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The paper [39] uses the Craig-Wayne-Bourgain method to construct solutions of an elliptic problem involving parameters. The results of [39] include regularity assumptions on the perturbation and involve excluding parameters. The paper [39] also constructs response solutions to a quasi-periodically perturbed (ill-posed evolution) problem. In this paper, we use several classical methods (freezing of coefficients, alternative methods for nonlinear elliptic equations) to extend the results of [39]. We weaken the regularity assumptions on the perturbation and we describe the phenomena that happens for all parameters. In the ill-posed problem, we use a recently developed time-dependent center manifold theorem which allows to reduce the problem to a finite-dimensional ODE with quasi-periodic dependence on time. The bounded and sufficiently small solutions of these ODE give solutions of the ill-posed PDE. [ABSTRACT FROM AUTHOR]

In this short communication the solution of a steady state model of a sextuple evaporator system used in a paper industry has been presented. With the help of the residual values of model equations (a set of 12 nonlinear algebraic equations) it has been shown that the presently obtained results are superior to the previously published results. The importance of the number of significant digits in calculation are also briefly discussed. [ABSTRACT FROM AUTHOR]

This paper focusses on the nonlinear fractional Kadomtsev-Petviashvili (FKP) equation in space-time, employing the conformable fractional derivative (CFD) approach. The main objective of this paper is to examine the application of the (G'/G)-expansion method in order to find analytical solutions to the FKP equation. The (G'/G)-expansion method is a powerful tool for constructing traveling wave solutions of nonlinear evolution equations. However, its application to the FKP equation remains relatively unexplored. By employing traveling wave transformation, the FKP equation was transformed into an ordinary differential equation (ODE) to acquire exact wave solutions. A range of exact analytical solutions for the FKP equation is obtained. Graphical illustrations are included to elucidate the physical characteristics of the acquired solutions. To demonstrate the impact of the fractional operator on results, the acquired solutions are exhibited for different values of the fractional order α, with a comparison to their corresponding exact solutions when taking the conventional scenario where α equals 1. The results indicate that the (G'/G)-expansion method serves as an efficient method and dependable in solving the nonlinear FKP equation. [ABSTRACT FROM AUTHOR]

A prediction of rod wave type with great precision is extremely important in theoretical analysis and practical applications. Besides the well-known periodic motion and resonance, this paper studies the rod wave in a fractal space, and a fractal solitary wave is unlocked by the variational approach, the results reveal that the rod strain non-linearity and fractal dimensions affect greatly the wave travelling properties. This paper offers a new window for identifying a solitary wave from periodic motion easily and accurately. [ABSTRACT FROM AUTHOR]

In this paper, the Cauchy problem for the nonlinear Schrodinger system {idtu1(x,t)== ∆u1(x, t)− |u1(x, t)| p−1u1(x, t)− |u2(x, t)| p−1u1(x, t), idtu2(x, t) = ∆u2(x, t) − |u2(x, t)| p−1u2(x, t) − |u1(x, t)| p−1u2(x, t), was investigated in d space dimensions. For 1 < p ≤ 1 + 2/d, the nonexistence of asymptotically free solutions for the nonlinear Schrödinger system was proved based on mathematical analysis and scattering theory methods. The novelty of this paper was to give the proof of pseudo-conformal identity on the nonlinear Schrödinger system. The present results improved and complemented these of Bisognin, Sepúlveda, and Vera(Appl. Numer. Math. 59(9)(2009): 2285–2302), in which they only proved the nonexistence of asymptotically free solutions when d = 1, p = 3. [ABSTRACT FROM AUTHOR]

Tautenhahn (2002) studied the Lavrentiev regularization method (LRM) to approximate a stable solution for the ill-posed nonlinear equation κ (u) = v , where κ : D (κ) ⊆ X ⟶ X is a nonlinear monotone operator and X is a Hilbert space. The operator in the example used in Tautenhahn's paper was not a monotone operator. So, the following question arises. Can we use LRM for ill-posed nonlinear equations when the involved operator is not monotone? This paper provides a sufficient condition to employ the Lavrentiev regularization technique to such equations whenever the operator involved is non-monotone. Under certain assumptions, the error analysis and adaptive parameter choice strategy for the method are discussed. Moreover, the developed theory is applied to two well-known ill-posed problems—inverse gravimetry and growth law problems. [ABSTRACT FROM AUTHOR]

In this paper, our study is divided into two parts. The first part involves analyzing a coupled system of beam deflection type that involves nonlinear equations with sequential Caputo derivatives. The also system incorporates the Caputo derivatives in the initial conditions, which adds a layer of complexity and realism to the problem. We focus on proving the existence of a unique solution for this system, and highlighting the robustness and applicability of fractional derivatives in modeling complex physical phenomena. In the second part of the paper, we employ conformable fractional derivatives, as defined by Khalil, to examine another system consisting of two coupled evolution equations. By the Tanh method, we derive new progressive waves. The connection between these two parts lies in the use of fractional calculus to extend and enhance classical problems. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence result of solutions for fuzzy nonlinear fractional differential equations involving Caputo differentiability of an arbitrary order 0 < q < 1. As application, an example is included to show the applicability of our result. [ABSTRACT FROM AUTHOR]

The classical linear discriminant analysis (LDA) algorithm has three primary drawbacks, i.e., small sample size problem, sensitivity to noise and outliers, and inability to deal with multi-modal-class data. This paper reviews LDA technology and its variants, covering the taxonomy and characteristics of these technologies and comparing their innovations and developments in addressing these three shortcomings. Additionally, we describe the application areas and emphasize the kernel extensions of these technologies to solve nonlinear problems. Most importantly, this paper presents perspectives on future research directions and potential research areas in this field. [ABSTRACT FROM AUTHOR]

The present paper dealt with the study of solitons of Lorentzian Kähler space-time manifolds. In this paper, we have discussed different conditions for solitons to be steady, expanding, or shrinking in terms of isotropic pressure, the cosmological constant, energy density, nonlinear equations, and gravitational constant in pseudo-quasi-conformally flat and pseudo-W8 flat Lorentzian Kahler space-time manifolds. [ABSTRACT FROM AUTHOR]

The development of machine tools in the last twenty years includes, among other things, the application of mechanisms with a non-linear kinematic structure as the mechanical basis of machines. This results in significant improvements in kinematic characteristics and problems related to non-linear dependencies of the accuracy of the drive elements and the realization of movement in the machine's external coordinates. The paper presents an approach to machine tool calibration based on the original O-X glide mechanism based on the ISO 230-4 standard with the mono- and bi-directional compensation of systematic errors and adaptation to the specifics of the mechanism's kinematics. A machine tool prototype was designed and built for the research presented in the paper. The obtained results indicate the possibility of applying the existing recommendations and standards for testing the accuracy of machine tools with the need to correct the methodology by using linear and non-linear kinematic structures in machine tools. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to reduce the dimensionality of unknown coefficient vectors of finite-element (FE) solutions in two-grid (CN) FE (TGCNFE) format for the nonlinear unsaturated soil water flow problem by using a proper orthogonal decomposition (POD) and to design a new reduced-dimension iteration TGCNFE (RDITGCNFE). For this objective, a new time semi-discrete CN (TSDCN) scheme for the nonlinear unsaturated soil water flow problem is first designed and the existence, stability, and error estimates of TSDCN solutions are demonstrated. Subsequently, a new TGCNFE format for the nonlinear unsaturated soil water flow problem is designed and the existence, unconditional stability, and error estimates of TGCNFE solutions are demonstrated. Next, a new RDITGCNFE format with the same FE basis functions as the TGCNFE format is built by the POD method and the existence, unconditional stability, and error estimates of RDITGCNFE solutions are discussed. Ultimately, the rightness of theory results and the superiority of the RDITGCNFE format are verified by two sets of numerical tests. It is worth noting that the RDITGCNFE format differs completely from all previous reduced-dimension methods, including the authors' previous works. Therefore, the study of this paper can not only provide a new theoretical method for the dimensionality reduction of numerical models for nonlinear problems but also provide an algorithm implementation technology for the numerical simulation of practical engineering problems. [ABSTRACT FROM AUTHOR]

To address the challenge of accurately locating unmanned aerial vehicles (UAVs) in situations where radar tracking is not feasible and visual observation is difficult, this paper proposes an innovative acoustic source localization method based on improved Empirical Mode Decomposition (EMD) within an adaptive frequency window. In this study, the collected flight signals of UAVs undergo smoothing filtering. Additionally, Robust Empirical Mode Decomposition (REMD) is applied to decompose the signals into Intrinsic Mode Function (IMF) components for spectrum analysis. We introduce a sliding frequency window with adjustable bandwidth, which is automatically determined using a Grey Wolf Optimizer (GWO) with a sliding index. This window is used to lock and extract specific frequencies from the IMFs. Based on predefined criteria, the extracted IMF components are reconstructed, and trigger signal times are analyzed and recorded from these reconstructed IMFs. The time differences between sensor receptions are then calculated. Furthermore, this study introduces the Chan–Taylor localization algorithm based on weighted least squares. This advanced algorithm takes sensor time delay parameters as input and solves a set of nonlinear equations to determine the target's location. Simulations and real-world signal tests are used to validate the robustness and performance of the proposed method. The results indicate that the localization error remains below 5% within a 15 m × 15 m measurement area. This provides an efficient and real-time method for detecting the location of small UAVs. [ABSTRACT FROM AUTHOR]

This paper studies the existence of uniform attractors for 3D micropolar equation with damping term. When β>3, with initial data (uτ,ωτ)∈V1×V2 and external forces (f1,f2)H(f01× H(f02), some uniform estimates of the solution in different function spaces are given. Based on these uniform estimates, the ((V1×V2)×(H(f01)×H(f02)),V1×V2)-continuity of the family of processes {U(f1,f2)(t,τ)}t≥τ is demonstrated. Meanwhile, the (V1×V2,H²(Ω)×H²(Ω))-uniform compactness of {U(f1,f2)(t,τ)}t≥τ is proved. Finally, the existence of a (V1×V2,V1×V2)-uniform attractor and a (V1× V2,H²(Ω)×H²(Ω))-uniform attractor are obtained. Furthermore, the (V1×V2,V1×V2)-uniform attractor and the (V1×V2,H²(Ω)×H²(Ω))-uniform attractor are verified to be the same. [ABSTRACT FROM AUTHOR]

In this paper we study the following nonlinear fractional Hartree (or Choquard-Pekar) equation (− Δ) s u + μ u = ( I α * F (u)) F ′ (u) in R N , (*) where μ > 0, s ∈ (0, 1), N ≥ 2, α ∈ (0, N), I α ∼ 1 | x | N − α is the Riesz potential, and F is a general subcritical nonlinearity. The goal is to prove existence of multiple (radially symmetric) solutions u ∈ H s ( R N ) , by assuming F odd or even: we consider both the case μ > 0 fixed and the case ∫ R N u 2 = m > 0 prescribed. Here we also simplify some arguments developed for s = 1 (S. Cingolani, M. Gallo, and K. Tanaka, "Multiple solutions for the nonlinear Choquard equation with even or odd nonlinearities," Calc. Var. Partial Differ. Equ., vol. 61, no. 68, p. 34, 2022). A key point in the proof is given by the research of suitable multidimensional odd paths, which was done in the local case by Berestycki and Lions (H. Berestycki and P.-L. Lions, "Nonlinear scalar field equations II: existence of infinitely many solutions," Arch. Ration. Mech. Anal., vol. 82, no. 4, pp. 347–375, 1983); for (*) the nonlocalities play indeed a special role. In particular, some properties of these paths are needed in the asymptotic study (as μ varies) of the mountain pass values of the unconstrained problem, then exploited to describe the geometry of the constrained problem and detect infinitely many normalized solutions for any m > 0. The found solutions satisfy in addition a Pohozaev identity: in this paper we further investigate the validity of this identity for solutions of doubly nonlocal equations under a C1-regularity. [ABSTRACT FROM AUTHOR]

In this paper, we consider the general dual fractional parabolic problem ∂ t α u (x , t) + L u (x , t) = f (t , u (x , t)) in R n × R. We show that the bounded entire solution u satisfying certain one-direction asymptotic assumptions must be monotone increasing and one-dimensional symmetric along that direction under an appropriate decreasing condition on f. Our result here actually solves a well-known problem known as Gibbons' conjecture in the setting of the dual fractional parabolic equations. To overcome the difficulties caused by the nonlocal divergence type operator L and the Marchaud time derivative ∂ t α , we introduce several new ideas. First, we derive a general weighted average inequality corresponding to the nonlocal operator L , which plays a fundamental bridging role in proving maximum principles in unbounded domains. Then we combine these two essential ingredients to carry out the sliding method to establish the Gibbons' conjecture. It is worth noting that our results are novel even for a special case of L , the fractional Laplacian (−Δ)s, and the approach developed in this paper will be adapted to a broad range of nonlocal parabolic equations involving more general Marchaud time derivatives and more general non-local elliptic operators. [ABSTRACT FROM AUTHOR]

Consider an infinite-dimentional Banach space denoted as X, and designate ∂(X) as the algebra of all bounded linear operators on X. Moreover, let s (A) denote the spectrum of ..., and ...(A)) indicate the boundary of s (A). A map ... is termed a ρ-spectrum if ... (A) for all A ... In this paper, we characterize all surjective maps f1 and f2 on ρ(X) satisfying ... = (AB) for all A,B(X). [ABSTRACT FROM AUTHOR]

In this paper, the semi-local convergence of the Cordero's sixth-order iterative method in Banach space was proved by the method of recursion relation. In the process of proving, the auxiliary sequence and three increasing scalar functions can be derived using Lipschitz conditions on the first-order derivatives. By using the properties of auxiliary sequence and scalar function, it was proved that the iterative sequence obtained by the iterative method was a Cauchy sequence, then the convergence radius was obtained and its uniqueness was proven. Compared with Cordero's process of proving convergence, this paper does not need to ensure that G (s) is continuously differentiable in higher order, and only the first-order Fréchet derivative was used to prove semi-local convergence. Finally, the numerical results showed that the recursion relationship is reasonable. [ABSTRACT FROM AUTHOR]

In this paper, we have constructed a family of three-step methods with sixth-order convergence and a novel approach to enhance the convergence order p of iterative methods for systems of nonlinear equations. Additionally, we propose a three-step scheme with convergence order p + 3 (for p ≥ 3 ) and have extended it to a generalized (m + 2) -step scheme by merely incorporating one additional function evaluation, thus achieving convergence orders up to p + 3 m , m ∈ N . We also provide a thorough local convergence analysis in Banach spaces, including the convergence radius and uniqueness results, under the assumption of a Lipschitz-continuous Fréchet derivative. Theoretical findings have been validated through numerical experiments. Lastly, the performance of these methods is showcased through the analysis of their basins of attraction and their application to systems of nonlinear equations. In this paper, we have constructed a family of three-step methods with sixth-order convergence and a novel approach to enhance the convergence order of iterative methods for systems of nonlinear equations. Additionally, we propose a three-step scheme with convergence order (for ) and have extended it to a generalized -step scheme by merely incorporating one additional function evaluation, thus achieving convergence orders up to , . We also provide a thorough local convergence analysis in Banach spaces, including the convergence radius and uniqueness results, under the assumption of a Lipschitz-continuous Fréchet derivative. Theoretical findings have been validated through numerical experiments. Lastly, the performance of these methods is showcased through the analysis of their basins of attraction and their application to systems of nonlinear equations. [ABSTRACT FROM AUTHOR]

The compound continuum robot employs both concentric tube components and cable-driven continuum components to achieve its complex motions. Nevertheless, the interaction between these components causes coupling, which inevitably leads to reduced accuracy. Consequently, researchers have been striving to mitigate and compensate for this coupling-induced error in order to enhance the overall performance of the robot. This paper leverages the coupling between the components of the compound continuum robot to accomplish specific surgical procedures. Specifically, the internal concentric tube component is utilized to induce motion in the cable-driven external component, which generates coupled motion under the constraints of the cable. This approach enables the realization of high-precision surgical operations. Specifically, a kinematic model for the proposed robot is established, and an inverse kinematic algorithm is developed. In this inverse kinematic algorithm, the solution of a highly nonlinear system of equations is simplified into the solution of a single nonlinear equation. To demonstrate the effectiveness of the proposed approach, simulations are conducted to evaluate the efficiency of the algorithm. The simulations conducted in this study indicate that the proposed inverse kinematic (IK) algorithm improves computational speed by a significant margin. Specifically, it achieves a speedup of 2.8 × 103 over the Levenberg–Marquardt (LM) method. In addition, experimental results demonstrate that the coupled-motion system achieves high levels of accuracy. Specifically, the repetitive positioning accuracy is measured to be 0.9 mm, and the tracking accuracy is 1.5 mm. This paper is significant for dealing with the coupling of the compound continuum robot. [ABSTRACT FROM AUTHOR]

With the popularity of shared bicycles in urban areas, more and more residents choose this fast and convenient mode of transportation for short-distance travel. By optimizing the layout of shared bicycle parking areas and delivery quantity, the investment cost of shared bicycle enterprises can be effectively reduced, and the convenience of residents' travel can be improved at the same time. In this paper, we develop a collaborative optimization model for the layout of the shared bicycle parking area and delivery quantity, aiming at minimizing the walking distance of residents and the investment cost of enterprises, while considering the constraints of the parking area's attractive range and the number of bicycles placed. Aiming at the characteristics of this mixed integer nonlinear problem, an improved genetic algorithm incorporating symmetric individual precision control mechanism is designed. Finally, taking the planned area between the Second Ring Road and the Third Ring Road in the northern part of Jin-niu District, Chengdu as the background, the proposed collaborative optimization model for the layout of shared bicycle parking areas and delivery quantity is applied to a real scene. The results show that after optimization, the number of parking areas is reduced by 2, and the total investment cost is reduced by about 12.2%. [ABSTRACT FROM AUTHOR]

In this paper, new Sturmian comparison results were obtained for linear and nonlinear hyperbolic equations on a rectangular prism. The results obtained for linear equations extended those given by Kreith [Sturmian theorems on hyperbolic equations, Proc. Amer. Math. Soc., 22 (1969), 277-281] in which the Sturmian comparison theorem for linear equations was obtained on a rectangular region in the plane. For the purpose of verification, an application was described using an eigenvalue problem. [ABSTRACT FROM AUTHOR]

In this paper, we consider a Li-Yau gradient estimate on the positive solution to the following nonlinear parabolic equation ∂/∂t f = Δf + af (ln f)p with Neumann boundary conditions on a compact Riemannian manifold satisfying the integral Ricci curvature assumption, where p ≥ 0 is a real constant. This contrasts Olivé's gradient estimate, which works mainly for the heat equation rather than nonlinear parabolic equations and the result can be regarded as a generalization of the Li-Yau [P. Li, S. T. Yau, On the parabolic kernel of the Schrödinger operator, Acta Math., 156 (1986), 153-201] and Olivé [X. R. Olive, Neumann Li-Yau gradient estimate under integral Ricci curvature bounds, Proc. Amer. Math. Soc., 147 (2019), 411-426] gradient estimates. [ABSTRACT FROM AUTHOR]

This paper describes a numerical algorithm for solving the classic problem of a plane strain (KGD) fracture propagating in an impermeable elastic medium with zero toughness. The method, which takes advantage of the self-similar nature of the solution, combines a domain-based scheme to solve the elasticity equations and a finite volume method to solve the nonlinear lubrication equation. This work represents a first step towards developing a model able to account for pore pressure diffusion in the medium and corresponding poroelastic effects, noting that these processes are more efficiently solved using a domain-based rather than a boundary integral method. To enhance the efficiency and accuracy of the numerical scheme, the far-field crack asymptotics is embedded in the discretized elastic relationship between the fluid pressure and the crack opening, while the coupled fluid-solid tip asymptote is enforced in a weak form when solving the nonlinear lubrication equation. The proposed technique yields results that closely match the analytical solution, even with a coarse mesh. This approach offers potential for addressing more complex hydraulic fracturing problems in the future. [ABSTRACT FROM AUTHOR]

Multiple hypersonic gliding vehicles' (HGVs') formation control problems with obstacle and collision avoidance are investigated in this paper, which is addressed in the stage of entry gliding. The originality of this paper stems from the formation control algorithm where constraints of dynamic pressure, heating rate, total aerodynamic load, control inputs, collision avoidance, obstacle avoidance, and the terminal states are considered simultaneously. The algorithm implements a control framework designed to be of two terms: distributed virtual controller and actual control input solver. The distributed virtual controller is based on distributed model predictive control with synchronous update strategy, where the virtual control signals are derived by the optimization simultaneously at each time step for each HGV under directed communication topology. Subsequently, according to the virtual control signals obtained, a coupled nonlinear equation set is solved to get actual control signals: each HGV's bank angle together with the angle of attack. The actual control input solver adopts a feasible solution process to calculate the actual control signals while dealing with constraints. Finally, extensive numerical simulations are implemented to unveil the proposed algorithm's performance and superiority. [ABSTRACT FROM AUTHOR]

Whether the 3D incompressible Navier–Stokes equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in nonlinear PDEs. In this paper, we present some new numerical evidence that the incompressible axisymmetric Navier–Stokes equations with smooth initial data of finite energy seem to develop potentially singular behavior at the origin. This potentially singular behavior is induced by a potential finite time singularity of the 3D Euler equations that we reported in a companion paper published in the same issue, see also Hou (Potential singularity of the 3D Euler equations in the interior domain. arXiv:2107.05870 [math.AP], 2021). We present numerical evidence that the 3D Navier–Stokes equations develop nearly self-similar singular scaling properties with maximum vorticity increased by a factor of 10 7 . We have applied several blow-up criteria to study the potentially singular behavior of the Navier–Stokes equations. The Beale–Kato–Majda blow-up criterion and the blow-up criteria based on the growth of enstrophy and negative pressure seem to imply that the Navier–Stokes equations using our initial data develop a potential finite time singularity. We have also examined the Ladyzhenskaya–Prodi–Serrin regularity criteria (Kiselev and Ladyzhenskaya in Izv Akad Nauk SSSR Ser Mat 21(5):655–690, 1957; Prodi in Ann Math Pura Appl 4(48):173–182, 1959; Serrin in Arch Ration Mech Anal 9:187–191, 1962) that are based on the growth rate of L t q L x p norm of the velocity with 3 / p + 2 / q ≤ 1 . Our numerical results for the cases of (p , q) = (4 , 8) , (6 , 4) , (9 , 3) and (p , q) = (∞ , 2) provide strong evidence for the potentially singular behavior of the Navier–Stokes equations. The critical case of (p , q) = (3 , ∞) is more difficult to verify numerically due to the extremely slow growth rate in the L 3 norm of the velocity field and the significant contribution from the far field where we have a relatively coarse grid. Our numerical study shows that while the global L 3 norm of the velocity grows very slowly, the localized version of the L 3 norm of the velocity experiences rapid dynamic growth relative to the localized L 3 norm of the initial velocity. This provides further evidence for the potentially singular behavior of the Navier–Stokes equations. [ABSTRACT FROM AUTHOR]

In this paper, a new class of extended (2+1)-dimensional Ito equations is investigated for its group invariant solutions. The Lie symmetry method is employed to transform the nonlinear Ito equation into an ordinary differential equation. The general solution of the solvable linear differential equation with different parameters is obtained, and the plot of the solvable linear differential equation is given. A power series solution for the equation is then derived. Furthermore, a conservation law for the equation is constructed by utilizing a new Ibragimov conservation theorem. [ABSTRACT FROM AUTHOR]

A single elliptical or ellipsoidal inclusion with an arbitrary uniform eigenstrain is known to achieve a constant stress field when embedded in an elastic medium provided the edge of the medium is sufficiently far from the inclusion (i.e. the interaction between the inclusion and the edge of the medium is negligible). In this paper, we aim to answer the question as to whether there exists an inclusion of certain configuration (with a uniform eigenstrain) that remains to possess a constant stress when embedded in a bounded medium whose edge interacts significantly with it. Specifically, we consider the anti-plane shear case of an inclusion with a uniform eigenstrain in a circular medium with a traction-free edge. We derive a sufficient and necessary condition ensuring the uniformity of the stress within the inclusion, which further leads to a nonlinear system of equations with respect to an infinite group of parameters characterizing the shape of the inclusion. We obtain convergent solutions for the truncated version of the nonlinear system using numerical techniques, and illustrate the corresponding shape of the inclusion in a few numerical examples. Our results for the case corresponding to small inclusion size and small edge-inclusion distance (relative to the radius of the medium) are well-consistent with the existing results for an inclusion with uniform stress in a semi-infinite medium with a traction-free surface, while those for centrally placed inclusions achieving uniform stress capture the classical case of centric circular inclusion accurately. The results presented in this paper provide a strong evidence for the existence of inclusions possessing uniform stress in an elastic bounded domain subjected to common external boundary conditions under anti-plane shear deformation. [ABSTRACT FROM AUTHOR]

This research article addresses a nonclassical initial boundary value problem characterized by a non-local constraint within the framework of a pseudo-hyperbolic equation. Employing rigorous analytical techniques, the paper establishes the existence, uniqueness, and continuous dependence of a strong solution to the problem at hand. With respect to the associated linear problem, the uniqueness of its solution is ascertained through an energy inequality, which provides an a priori bound for the solution. Moreover, the solvability of this linear problem is verified by proving that the operator range engendered by the problem is indeed dense. Extending the analysis to the nonlinear problem, an iterative methodology is utilized. This approach is predicated on the insights gained from the linear problem and facilitates the demonstration of both the existence and uniqueness of a solution for the nonlinear problem under study. Consequently, the paper contributes a robust mathematical framework for solving both linear and nonlinear variants of complex initial boundary value problems with non-local constraints. [ABSTRACT FROM AUTHOR]

: Non-linear neutrosophic numbers (NLNNs) are different kinds of neutrosophic numbers with at least one non-linear membership function (either of truthiness, falsity or indeterminacy part) of the information. Furthermore, a linear programming problem with non-linear neutrosophic numbers as coefficients/parameters is a special type of programming problem known as a non-linear linear programming problem (NLN-LPP). This paper elaborates on the concepts of non-linear neutrosophic number (NLNN) sets, different forms of non-linear neutrosophic numbers (NLNNs),α, β, γ cuts on non-linear neutrosophic numbers (NLNNs), possibility mean, possibility standard deviation, and possibility variance of non-linear neutrosophic numbers (NLNNs). In this paper, we also propose the solution technique for non-linear neutrosophic linear programming problems (NLN-LPPs) in which all coefficients/parameters are non-linear neutrosophic numbers (NLNNs). In this continuation, we suggest a new modified possibility score function for non-linear NNs in terms of possibility means and possibility standard deviations of non-linear neutrosophic numbers (NLNNs) for better use of all parts of information. This modified score function is used to convert non-linear neutrosophic number (NLNN) coefficients/parameters of non-linear neutrosophic linear programming problem (NLN-LPP) into equivalent crisp values. Thereafter, the equivalent crisp problem is solved with the usual method to obtain the optimal solution of non-linear neutrosophic linear programming problem (NLN-LPP). The proposed solution algorithm is unique and new for solving non-linear neutrosophic linear programming problems. A numerical example is solved with the proposed algorithm to legitimate the research output. A case study is also discussed to show its applicability in solving real-life problems. [ABSTRACT FROM AUTHOR]

Loss of electrochemical surface area in proton-exchange membrane is of large practical importance, since membrane degradation largely affects the durability and life of fuel cells. In this paper, the electrokinetic model developed by Holby and Morgan is considered. The paper describes degradation mechanisms in membrane catalyst presented by platinum dissolution, platinum diffusion, and platinum oxide formation. A one-dimensional model is governed by nonlinear reaction–diffusion equations given in a cathodic catalyst layer using Butler–Volmer relationships for reaction rates. The governing system is endowed with initial conditions, mixed no-flux boundary condition at the interface with gas diffusion layer, and a perfectly absorbing condition at the membrane boundary. In cyclic voltammetry tests, a non-symmetric square waveform is applied for the electric potential difference between 0.6 and 0.9 V held for 10 and 30 s, respectively, according to the protocol of European Fuel Cell and Hydrogen Joint Undertaking. Aimed at mitigation strategies, the impact of cycling operating conditions and model parameters on the loss rate of active area is investigated. The global behavior with respect to variation of parameters is performed using the method of sensitivity analysis. Finding feasible and unfeasible values helps to determine the range of test parameters employed in the model. Comprehensive results of numerical simulation tests are presented and discussed. [ABSTRACT FROM AUTHOR]

The parameter extraction of Photovoltaic (PV) cell and module is a necessary to simulate and evaluate the performance of the PV system. The parameter extraction is a complex and challenging task due to its non-linear nature. Researchers are used several metaheuristic algorithms to solve the non-linear problem of parameter extraction. However, the demand for most accurate and reliable methods is increasing to get precise estimation of parameters. In this paper, a novel hybrid optimization algorithm is proposed based on the Spotted-Hyena optimization (SHO) and Ant Lion Optimization (ALO). The hybrid method is called as Spotted Hyena - Ant Lion (SH-AL) optimization. The optimization algorithm is applied in two stages. In stage 1, essential parameters are identified and extracted using SHO and passed to stage 2. In stage 2, identified parameters are optimized using ALO for accurate model of PV cell. Different type of PV cells such as thin film, mono and multi crystalline are examined under various irradiance conditions to extract the parameters. The proposed algorithm is validated by comparing the results with other algorithms and proposed algorithm is proved its superiority. [ABSTRACT FROM AUTHOR]

In this paper, a non-classical symmetry method for obtaining the symmetries of differential–difference equations is proposed. The non-classical symmetry method introduces an additional constraint known as the invariant surface condition, which is applied after the infinitesimal transformation. By solving the governing equations that satisfy this condition, we can obtain the corresponding reduced equation. This allows us to determine the non-classical symmetry of the differential–difference equation. This method avoids the complicated calculation involved in extending the infinitesimal generator and allows for a wider range of symmetry forms. As a result, it enables the derivation of a greater number of differential–difference equations. In this paper, two kinds of (2+1)-dimensional Toda-like lattice equations are taken as examples, and their corresponding symmetric and reduced equations are obtained using the non-classical symmetry method. [ABSTRACT FROM AUTHOR]

In this paper, the two-dimensional nonlinear elliptic equation with the boundary integral condition depending on two parameters is solved by finite difference method. The main aim of this paper is to investigate the conditions under those all eigenvalues of corresponding difference eigenvalue problem are positive. For this purpose, we investigate the spectrum structure of one-dimensional difference eigenvalue problem with integral condition. In particular, conditions of the existence and some properties of negative eigenvalue are analyzed in details. [ABSTRACT FROM AUTHOR]

This paper deals with the following nonlinear elliptic equation: − Δ u = Q (| y ′ | , y ″) u N + 2 N − 2 , u > 0 , in R N , u ∈ D 1 , 2 (R N) , where (y ′ , y ″) ∈ R 2 × R N − 2 , N ≥ 5 , Q (| y ′ | , y ″) is a bounded nonnegative function in R 2 × R N − 2 . By using the local Pohozaev identities we prove a nondegeneracy result for the positive solutions constructed in (Peng et al. in J. Differ. Equ. 267:2503–2530, 2019). [ABSTRACT FROM AUTHOR]

Multi‐object tracking in autonomous driving is a non‐linear problem. To better address the tracking problem, this paper leveraged an unscented Kalman filter to predict the object's state. In the association stage, the Mahalanobis distance was employed as an affinity metric, and a Non‐minimum Suppression method was designed for matching. With the detections fed into the tracker and continuous 'predicting‐matching' steps, the states of each object at different time steps were described as their own continuous trajectories. We conducted extensive experiments to evaluate tracking accuracy on three challenging datasets (KITTI, nuScenes and Waymo). The experimental results demonstrated that our method effectively achieved multi‐object tracking with satisfactory accuracy and real‐time efficiency. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of unified interpolative contractions of the Reich–Rus–Ćirić type and give some results about the fixed points for these mappings in the framework of relational metric spaces. We present examples where the results of some previous research are not relevant. Also, we apply our results to solving problems related to nonlinear matrix equations, emphasizing their practical importance. [ABSTRACT FROM AUTHOR]

In this paper, we examine the existence and uniqueness of solutions for a system of the first-order q-difference equations with multi-point and q-integral boundary conditions using various fixed point (fp) theorems. Also, we give two examples to support our results. [ABSTRACT FROM AUTHOR]

By simultaneously transmitting multiple different waveform signals, a multiple-input multiple-output (MIMO) radar possesses higher degrees of freedom and potential in many aspects compared to a traditional phased-array radar. The spatial–temporal characteristics of waveforms are the key to determining their performance. In this paper, a transmitting waveform design method based on spatial–temporal joint (STJ) optimization for a MIMO radar is proposed, where waveforms are designed not only for beam-pattern matching (BPM) but also for minimizing the autocorrelation sidelobes (ACSLs) of the spatial synthesis signals (SSSs) in the directions of interest. Firstly, the STJ model is established, where the two-step strategy and least squares method are utilized for BPM, and the L2p-Norm of the ACSL is constructed as the criterion for temporal characteristics optimization. Secondly, by transforming it into an unconstrained optimization problem about the waveform phase and using the gradient descent (GD) algorithm, the hard, non-convex, high-dimensional, nonlinear optimization problem is solved efficiently. Finally, the method's effectiveness is verified through numerical simulation. The results show that our method is suitable for both orthogonal and partial-correlation MIMO waveform designs and efficiently achieves better spatial–temporal characteristic performances simultaneously in comparison with existing methods. [ABSTRACT FROM AUTHOR]

This paper explores resilient control problems for discrete-time nonlinear semi-Markov jump systems characterized by incremental quadratic constraints. Considering the system's uncertainties and external environmental factors, mode-dependent resilient controllers are developed to ensure the system's mean-square stability. A proposed event-triggering mechanism is suggested to alleviate the communication burden within the system. Additionally, the system's nonlinearity is characterized by using incremental quadratic constraints to derive a less conservative feasible solution. Sufficient conditions for the system's mean-square stability are established by employing the Lyapunov stability theory. Finally, a numerical simulation example is given to prove the conclusion's validity. [ABSTRACT FROM AUTHOR]

The optimal control problems of degenerate parabolic equations have many applications in economics, physics, climatology, and so on. Motivated by the applications, we consider the optimal control problems of a class of nonlinear degenerate parabolic equations in this paper. The main result is that we deduce the first order necessary condition for the optimal control problem of nonlinear degenerate parabolic equations by variation method. Moreover, we investigate the uniqueness of the solutions to the optimal control problems. For the linear equations, we obtain the global uniqueness, while for the nonlinear equations, we obtain only the local uniqueness. Finally, we give a numerical example to validate the theoretical results. [ABSTRACT FROM AUTHOR]

Full-waveform inversion (FWI) is an effective method for imaging subsurface properties using sparsely recorded data. It involves solving a wave propagation problem to estimate model parameters that accurately reproduce the data. Recent trends in FWI have seen a renewed interest in extended methodologies, among which source extension methods leveraging reconstructed wavefields to solve penalty or augmented Lagrangian (AL) formulations have emerged as robust algorithms, even for inaccurate initial models. Despite their demonstrated robustness on synthetic data, challenges remain, such as the lack of a clear physical interpretation and reliance on difficult-to-compute least-squares (LS) wavefields. Moreover, the literature lacks a general and through comparison of these source extension methods with each other and with the standard FWI. This paper is divided into three critical parts. In the first, a novel formulation of these methods is explored within a unified Lagrangian framework. This novel perspective permits the introduction of alternative algorithms that use LS multipliers instead of wavefields. These multiplier-oriented variants appear as regularizations of the standard FWI, are suitable to the time domain, offer tangible physical interpretations, and foster enhanced convergence efficiency. The second part of the paper delves into understanding the underlying mechanisms of these techniques. This is achieved by solving the associated non-linear equations using iterative linearization and inverse scattering methods. The paper provides insight into the role and significance of Lagrange multipliers in enhancing the linearization of the equations. It explains how different methods estimate multipliers or make approximations to increase computing efficiency. Additionally, it presents a new physical understanding of the Lagrange multiplier used in the AL method, highlighting how important it is for improving algorithm performance when compared to penalty methods. In the final section, the paper presents numerical examples that compare different methods within a unified iterative algorithm, utilizing benchmark Marmousi and 2004 BP salt models. [ABSTRACT FROM AUTHOR]

This paper discusses the existence of positive radial symmetric solutions of the nonlinear biharmonic equation ▵ 2 u = f (u , ▵ u) on an annular domain Ω in R N with the Navier boundary conditions u | ∂ Ω = 0 and ▵ u | ∂ Ω = 0 , where f : R + × R − → R + is a continuous function. We present some some inequality conditions of f to obtain the existence results of positive radial symmetric solutions. These inequality conditions allow f (ξ , η) to have superlinear or sublinear growth on ξ , η as | (ξ , η) | → 0 and ∞. Our discussion is mainly based on the fixed-point index theory in cones. [ABSTRACT FROM AUTHOR]

The paper aims to demonstrate that a linear expansion in a unital two-dimensional algebra can generate integrable couplings, proposing a novel approach for their construction. The integrable couplings presented encompass a range of perturbation equations and nonlinear integrable couplings. Their corresponding Lax pairs and hereditary recursion operators are explicitly detailed. Concrete applications to the KdV equation and the AKNS system of nonlinear Schrödinger equations are extensively explored. [ABSTRACT FROM AUTHOR]

To address the issue of system parameter variations during the operation of a maritime light vessel rudder permanent magnet synchronous motor (PMSM), an extended Kalman particle filter (EKPF) algorithm that combines a particle filter (PF) with an extended Kalman filter (EKF) is proposed in this paper. This approach enables the online identification of motor resistance and inductance. For highly nonlinear problems that are challenging for traditional methods such as Kalman filtering, this algorithm is typically a statistical and effective estimation method that usually yields good results. Firstly, a standard linear discrete parameter identification model is established for a PMSM. Secondly, the PF algorithm based on Bayesian state estimation as a foundation for subsequent research is derived. Thirdly, the advantages and limitations of the PF algorithm are analyzed, addressing issues such as sample degeneracy, by integrating it with the Kalman filtering algorithm. Specifically, the EKPF algorithm for online parameter identification is employed. Finally, the identification model within MATLAB/Simulink is constructed and the simulation studies are executed to ascertain the viability of our suggested algorithm. The outcomes from these simulations indicate that the proposed EKPF algorithm identifies resistance and inductance values both swiftly and precisely, markedly boosting the robustness and enhancing the control efficacy of the PMSM. [ABSTRACT FROM AUTHOR]

This paper analyzes a novel use of the shifted Fibonacci polynomials (SFPs) to treat the time-fractional Burgers equation (TFBE). We first develop the fundamental formulas of these polynomials, which include their power series representation and the inversion formula. We establish other new formulas for the SFPs, including integer and fractional derivatives, in order to design the collocation approach for treating the TFBE. These derivative formulas serve as tools that aid in constructing the operational metrics for the integer and fractional derivatives of the SFPs. We use these matrices to transform the problem and its underlying conditions into a system of nonlinear equations that can be treated numerically. An error analysis is analyzed in detail. We also present three illustrative numerical examples and comparisons to test our proposed algorithm. These results showed that the proposed algorithm is advantageous since highly accurate approximate solutions can be obtained by choosing a few terms of retained modes of SFPs. [ABSTRACT FROM AUTHOR]

This paper is concerned with nonlocal fractional p-Laplacian Schrödinger–Hardy-type equations involving concave–convex nonlinearities. The first aim is to demonstrate the L ∞ -bound for any possible weak solution to our problem. As far as we know, the global a priori bound for weak solutions to nonlinear elliptic problems involving a singular nonlinear term such as Hardy potentials has not been studied extensively. To overcome this, we utilize a truncated energy technique and the De Giorgi iteration method. As its application, we demonstrate that the problem above has at least two distinct nontrivial solutions by exploiting a variant of Ekeland's variational principle and the classical mountain pass theorem as the key tools. Furthermore, we prove the existence of a sequence of infinitely many weak solutions that converges to zero in the L ∞ -norm. To derive this result, we employ the modified functional method and the dual fountain theorem. [ABSTRACT FROM AUTHOR]

Orifice Hollow Cathodes are electric devices necessary for the functioning of common plasma thrusters for space applications. Their reliability mainly depends on the success of a spacecraft's mission equipped with electric propulsion. The development of plasma models is crucial in the evaluation of plasma properties within the cathodes that are difficult to measure due to the small dimensions. Many models, based on non-linear systems of plasma equations, have been proposed in the openiterature. These are solved commonly by means of iterative procedures. This paper investigates the possibility of solving them by means of the Particle Swarm Optimization method. The results of the validation tests confirm the expected trends for all the unknowns; the confidence bound of the discharge current as a function of mass flow rate is very narrow (2 ÷ 5) V); moreover, the results match very well the experimental data except at theowest mass flow rate (0.08 mg/s) and discharge current (1A), where the computations underpredict the discharge current to the utmost by 40%. The highest data dispersion regards the plasma density in the emitter region (±20% of the average value) and the wall temperatures (±50 K with respect to the average values) of the orifice and insert; those of the others variables are very tiny. [ABSTRACT FROM AUTHOR]

High‐resolution modeling of the flow dynamics in fractured soils is highly complex and computationally demanding as it requires precise geometrical description of the fractures in addition to resolving a multiphase free‐flow problem inside the fractures. In this paper, we present an idealized model for saturated–unsaturated flow in fractured soils that preserves the core aspects of fractured flow dynamics using an explicit representation of the fractures. The model is based on Richards' equation in the matrix and hydrostatic equilibrium in the fractures. While the first modeling choice is standard, the latter is motivated by the difference in flow regimes between matrix and fractures, that is, the water velocity inside the fractures is considerably larger than in the soil even under saturated conditions. On matrix/fracture interfaces, the model permits water exchange between matrix and fractures only when the capillary barrier offered by the presence of air inside the fractures is overcome. Thus, depending on the wetting conditions, fractures can either act as impervious barriers or as paths for rapid water flow. Since in numerical simulations each fracture face in the computational grid is a potential seepage face, solving the resulting system of nonlinear equations is a nontrivial task. Here, we propose a general framework based on a discrete‐fracture matrix approach, a finite volume discretization of the equations, and a practical iterative technique to solve the conditional flow at the interfaces. Numerical examples support the mathematical validity and the physical applicability of the model. Core Ideas: Due to the presence of air in dry fractured soils, fractures of large aperture act as capillary barriers.If the capillary barriers are overcome, fractures become fast‐flowing paths for water to travel downward.A flow model based on Richards' equation in the soil and instantaneous ponding in the fractures is proposed.The model is numerically consistent and its physical applicability is showcased in two‐dimensional simulations. [ABSTRACT FROM AUTHOR]