the uniform continuity and strong continuity of the semigroups of Katugampola integral operators.Using the Krasnoselskii ′s fixed point theorem, we study the nonlocal problem of Katugampola fractional differential equations with multi-point integral boundary conditions and obtain the uniqueness of the solution.

We study in this article the blow-up of solutions to a coupled semilinear wave equations which are characterized by linear damping terms in the scale-invariant regime, time-derivative nonlinearities, mass and Tricomi terms. The latter are specifically of great interest from both physical and mathematical points of view since they allow the speeds of propagation to be time-dependent ones. However, we assume in this work that both waves are propagating with the same speeds. Employing this fact together with other hypotheses on the aforementioned parameters (mass and damping coefficients), we obtain a new blow-up region for the system under consideration, and we show a lifespan estimate of the maximal existence time.

The asymptotic stability of the one-dimensional attraction-repulsion chemotaxis system is concerned with in this paper. First of all, we get the globally bounded classical solutions of the system with initial data ( u 0 , v 0 , ω 0 ) ∈ ( W 1 , ∞ ( Ω ) ) 3 . Next, by constructing an appropriate Lyapunov function, we also show that the solutions converge exponentially to the constant steady state ( u ̵̄ 0 , α β u ̵̄ 0 , γ δ u ̵̄ 0 ) if ξγ χα > max { β δ , δ β } . It is worth mentioning that Jin and Wang (2015) showed that the solutions converge algebraically to the constant steady state under conditions β= δ and ξγ− χα<0, and our results filled the gap in the ξγ− χα>0.

javascript:void(0) This study presents a novel predator-prey model that integrates disease dynamics within the prey population, using a Susceptible-Infected (SI) framework. The model introduces two control variables: one aimed at reducing disease transmission among prey and another focused on managing predator population levels. A logarithmic penalty cost function is employed to penalize reductions in prey and predator populations, reflecting the heightened ecological risks associated with low population sizes. The system is described by a set of differential equations that model the interactions between susceptible prey, infected prey, and predators under the influence of these control measures. The goal is to minimize the cost function over a specified time horizon, ensuring ecosystem stability while effectively managing disease spread and predator numbers. Our results provide insights into optimal control strategies that maintain ecological balance, offering a framework applicable to real-world wildlife management and conservation scenarios.

not-yet-known not-yet-known not-yet-known unknown The study of parameter-dependent partial differential equations (parametric PDEs) with countably many parameters has been actively studied for the last few decades. In particular, it has been well known that a certain type of parametric holomorphy of the PDE solutions allows the application of deep neural networks without encountering the curse of dimensionality. This paper aims to propose a general framework for verifying the desired parametric holomorphy by utilizing the bounds on parametric derivatives. The framework is illustrated with examples of parametric elliptic eigenvalue problems (EVPs), encompassing both linear and semilinear cases. As the results, it will be shown that the ground eigenpairs have the desired holomorphy. Furthermore, under the same conditions, improved bounds for the mixed derivatives of the ground eigenpairs are derived. These bounds are well known to take a crucial role in the error analysis of quasi-Monte Carlo methods.

The transmission of monkeypox is studied using a stochastic model taking into account the biological aspects, the contact mechanisms and the demographic factors together with the intrinsic uncertainties. Our results provide insight into the interaction between stochasticity and biological elements in the dynamics of monkeypox transmission. The rigorous mathematical analysis determines threshold parameters for disease persistence. For the proposed model, the existence of a unique global almost sure non-negative solution is proven. Conditions leading to disease extinction are established. Asymptotic properties of the model are investigated such as the speed of transmission.

In this paper, we conduct a thorough analysis for a group of stochastic Hilfer fractional delay differential equations (SHFDDEs) with Lipschitz parameters. By introducing innovative theoretical conditions, we successfully establish the existence and uniqueness solutions using the Carath e ̵́ odory approximation approach. Furthermore, we deduce the average principle for accordingly system by employing H o ̵̈ lder’s inequality, Jensen’s inequality, It o ˆ isometric distance, and Gronwall’s inequality. Ultimately, two demonstrative examples are undertaken to substantiate the efficacy and practical applicability of our findings.

We present and analyze a second order numerical scheme with variable time steps for a liquid thin film coarsening model, which is a Cahn-Hilliard-type equation with a singular Leonard-Jones energy potential. The fully discrete scheme is mainly based on the Backward differentiation formula (BDF) method in time derivation combined with the finite difference method in spacial discretization. A second order viscous regularization term is added at the discrete level to guarantee the energy dissipation property under the condition that r ≤ r max . The uniquely solvable and positivity-preserving properties of the numerical solution are established at a theoretical level. In addition, based on the strict separation property of the numerical solution obtained by using the technique of combining the rough and refined error estimates, the optimal rate convergence analysis in ℓ ∞ ( 0 , T ; H h − 1 ) norm is established when τ≤ Ch by using the technique of the discrete orthogonal convolution(DOC) kernels. Finally, several numerical experiments are carried out to validate the theoretical results.

We show that any bounded smooth helically symmetric solution ( u , h ) in R 3 must be constant vectors. This is an extension of previous result[17, Theorem 1.1] from Navier-Stokes system to MHD system. The proof relies on establishing a Saint-Venant type estimate to characterize the growth of Dirichlet integral of nontrivial solutions.

We consider a class of p-Laplacian quasilinear Schrödinger equations { − ∆ p u − p 2 p − 1 u ∆ p ( u 2 )= λ u − γ + u q in Ω , u > 0 in Ω , u = 0 on ∂ Ω , where Ω ⊂ R N is a bounded domain with regular boundary, 1 ∞, 0 1, 2 p − 1 < q ≤ 2 · p ∗ − 1 for p≤ N, 2 p−1 ∞ for p>N, where p ∗ = Np N − p if 1 , p ∗ ∈ ( p , ∞ ) is arbitrarily large if p= N, p ∗ = ∞ if p>N. We establish global existence and multiplicity results of positive solutions via a new strong comparison principle and a regularity result for weak solutions.

In this paper, we proposed a fully discrete projection method with modular grad-div stabilization for solving the time-dependent inductionless magnetohydrodynamic equations. The method incorporates a minimally intrusive module into the classical projection method, serving as a post-processing step, thereby enhancing solution accuracy and improving mass conservation. Concurrently, a decoupled strategy is employed to separate the magnetic and fluid field functions from the original system. Therefore, at each time step, we only need to solve several linear sub-systems for which the numerical solutions can be obtained efficiently. For spatial discretization, the current density and electric potential are discretized by H ( div , Ω ) × L 2 ( Ω ) -conforming finite element pair, which ensures that the discrete current density is exactly divergence-free. Therefore, the designed numerical scheme maintains the features of linearization, decoupling, unconditional energy stability, charge conservation, and improved mass conservation. The unconditional energy stability and convergence of the algorithm are analyzed and derived. Numerical results are presented to verify that the algorithm exhibit robustness with respect to the stabilization parameters and demonstrate the performance of the scheme, particularly with respect to its stability and accuracy.

Fractals have been studied in areas such as mathematics, physics, chemistry, social sciences, computing, economics and biology. Fractal networks have many interesting properties, such as recursive selfsimilarity, that are present in many real networks. In this paper we study the geometrical and topological properties of fractal networks (Sierpiński triangle, the Sierpiński carpet and the Koch snowflake). Furthermore, we establish relationships, in the fractal structures studied, between the geometric properties associated with hyperbolicity and the topological indices.

Models of infectious disease dynamics suggest that treatment, vaccination, and isolation are required for the control of infectious diseases. Considering that vaccination is one of the most effective methods to control infectious diseases, it is often not possible to rapidly vaccinate all susceptible populations in the early stages of the spread of infectious diseases due to the limitation of the number of vaccines, insufficient medical personnel, or the slow progress of vaccination efforts. Our simulation analysis by building an SVIWRD model found that the degree of negative impact of infectious diseases shown when young and old people were divided into two populations and vaccinated at different rates was different.Therefore, for the current problem of continued spread of COVID-19, we consider the infectious disease dynamics model to achieve the goal of making the risk of infection of COVID-19 lower by controlling the proportion of vaccination of elderly and young people. In this paper, we divided young and old people into two groups, established an SVIWRD model, performed single-objective optimization using Pontryagin's maximum principle, and used the Runge-Kutta method for numerical calculation and simulation, so as to arrive at a certain vaccination ratio that plays the effect of reduced negative impact of COVID-19.

For linked Fractional modified Korteweg-de Vries (mKdV) systems, in which the coefficient is a time-dependent variable, we investigate the exact multiple soliton solutions. Based on the similarity transformation and Hirota bilinear technique, we report both multiple wave kink and wave single kink solutions for two different models of fractional mKdV with time dependent variable coefficient. We use the fractional Hirota bilinear technique to compute analytical solutions for modified coupled space–time–fractional KdV systems. We construct many kink waves for the proposed fractional differential models that are being studied. For the treatment of nonlinear differential models of integer and fractional orders, the Hirota bilinear technique provides a straightforward and promising method. Recently, researchers have been using symbolic computation—like maple—to perform these calculations. We investigated if the results demonstrate the simplicity, effectiveness, and ease of computation of the approach for a range of engineering and physics models. The flexible and random selection of the fractional orders allows us to build deeper structures. Soliton modifications based on fractional order changes enable further applications in the applied science

This research deals with the Kaup-Newell model (KNM), a class of nonlinear Schrödinger equations with important applications in plasma physics and nonlinear optics. Soliton solutions are essential for analyzing nonlinear wave behaviors in different physical systems, and the KNM is also significant in this context. The model’s ability to represent sub-picosecond pulses makes it a significant tool for the research of nonlinear optics and plasma physics. Overall, the KNM is an important research domain in these areas, with ongoing efforts focused on understanding its various solutions and potential applications. A new version of the generalized exponential rational function method (nGERFM) and ( G ′ G 2 ) -expansion function methods are utilized to discover diverse soliton solutions. The nGERFM facilitates the generation of multiple solution types, including singular, shock, singular periodic, exponential, combo trigonometric, and hyperbolic solutions in mixed forms. Thanks to ( G ′ G 2 ) -expansion function method, we obtain trigonometric, hyperbolic, and rational solutions. The modulation instability (MI) of the proposed model is examined, with numerical simulations complementing the analytical results to provide a better understanding of the solutions’ dynamic behavior. These results offer a foundation for future research, making the solutions effective, manageable, and reliable for tackling complex nonlinear problems. The methodologies used in this study are robust, influential, and practicable for diverse nonlinear partial differential equations (NLPDEs); to our knowledge, for this equation, these methods of investigation have not been explored before. The accuracy of each solution has been verified using the Maple software program.

In this manuscript, we elucidate essential connections associated with the ( α , k ) − gamma and ( α , k ) − beta functions, initially introduced by Sarikaya et al. in their work as referenced in [14]. Our investigation includes the establishment of numerous conformable fractional integral inequalities, extending those previously established for the k−gamma and k−beta functions. Furthermore, we provide rigorous proofs affirming the log-convex nature of both the ( α , k ) − gamma and ( α , k ) − beta functions.

We consider conditions for the decay in time of solutions of non-homogenous hyperbolic equations. It is proven that solutions of the equations go to 0 in L 2 at infinity if and only if an equation’s right-hand side uniquely determines the initial conditions in a certain way. We also obtain that a hyperbolic equation has a unique solution that fades when t→∞.

Efficient handling of perishable medication inventory is crucial for maintaining healthcare system operations. It prioritizes patient needs while also aiming to minimize costs and environmental impact. This study enhanced a production inventory model to address the distinct challenges like deterioration, preservation technology, and carbon emissions associated with perishable medications. With these considerations, a mathematical model has developed an optimal production inventory that balances cost efficiency with environmental sustainability. Finally, the sensitivity analysis and results validated the model's effectiveness across diverse scenarios.