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

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.

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.

Bacteriophages, or phages (viruses of bacteria), play significant roles in shaping the diversity of bacterial communities within the human gut. A phage-infected bacterial cell can either immediately undergo lysis (virulent/lytic infection) or enter a stable state within the host as a prophage (lysogeny) until a trigger event, called prophage induction, initiates the lysis process. We develop an approach based on a model structured in terms of time since bacterial infection. We derive important threshold parameters for the asymptotic dynamics of the system and demonstrate that the model’s qualitative behavior can range from the extinction of all bacterial strains to the persistence of a single strain (either lysogen or non-lysogen bacteria) or the coexistence of all strains at a positive steady state. We highlight the existence of critical induction rate values that lead to the coexistence of all states through periodic oscillations. We also conduct a global sensitivity analysis for an effective bacterial clearance. In scenarios where antibiotics are not sufficiently effective, we identify four key phage parameter traits: (i) the phage induction probability, describing the capacity of prophages to be induced, (ii) the probability of absorption, describing the phages’ ability to invade susceptible bacteria, (iii) the reproduction number of susceptible bacteria in the absence of antibiotics, and (iv) the latent period, describing the time since absorption. The obtained results emphasize the effective therapeutic potential of selected phages.

In this work, we introduce a new form of the quaternionic fractional uncertainty relation within the framework of quaternionic quantum mechanics. This is closely associated with the Li-Ostoja-Starzewski fractional gradient operator, characterized by an order range of 0 ≤1. We explore a novel Quaternionic Schrödinger equation and its specific implications, particularly addressing solutions that lead to the emergence of position-dependent mass. Additionally, we validate the theory by comparing it against the observed maximum wavelengths in the 1,3,5-hexatriene molecule.

One of the crucial and major issues in engineering, particularly in signal processing, is signal reconstruction. Frames are flexible tools that helps to reconstruct signals/vectors in a stable way. Frame theory was introduced for Hilbert spaces by Duffin et al. in 1952. If it happens that we only have the intensity measurements or the phaseless measurements of the lost signal then reconstructing the original signal becomes difficult. In such cases, phase retrieval sequences play essential roles to reconstruct or regain the signal from its intensity measurements or phaseless measurements. In 2006, Balan et al. introduced phase retrievable frames for Hilbert spaces. Phase retrieval has received significant attention in various fields, including image processing and signal reconstruction. Similar to phase retrieval, norm retrieval sequences help to regain the norm of the original signal. In 2015 Bahmanpour et al. introduced norm retrieval sequences for Hilbert spaces. We provide a thorough overview of phase retrieval and norm retrieval by vectors and subspaces in separable Hilbert space. We also highlight and discuss the recent developments in phase retrieval and norm retrieval of partially lost signals.

This paper proves the existence of nontrivial solution for two classes of quasilinear systems of the type { − ∆ Φ 1 u = F u ( x , u , v )+ λ R u ( x , u , v ) in Ω − ∆ Φ 2 v = − F v ( x , u , v ) − λ R v ( x , u , v ) in Ω u = v = 0 on ∂ Ω where λ>0 is a parameter, Ω is a bounded domain in R N ( N≥2) with smooth boundary ∂Ω. The first class we drop the ∆ 2 -condition of the functions Φ ̵̃ i ( i=1 ,2) and assume that F has a double criticality. For this class, we use a linking theorem without the Palais-Smale condition for locally Lipschitz functionals combined with a concentration–compactness lemma for nonreflexive Orlicz-Sobolev space. The second class, we relax the ∆ 2 -condition of the functions Φ i ( i=1 ,2). For this class, we consider F=0 and λ=1 and obtain the proof based on a saddle-point theorem of Rabinowitz without the Palais-Smale condition for functionals Fréchet differentiable combined with some properties of the weak ∗ topology.

In this paper, we use a stochastic partial differential equation (SPDE) as a model for the density of a population under the influence of random external forces/stimuli given by the environment. We study statistical properties for two crucial parameters of the SPDE that describe the dynamic of the system. To do that we use the Galerkin projection to transform the problem, passing from the SPDE to a system of independent SDEs; in this manner, we are able to find the Maximum likelihood estimator of the parameters. We validate the method by using simulations of the SDEs. We prove consistency and asymptotic normality of the estimators; the latter is showed using the Malliavin-Stein method. We illustrate our results with numerical experiments.

In comparison to quaternions, split quaternions exhibit a more intricate algebraic structure, characterized by the presence of nontrivial zero factors. Furthermore, in various fields such as geometry and electromagnetism, split quaternions serve as more convenient research tools than quaternions. This paper aims to establish the necessary and sufficient conditions for the existence of (anti-) η-Hermitian solutions to a novel system of matrix equations formulated over split quaternions. We also provide the general expression of such solutions when the system is solvable, along with the least squares (anti-) η-Hermitian solution in the case of inconsistency. To illustrate the key findings of this paper, numerical examples are presented.

In the previous work [6], we classified the solutions to a family of systems of Klein-Gordon equations with non-negative energy below the ground state into two parts: one blows up in finite time while the other extends to a global solution. In the present work, we strengthen this result, showing that these global solutions are indeed scattering in the energy space. Here we adapted Kenig-Merle’s concentration-compactness approach to the system.

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We investigate the motion of closed, smooth non-self-intersecting curves that evolve in space R 3 . The geometric evolutionary equation for the evolution of the curve is accompanied by a parabolic equation for the scalar quantity evaluated over the evolving curve. We apply the direct Lagrangian approach to describe the geometric flow of 3D curves resulting in a system of degenerate parabolic equations. We prove the local existence and uniqueness of classical Hölder smooth solutions to the governing system of nonlinear parabolic equations. A numerical discretization scheme has been constructed using the method of flowing finite volumes. We present several numerical examples of the evolution of curves in 3D with a scalar quantity. In this paper, we analyze the flow of curves with no torsion evolving in rotating and parallel planes. Next, we present examples of the evolution of curves with initially knotted and unknotted curves.

Marriage and divorce are pivotal aspects of societal dynamics, influencing family structures and individual well-being. This research introduces an advanced mathematical model to comprehend the intricate dynamics of marital relationships. Unlike conventional models, which often overlook the unpredictability inherent in human interactions, our model integrates stochastic elements and real-world factors such as mental distress and family issues. Mathematical modeling unveils the dynamics of marriage and divorce, guiding interventions to foster relationship stability and societal cohesion amidst multifaceted influences. However, the deterministic approach is limited due to the randomness of virus behavior. To address this issue, a more realistic stochastic model for computer viruses is considered, which can better capture the unpredictability of virus spread and establish effective preventive measures. The positivity and uniqueness of the model are checked, and the extinction and persistence theorems are proved using Ito's method. Numerical simulation shows that when the value of increases the divorces increases in the population while by increasing the value of the broken individuals recovered from their conflict and came back to its married lifestyle.

In this paper, we study the initial-boundary value problem of three-dimensional viscous, compressible, and heat conductive magnetohydrodynamic equations. Local existence and uniqueness of strong solutions is established with any such initial data that the initial compatibility conditions do not be required. The analysis is based on some suitable prior estimates for the strong coupling term u·∇ H and strong nonlinear term curl H × H . Our proof of the existence and uniqueness of solutions is in the Lagrangian coordinates first and then transformed back to the Euler coordinates.

The goal of this note is to study the spectrum of a self-adjoint convolution operator in L 2 ( R d ) with an integrable kernel that is perturbed by an essentially bounded real-valued potential tending to zero at infinity. We show that the essential spectrum of such operator is the union of the spectrum of the convolution operator and of the essential range of the potential. Then we provide several sufficient conditions for the existence of a countable sequence of discrete eigenvalues. For operators having non-connected essential spectrum we give sufficient conditions for the existence of discrete eigenvalues in the corresponding spectral gaps.

In this article, a two prey-one predator model in which prey and predator disperse simultaneously in a heterogeneous environment with n patches is proposed and analyzed. We prove that the solution of the system is positive and uniformly ultimately bounded. Meanwhile, we use the monotonic theory of spectral bounds to investigate the effect of the dispersal rate on population dynamics. To be precise, we discuss the stability behaviour for the trivial equilibrium and semitrivial equilibrium as well as the uniform persistence of the system. Furthermore, we prove the global asymptotic stability of the positive equilibrium by constructing a global Lyapunov function which applies the results from graph theory. Some numerical simulations are provided to show the effectiveness of the theoretical results.

The paper aims to apply the trigintaduonion spaces to explore the physical properties of four interactions simultaneously, including the electromagnetic fields, gravitational fields, weak nuclear fields, and strong nuclear fields. J. C. Maxwell first applied the algebra of quaternions to study the physical properties of electromagnetic fields. It inspired some subsequent scholars to introduce the quaternions, octonions, sedenions, and trigintaduonions to research the electromagnetic fields, gravitational fields, weak nuclear fields, strong nuclear fields, quantum mechanics, gauge fields, and curved spaces and so forth. The algebra of trigintaduonions is able to discuss the physical quantities of four interactions, including the field potential, field strong, field source, linear momentum, angular momentum, torque, and force. In the field theories described with the algebra of trigintaduonions, the weak nuclear field is composed of three types of fundamental fields. These three fundamental fields, related to weak nuclear fields, can describe the physical properties of weak nuclear fields collectively. This is consistent with the conclusion of the electroweak theory. Meanwhile the strong nuclear field consists of three types of fundamental fields. These three fundamental fields relevant to strong nuclear fields may investigate the physical properties of strong nuclear fields mutually. It is coincident with the deduction of quark theory. According to the properties of trigintaduonions, one can deduce the Yang-Mills equation related to the gauge fields. It means that the electromagnetic field occupies a quaternion space. The gravitational field owns one different quaternion space. The weak nuclear fields occupy three mutually independent quaternion spaces. The properties of weak nuclear fields are different from those of electromagnetic fields or gravitational fields. According to the multiplicative closure, the strong nuclear fields also own three quaternion spaces independent of each other. These explorations further deepen the understanding of the physical properties of weak and strong nuclear fields.

In this paper, we study the stability and convergence of a conservative Crank-Nicolson finite difference scheme applied to the Korteweg-De Vries (KdV) equation endowed with initial data. We design a three-point average scheme associated to the convective term and the dispersion term is discretized by certain discrete operators along with the Crank-Nicolson scheme for the temporal discretization to establish that the proposed scheme is L 2 -conservative. The convergence analysis reveals that utilizing inherent Kato’s local smoothing effect, the proposed scheme converges to a classical solution for sufficiently regular initial data u 0 ∈ H 3 ( R ) and to a weak solution in L 2 ( 0 , T ; L loc 2 ( R ) ) for non-smooth initial data u 0 ∈ L 2 ( R ) . Optimal convergence rates in both space and time for the devised scheme are derived. The theoretical results are justified through several numerical illustrations.