Search Results (387)

This work addresses a significant gap in the existing literature by developing integral representation formulas for extended quaternion-valued functions within the framework of Clifford analysis. While classical Cauchy-type and Borel–Pompeiu formulas are well established for complex and standard quaternionic settings, there is a lack of analogous tools for functions taking values in extended quaternion algebras such as split quaternions and biquaternions. The motivation is to extend the analytical power of Clifford analysis to these broader algebraic structures, enabling the study of more complex hypercomplex systems. The objectives are as follows: (i) to construct new Cauchy-type integral formulas adapted to extended quaternionic function spaces; (ii) to identify explicit kernel functions compatible with Clifford-algebra-valued integrands; and (iii) to demonstrate the application of these formulas to boundary value problems and potential theory. The proposed framework unifies quaternionic function theory and Clifford analysis, offering a robust analytic foundation for tackling higher-dimensional and anisotropic partial differential equations. The results not only enhance theoretical understanding but also open avenues for practical applications in mathematical physics and engineering. Full article

This paper establishes the global existence of solutions to a chemotaxis system with indirect signal production in the whole two-dimensional space. This system exhibits a mass threshold phenomenon governed by a critical mass $m_c=8\pi \delta$, where $\delta$ represents the decay rate of the static individuals. When the total initial mass $m={\int }_{R^2}{u}_{0}dx<m_c$, all solutions exist globally and remain bounded. In the critical case of $m=m_c$, the global existence or finite-time blow-up may occur depending on the initial conditions. The critical mass obtained in the whole space coincides with that previously derived in radially symmetric bounded domains. A key novelty lies in extending the analysis to the full plane, where the absence of compactness is overcome by constructing a suitable Lyapunov functional and employing refined Trudinger–Moser-type inequalities. Full article

In this paper, we are concerned with the Cauchy problem of the compressible Oldroyd-B model without stress diffusion in ${\mathbb{R}}^n(n=2,3)$. The absence of stress diffusion introduces significant challenges in the analysis of this system. By employing tools from harmonic analysis, particularly the Littlewood–Paley decomposition theory, we establish the global well-posedness of solutions with initial data in $L^p$ critical spaces, which accommodates the case of large, highly oscillating initial velocity. Furthermore, we derive the optimal time decay rates of the solutions by a suitable energy argument. Full article

The majority of the challenges faced in power system engineering are presented as constrained optimization functions, which are frequently characterized by their complicated architectures. Metaheuristics are mathematical techniques used to solve complicated optimization problems. One such technique, Artificial Rabbits Optimization (ARO), has been designed to address global optimization challenges. However, ARO has limitations in terms of search functionality, restricting its efficiency in dealing with constrained optimization environments. To improve ARO’s compatibility with a variety of challenging problems, this work proposes implementing the Cauchy mutation operator into the position-updating procedure during the exploration stage. Furthermore, a novel multi-mode control parameter is developed to facilitate a smooth transition between exploration and exploitation phases. The enhancements may boost the performance and serve as an effective optimization tool for tackling complex engineering tasks. The improved version is known as Cauchy Artificial Rabbits Optimization (CARO). The proposed CARO’s performance is evaluated using eleven power system challenges as part of the CEC2020 competition’s test set of real-world constrained problems. The experimental results demonstrate the practical applicability of the proposed CARO in engineering applications and provide areas for future investigation. Full article

Intelligent reflecting surfaces (IRSs) have attracted extensive attention in the design of future communication networks. However, their large number of reflecting elements still results in non-negligible power consumption and hardware costs. To address this issue, we previously proposed a green heterogeneous IRS (HE-IRS) consisting of both dynamically tunable elements (DTEs) and statically tunable elements (STEs). Compared to conventional IRSs with only DTEs, the unique DTE–STE integrated structure introduces new challenges in optimizing the positions and phase shifts of the two types of elements. In this paper, we investigate the element position and phase shift optimization problems in HE-IRS-assisted millimeter-wave systems. We first propose a particle swarm optimization algorithm to determine the specific positions of the DTEs and STEs. Then, by decomposing the phase shift optimization of the two types of elements into two subproblems, we utilize the manifold optimization method to optimize the phase shifts of the STEs, followed by deriving a closed-form solution for those of the DTEs. Furthermore, we propose a low-complexity phase shift optimization algorithm for both DTEs and STEs based on the Cauchy–Schwarz bound. The simulation results show that with the tailored element position and phase shift optimization algorithms, the HE-IRS can achieve a competitive performance compared to that of the conventional IRS, but with much lower power consumption. Full article

In this contribution, the classical Cauchy first-gradient elastic theory is used to solve the equilibrium problem of a bidimensional (2D) reinforced elastic structure under small displacements and strains. Such a 2D first-gradient continuum is embedded with a reinforcement, which is modeled as a zero-thickness interface endowed with the elastic properties of an extensional Euler–Bernoulli 1D beam. Modeling the reinforcement as an interface eliminates the need for a full geometric representation of the reinforcing bar with finite thickness in the 2D model, and the associated mesh discretization for numerical analysis. Thus, the effects of the 1D beam-like reinforcements are described through proper and generalized boundary conditions prescribed to contiguous continuum regions, deduced from a standard variational approach. The novelty of this work lies in the formulation of an interface model coupling 1D and 2D continua, based on weak formulation and variational derivation, capable of accurately capturing stress distributions without requiring full geometric resolution of the reinforcement. The proposed framework is therefore illustrated by computing, with finite element simulations, the response of the reinforced structural element under uniform bending. Numerical results reveal the presence of jumps for some stress components in the vicinity of the reinforcement tips and demonstrate convergence under mesh refinement. Although the reinforcement beams possess only axial stiffness, they significantly influence the equilibrium configuration by causing a redistribution of stress and enhancing stress transfer throughout the structure. These findings offer a new perspective on the effective modeling of fiber-reinforced structures, which are of significant interest in engineering applications such as micropiles in foundations, fiber-reinforced concrete, and advanced composite materials. In these systems, stress localization and stability play a critical role. Full article

This paper is concerned with the non-existence of a global solution to the initial value problem of the inhomogeneous damped wave equation with a nonlinear memory term and a nonlinear gradient term. The critical exponent and formation of singularity of solution are closely related to symmetry in the study of blow-up dynamics for nonlinear wave equations, which provides a profound mathematical tool for analyzing the explosion of solutions within finite time. The proofs of blow-up results of solutions are based on the test function method, where the test function is variable separated. The influences of two types of damping terms, two types of nonlinearities, and an inhomogeneous term on exponents of the problem in blow-up cases are explained. It is worth pointing out that the inhomogeneous term in the problem is discussed with respect to the exponent $\sigma$ in three cases (namely, $\sigma =0$, $-1<\sigma <0$, and $\sigma >0$). As far as we know, the results in Theorems 1–4 are new. Full article

This paper establishes the well-posedness of Cauchy-type problems with non-symmetric initial conditions for nonlinear implicit Hilfer fractional differential equations of general fractional orders in weighted function spaces. Using fixed-point techniques, we first prove the existence of solutions via Schaefer’s fixed-point theorem. The uniqueness and Ulam–Hyers stability are then derived using Banach’s contraction principle. By introducing a novel singular-kernel Gronwall inequality, we extend the analysis to Ulam–Hyers–Rassias stability and continuous dependence on initial data. The theoretical framework is unified for general fractional orders and validated through examples, demonstrating its applicability to implicit systems with memory effects. Key contributions include weighted-space analysis and stability criteria for this class of equations. Full article

From collective intelligence to evolutionary computation and machine learning, symmetry can be leveraged to enhance algorithm performance, streamline computational procedures, and elevate solution quality. Grasping and leveraging symmetry can give rise to more resilient, scalable, and understandable algorithms. In view of the flaws of the original Sailfish Optimization Algorithm (SFO), such as low convergence precision and a propensity to get stuck in local optima, this paper puts forward an Osprey and Cauchy Mutation Integrated Sailfish Optimization Algorithm (OCSFO). The enhancements are mainly carried out in three aspects: (1) Using the Logistic map to initialize the sailfish and sardine populations. (2) In the first stage of the local development phase of sailfish individual position update, adopting the global exploration strategy of the Osprey Optimization Algorithm to boost the algorithm’s global search capability. (3) Introducing Cauchy mutation to activate the sailfish and sardine populations during the prey capture stage. Through the comparative analysis of OCSFO and seven other swarm intelligence optimization algorithms in the optimization of 23 classic benchmark test functions, as well as the Wilcoxon rank-sum test, it is evident that the optimization speed and convergence precision of OCSFO have been notably improved. To confirm the practicality and viability of the OCSFO algorithm, it is applied to solve the optimization problems of piston rods, three-bar trusses, cantilever beams, and topology. Through experimental analysis, it can be concluded that the OCSFO algorithm has certain advantages in solving practical optimization problems. Full article

This paper identifies a source term depending on spatial variable in a heat equation from just part of the boundary conditions. The measurement data are specified at an internal moment of time. The ill-posedness of the problem is higher than most of the previous source identification problems. This is because the problem becomes a noncharacteristic Cauchy problem for the heat equation if the source term is given, which is known as severely ill-posed. The method of fundamental solutions (MFS) in conjunction with the classical Tikhonov regularization method is proposed to reconstruct a stable approximation. The fundamental solutions for the heat equation are spherically symmetric in spatial variable and satisfy the equation automatically, and thus only the boundary conditions need to be satisfied. This characteristic allows the discretization to be performed only on boundary-like geometry and improve the computational efficiency. In this paper, several numerical examples are listed to show the feasibility and effectiveness of the suggested method. Full article

In this paper, for a mixed elliptic-hyperbolic type equation with various degeneration orders and singular coefficients, theorems of uniqueness and existence of the solution to the problem with a missing Tricomi condition on boundary characteristic and with an analog of Frankl condition on different parts of the cut boundary along the degeneration segment in the mixed domain are proved. On the degeneration line segment, a general conjugation condition is set, and on the boundary of the elliptic domain and degeneration segment, the Bitsadze–Samarskii condition is posed. The considered problem, based on integral representations of the solution to the Dirichlet problem (in elliptic part of the domain) and a modified Cauchy problem (in hyperbolic part of the domain), is reduced to solving a non-standard singular Tricomi integral equation with a non-Fredholm integral operator (featuring an isolated first-order singularity in the kernel) in non-characteristic part of the equation. Non-standard approaches are applied here in constructing the solution algorithm. Through successive applications of the theory of singular integral equations and then the Wiener–Hopf equation theory, the non-standard singular Tricomi integral equation is reduced to a Fredholm integral equation of the second kind, the unique solvability of which follows from the uniqueness theorem for the problem. Full article

In this paper, we investigated a three-dimensional incompressible fractional rotating magnetohydrodynamic (FrMHD) system by reformulating the Cauchy problem into its equivalent mild formulation and working in critical homogeneous Sobolev spaces. For this, we first established the existence and uniqueness of a global mild solution for small and divergence-free initial data. Moreover, our approach is based on proving sharp bilinear convolution estimates in critical Sobolev norms, which in turn guarantee the uniform analyticity of both the velocity and magnetic fields with respect to time. Furthermore, leveraging the decay properties of the associated fractional heat semigroup and a bootstrap argument, we derived algebraic decay rates and established the long-time dissipative behavior of FrMHD solutions. These results extended the existing literature on fractional Navier–Stokes equations by fully incorporating magnetic coupling and Coriolis effects within a unified fractional-dissipation framework. Full article

To address the high complexity layout optimization problem of an offshore wind and wave energy co-generation system, an improved seagull optimization algorithm-based method is proposed. Firstly, the levelized cost of electricity (LCOE) model, based on the whole-life-cycle cost, serves as the optimization objective. Therein, the synergistic effect between wind turbines and wave energy generators is taken into consideration to decouple the problem and establish a two-layer optimization framework. Secondly, the seagull optimization algorithm is enhanced by integrating three strategies: the nonlinear adjustment strategy for control factors, the Gaussian–Cauchy hybrid variational strategy, and the multiple swarm strategy, thereby improving the global search capability. Finally, a case study in the South China Sea validates the effectiveness of the model and algorithm. Using the improved algorithm, the optimal layout of the co-generation system and the optimal wind turbine parameters are obtained. The results indicate that the optimized system achieves a LCOE of 0.6561 CNY/kWh, which is 0.29% lower than that achieved by traditional algorithms. The proposed method provides a reliable technical solution for the economic optimization of the co-generation system. Full article

We construct the Green functions (or Feynman’s propagators) for the Schrödinger equations of the form $i{\psi }_t+{\displaystyle \frac{1}{4}}{\psi }_{xx}\pm t{x}^2\psi =0$ (for the wave function $\psi$ and its time (t) and x-space derivatives) in terms of Airy functions and solve the Cauchy initial value problem in the coordinate and momentum representations. Particular solutions of the corresponding nonlinear Schrödinger equations with variable coefficients are also found. A special case of the quantum parametric oscillator is studied in detail first. The Green function is explicitly given in terms of Airy functions and the corresponding transition amplitudes are found in terms of a hypergeometric function. The general case of the quantum parametric oscillator is considered then in a similar fashion. A group theoretical meaning of the transition amplitudes and their relation with Bargmann’s functions is established. The relevant bibliography, to the best of our knowledge, is addressed. Full article

This paper investigates the Cauchy problem for the full compressible Navier–Stokes–Korteweg equations, which model fluid dynamics with capillary properties in ${\mathbb{R}}^d(d\ge 3)$. And the global well-posedness and Gevrey analytic of strong solutions for the system are established in the $L^2-L^p$ type critical hybrid Besov space with $2\le p\le \frac{2d}{d-2}$ and $p<d$. Full article