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The critical adsorption conditions of polyelectrolytes (PEs) onto planar, cylindrical, and spherical surfaces were obtained by solving the Edwards equation using the Wentzel–Kramers–Brillouin (WKB) method. It demonstrated to provide a suitable analytical approach for all three geometries, in conformity with some experimental results for weakly charged micelles. However, our Monte Carlo simulations implementing approximate solutions of the nonlinear Poisson–Boltzmann equation for highly charged surfaces indicated recently the emergence of a limiting value of ionic strength due to a nonlinear dependence of the electrostatic (ES) potential on the surface-charge density σ. Beyond this limiting ionic strength, the PE adsorption no longer occurs, shifting the standard paradigm. In this work, we employ the concept of a renormalized charge and use the WKB method to study the effects of this nonlinearity on the critical adsorption conditions, density profile, and adsorbed layer of PE segments, all in comparison with the results of the linear Debye–Hückel (DH) approach. Charge renormalization makes it possible to use the known WKB solutions in the DH regime also for surfaces with high σ, introducing a saturation effect observed in the nonlinear case. The larger ES screening affects the density profile and the adsorbed layer of PEs, promoting a more dispersed distribution of PEs at higher surface-charge densities. Our analytical results for the critical adsorption curve reproduce the limiting ionic strength for high σ and also recover the DH regime at low σ. [ABSTRACT FROM AUTHOR]

Basis sets consisting of functions that form linearly independent products (LIPs) have remarkable applications in quantum chemistry but are scarce because of mathematical limitations. We show how to linearly transform a given set of basis functions to maximize the linear independence of their products by maximizing the determinant of the appropriate Gram matrix. The proposed method enhances the utility of the LIP basis set technology and clarifies why canonical molecular orbitals form LIPs more readily than atomic orbitals. The same approach can also be used to orthogonalize basis functions themselves, which means that various orthogonalization techniques may be viewed as special cases of a certain nonlinear optimization problem. [ABSTRACT FROM AUTHOR]

Simulating mixed-state evolution in open quantum systems is crucial for various chemical physics, quantum optics, and computer science applications. These simulations typically follow the Lindblad master equation dynamics. An alternative approach known as quantum state diffusion unraveling is based on the trajectories of pure states generated by random wave functions, which evolve according to a nonlinear Itô–Schrödinger equation (ISE). This study introduces weak first-order and second-order solvers for the ISE based on directly applying the Itô–Taylor expansion with exact derivatives in the interaction picture. We tested the method on free and driven Morse oscillators coupled to a thermal environment and found that both orders allowed practical estimation with a few dozen iterations. The variance was relatively small compared to the linear unraveling and did not grow with time. The second-order solver delivers a much higher accuracy and stability with bigger time steps than the first-order scheme, with a small additional workload. However, the second-order algorithm has quadratic complexity with the number of Lindblad operators as opposed to the linear complexity of the first-order algorithm. [ABSTRACT FROM AUTHOR]

In this paper, the optical soliton and solitary wave solutions of the (2 + 1) -dimensional Mel'nikov equation are investigated using the Kudryashov R function technique. The Kudryashov R function approach has various features that significantly facilitate symbolic computing, particularly for highly dispersive nonlinear equations. In computations, this approach has the benefit of not requiring the use of a certain function form. This approach gives an algorithm that is straightforward, efficient, and simple for finding solitary wave solutions. In addition, this approach is very influential and reliable when it comes to discovering hyperbolic function solutions of nonlinear equations. Many new hyperbolic function solutions have been obtained from the governing equation by using this technique. In addition, numerous types of soliton solutions describing various structures of optical solitons are retrieved. Using this method, breather, W-shaped, bell shaped, and bright soliton solutions have been generated from the governing equation. From the obtained results, it can be asserted that the applied approach may be a useful tool for addressing more highly nonlinear problems in various fields. By choosing particular values for the relevant parameters, the dynamic features of some breather, W-shaped, bell shaped and bright soliton solutions to the (2 + 1) -dimensional Mel'nikov equation have been displayed in 3D, 2D and contour graphs. [ABSTRACT FROM AUTHOR]

Mixed Finite Elements (FEs) with assumed stresses and displacements provide many advantages in analysing shell structures. They ensure good results for coarse meshes and provide an accurate representation of the stress field. The shell FEs within the family designated by the acronym Mixed Isostatic Self‐equilibrated Stresses (MISS) have demonstrated high performance in linear and nonlinear problems thanks to a self‐equilibrated stress assumption. This article extends the MISS family by introducing an eight nodes solid‐shell FE for the analysis of geometrically nonlinear structures. The element, named MISS‐4S, features 24 displacement variables and an isostatic stress representation ruled by 18 parameters. The displacement field is described only by translations, eliminating the need for complex finite rotation treatments in large displacements problems. A total Lagrangian formulation is adopted with the Green–Lagrange strain tensor and the second Piola–Kirchhoff stress tensor. The numerical results concerning popular shell obstacle courses prove the accuracy and robustness of the proposed formulation when using regular or distorted meshes and demonstrate the absence of any locking phenomena. Finally, convergences for pointwise and energy quantities show the superior performance of MISS‐4S compared to other elements in the literature, highlighting that an isostatic and self‐equilibrated stress representation, already used in shell models, also gives advantages for solid‐shell FEs. [ABSTRACT FROM AUTHOR]

This study introduces the nonlinear inertial amplifier resilient friction bearing isolators (NIARFBI). Another friction base isolator, namely the nonlinear inertial amplifier friction pendulum system (NIAFPS), is derived by eliminating the static damping from the governing equation of motion of the NIARFBI. The damping for NIAFPS is generated through the motion of the friction element of the isolator during the movement of the base layer. These novel isolators are placed at the base of the multiple degrees of freedom (MDOF) systems to reduce their dynamic responses. Every element of the highly nonlinear governing equations of motion of the dynamic systems isolated by each nonlinear isolator is linearized using the stochastic linearization approach. The specific mathematical formulation for the optimal design parameters for each nonlinear isolator is derived using the H2 optimization approach. In order to determine the dynamic responses analytically, the transfer function is established. The dynamic responses of the novel isolators are compared with the dynamic responses of the traditional isolators. Accordingly, the displacement reduction capacities of optimum NIARFBI and NIAFPS are significantly 93.60% and 70.82%, superior to TRFBI and TFPS. The acceleration reduction capacities of optimum NIARFBI and NIAFPS are 88.27% and 58.92% superior to TRFBI and TFPS. The recently developed expressions for this study are all mathematically accurate and applicable for practical applications. [ABSTRACT FROM AUTHOR]

The authors performed the optimizations of an auxetic tube considering different structural responses: mass, critical buckling load, natural frequency, Poisson's ratio, and maximum compression load compression. The response surface methodology was applied to generate a metamodeling with a set of non-linear equations. All data were generated via numerical modeling after experimental validation. The Lichtenberg Algorithm was used to find the best possible configurations. This paper shows something unprecedented in the literature when evaluating the static and modal performance of an auxetic tube in a multi-objective problem. The results showed an improvement of up to 43% compared to the initial model. [ABSTRACT FROM AUTHOR]

The aim of this work is to analyze the effect of different materials on the folding of ultrathin tape spring hinges. This mechanism is modeled using the Finite Element Method (FEM) combined with the Carrera Unified Formulation (CUF) for the development of refined 2D shell models. These models can deal with different materials and different theoretical approximations can be introduced in an automatic way thanks to the hierarchical capability of the CUF formalism, according to which the degree of refinement of the mathematical model is an input to the analysis. The nonlinear governing equations are developed using a Total Lagrangian formulation and solved via a Newton-Raphson linearization technique with arc-length type constraints. A validation of the proposed model is carried out by comparing the results with those available in the literature involving a cured T300-1k/913 tow and HexPly 913 resin. After a convergence analysis, the mathematical model proves to ensure a good fit. This model is used for the final analysis with different materials, i.e. isotropic metallic with hardened steel tape spring and unidirectional T300 graphite fibre/epoxy prepreg with 34% resin content by weight. The results, in the form of moment angle curves, are reported for two different values of the thickness of the structure. [ABSTRACT FROM AUTHOR]

Significance: The thermal energy transfer in nanofluid flow over an exponentially stretching surface has crucial practical configurations in various industrial processes, and it has potential applications in heat exchangers, chemical engineering, energy harvesting, and material processing. Purpose: This study is devoted to exploring the features of free convection in the thermally stratified, unsteady flow of Casson fluid over an inclined, exponentially stretching surface. Moreover, the implications of nonlinear thermal radiation, activation energy, and thermal/salute stratification effects are examined over the thermal energy transport distributions. Diffusion-thermo and thermo diffusion impressions are also taken into consideration. Methodology: By introducing reasonable transformations, partial differential equations are altered into ordinary differential equations. A nonlinear system of differential equations is solved numerically by employing the Midrich numerical technique. Findings: The impacts of diverse fluid parameters like the Soret/Dufour number, temperature difference parameter, radiation parameter, thermal/salute stratification parameter, magnetic parameter, and Prandtal number are assessed and depicted in plots by explaining the physical justifications of each parameter. Also, numerical values of sink friction and local Nusselt and Sherwood numbers are computed and examined for different values of pertinent variables involved in the problems. It is found that the rate of thermal energy transport is significantly enhanced by the larger estimation of the radiation parameter. Furthermore, it is perceived that the escalation in the temperature ratio constant leads to increased thermal convection in the fluid, while the larger thermal stratification constant decays the rate of heat transport in the fluid. Additionally, the rate of thermal transport is de-escalated due to the escalation in the intensity of thermal stratification parameter. [ABSTRACT FROM AUTHOR]

This article proposes a novel thin-walled polyhedral functionally graded porous (FGP) liner consolidated by graphene nanofillers (GNF) to rehabilitate the cracked pipeline. A buckle propagation may occur since the liner is subjected to hydrostatic pressure and temperature variational field simultaneously. A cosine function is developed to describe the deformed shape of the liner. Both pores and the GNF are distributed symmetrically on the cross-section of the liner, which also has a polyhedral profile to improve its bending stiffness. The nonlinear equilibrium equations are accurately obtained by introducing the thin-walled shell theory and the principle of minimum potential energy. The critical buckling pressure is obtained by solving the nonlinear equilibrium equations. Analytical solutions are compared successfully with other closed-form results to accomplish the verification when the FGP-GNF polyhedral liner reduces to an isotropic circular one. Furthermore, one improvement factor is proposed to analyze the effect of different polyhedral shapes on the buckling pressure of the encased polyhedral FGP-GNF liner. Finally, parameters are evaluated to further understand the buckling behavior of the FGP-GNF polyhedral liner, including the number of polygons, porosity coefficient, mass fraction, geometric shape of the GNF, radius-to-thickness ratio, temperature field, etc. [ABSTRACT FROM AUTHOR]

We study asymptotic behaviour of positive ground state solutions of the nonlinear Choquard equation (P ε) − Δ u + ε u = (I α ⁎ | u | p) | u | p − 2 u + | u | q − 2 u , in R N , where N ≥ 3 is an integer, p ∈ [ N + α N , N + α N − 2 ] , q ∈ (2 , 2 N N − 2) , I α is the Riesz potential of order α ∈ (0 , N) and ε > 0 is a parameter. We show that as ε → 0 (resp. ε → ∞), the ground state solutions of (P ε) , after appropriate rescalings dependent on parameter regimes, converge in H 1 (R N) to particular solutions of five different limit equations. We also establish a sharp asymptotic characterisation of such rescalings, and the precise asymptotic behaviour of u ε (0) , ‖ ∇ u ε ‖ 2 2 , ‖ u ε ‖ 2 2 , ∫ R N (I α ⁎ | u ε | p) | u ε | p and ‖ u ε ‖ q q , which depend in a non-trivial way on the exponents p , q and the space dimension N. Further, we discuss a connection of our results with a mass constrained problem, associated to (P ε) with normalization constraint ∫ R N | u | 2 = c 2. As a consequence of the main results, we obtain the existence, multiplicity and precise asymptotic behaviour of positive normalized solutions of such a problem as c → 0 and c → ∞. [ABSTRACT FROM AUTHOR]

We prove the well posedness in weighted Sobolev spaces of certain linear and nonlinear elliptic boundary value problems posed on convex domains and under singular forcing. It is assumed that the weights belong to the Muckenhoupt class A p with p ∈ (1 , ∞). We also propose and analyze a convergent finite element discretization for the nonlinear elliptic boundary value problems mentioned above. As an instrumental result, we prove that the discretization of certain linear problems are well posed in weighted spaces. [ABSTRACT FROM AUTHOR]

Using the Darboux transformation method, the general Lax equation is solved and a collection of new exact solutions together with one-soliton solutions, singular one-soliton solutions, periodic solutions, singular periodic solution, two-soliton solutions, singular two-soliton solutions, two-periodic solutions and singular two-periodic solutions is obtained. Using traveling wave transformation, the Lax equation is transfigured to a conservative dynamical system (CDS) of dimension four with three equilibrium points involving two parameters γ and v. The CDS has various quasi-periodic motions for fixed values of the parameters γ and v at different initial conditions. Furthermore, effects of the parameters γ and v are shown on the quasiperiodic motions of the CDS by means of phase sections and time series plots. This approach can be applied to a heterogeneity of nonlinear model equations or partial differential equations for describing their inherent nonlinear phenomena. [ABSTRACT FROM AUTHOR]

The Korteweg–de Vries equation is a fundamental nonlinear equation that describes solitons with constant velocity. On the contrary, here we show that this equation also presents accelerated wavepacket solutions. This behavior is achieved by putting the KdV equation in terms of the Painlevé I equation. The accelerated waveform solutions are explored numerically showing their accelerated behavior explicitly. [ABSTRACT FROM AUTHOR]

In this work, we continue the development of methods for constructing Lax pairs and recursion operators for nonlinear integrable hyperbolic equations of soliton type, previously proposed in the work of Habibullin et al (2016 J. Phys. A: Math. Theor. 49 035202). This approach is based on the use of the well-known theory of Laplace transforms. The article completes the proof that for any known integrable equation of sine-Gordon type, the sequence of Laplace transforms associated with its linearization admits a third-order finite-field reduction. It is shown that the found reductions are closely related to the Lax pair and recursion operators for both characteristic directions of the given hyperbolic equation. Previously unknown Lax pairs and recursion operators were constructed. [ABSTRACT FROM AUTHOR]

We consider an elliptic problem with nonlinear boundary condition involving nonlinearity with superlinear and subcritical growth at infinity and a bifurcation parameter as a factor. We use re-scaling method, degree theory and continuation theorem to prove that there exists a connected branch of positive solutions bifurcating from infinity when the parameter goes to zero. Moreover, if the nonlinearity satisfies additional conditions near zero, we establish a global bifurcation result, and discuss the number of positive solution(s) with respect to the parameter using bifurcation theory and degree theory. [ABSTRACT FROM AUTHOR]

In this work, we give a qualitative study on the existence of positive solutions for local and nonlocal elliptic integro-differential quasilinear systems, by using the concept of super- and subsolutions combined with Schauder's fixed point theorem. [ABSTRACT FROM AUTHOR]

The typical treatment for tumor involves a combination of radiation therapy and chemotherapy. In this study, we explore a mathematical model based on proliferation and diffusion, considering the impacts of both radiotherapeutic and chemotherapeutic treatments. The mathematical model is formulated as a system of partial differential equations, accompanied by initial, boundary, and free boundary conditions. Existence and uniqueness of this mathematical problem are obtained by developing an iteration algorithm in this paper. [ABSTRACT FROM AUTHOR]

A nonstandard optimized packing convex polygons in a rectangular container is considered. The shapes of the polygons are not fixed: the polygons can be compressed/stretched in certain limits along the principal axes, but their areas remain constant. The polygons must be placed completely in the container without overlapping under free translations and rotations. The objective is to define the shapes of the polygons and their locations to minimize the height of the container. Non-overlapping and containment conditions for the stretched polygons are presented using the phi-function technique. The overall packing problem is formulated in the form of a nonconvex nonlinear programming problem. A solution approach is proposed based on the multistart strategy. To illustrate the main steps of the solution technique a numerical example is provided. Directions for future research are discussed. Our interest in this class of packing problems is motivated by studying properties of porous media under external force. Elements of porous media can be deformed under pressure, but the mass of each particle is conserved. In two-dimensional case this corresponds to the area conservation. [ABSTRACT FROM AUTHOR]

A numerical optimization technique used to address nonlinear programming problems is the trust region (TR) method. TR uses a quadratic model, which may represent the function adequately, to create a neighbourhood around the current best solution as a trust region in each step, rather than searching for the original function's objective solution. This allows the method to determine the next local optimum. The TR technique has been utilized by numerous researchers to tackle multi‐objective optimization problems (MOOPs). But there is not any publication that discusses the issue of applying a chaotic search (CS) with the TR algorithm for solving multi‐objective (MO) problems. From this motivation, the main contribution of this study is to introduce trust‐region (TR) technique based on chaotic search (CS) for solving MOOPs. First, the reference point interactive approach is used to convert MOOP to a single objective optimization problem (SOOP). The search space is then randomly initialized with a set of initial points. Second, in order to supply locations on the Pareto boundary, the TR method solves the SOOP. Finally, all points on the Pareto frontier are obtained using CS. A range of MO benchmark problems have demonstrated the efficiency of the proposed algorithm (TR based CS) in generating Pareto optimum sets for MOOPs. Furthermore, a demonstration of the suggested algorithm's ability to resolve real‐world applications is provided through a practical implementation of the algorithm to improve an abrasive water‐jet machining process (AWJM). [ABSTRACT FROM AUTHOR]

The vibration character of piezoelectric semiconductor nano-plate is important for understanding the multi-fields coupling and optimizing the serving behavior. In this article, the nonlinear vibration of five-layered functionally graded piezoelectric semiconductor nano-plate based on Pasternak foundation is studied, and the nonlocal theory is used to analyze the size-dependent dynamic response of nano-plate. Combining the Von karman's nonlinear deformation theory and the constitutive of piezoelectric semiconductor layers, Hamilton's principle is introduced to derive the nonlinear governing equations. To solve the nonlinear governing equations, the updated iteration method for this problem is proposed. Through numerical examples, it is found that the initial electron concentration in piezoelectric semiconductor layer, the nonlocal parameter, the functionally graded index in the plate, the elastic modulus of Pasternak foundation and the geometrical parameters can be designed to effectively tune the dynamic nonlinear frequency and damping of layered functionally graded piezoelectric semiconductor nanoplate. The interaction of these parameters is also analyzed in detail. Comparison with existing numerical results is also presented. A new way is provided to tune the nonlinear vibration of multi-layered piezoelectric semiconductor nano-plate. [ABSTRACT FROM AUTHOR]

This paper proposes a base excited, spring coupled double cantilever beam with attached tip mass based piezoelectric energy harvester. The dimensions of the beams are so chosen that the system experiences 1:1 internal resonance. The governing coupled electro-mechanical equations of motion of the system are obtained by using the Lagrange principle which is reduced to the temporal form by using generalized Galerkin's method. The governing temporal equations of motion are in the form of that of a parametrically excited coupled system with geometric and inertial nonlinearities. These nonlinear equations are solved using the method of multiple scales and the results are compared with those solved by numerical method (4th order Runge Kutta method) for principal parametric resonance conditions. Parametric studies are conducted by varying the tip masses of the beams, coupled spring stiffness, load resistance, external excitation frequency and excitation amplitude. Results show that by introducing spring coupling, the system operational frequency range increases significantly and also energy transfer process takes place due to the internal resonance. Significant enhancement of voltage and power are observed in the present work in comparison to the earlier studied equivalent single beam-based energy harvester. [ABSTRACT FROM AUTHOR]

This paper presents a precise and effective formulation for modeling and simulating impact with friction for planar multibody systems. Nonlinear equations of motion are formulated using normal impulse at contact point as a time-like independent variable. The derived equations demonstrate that sliding during impact period could persist, or it could cease leading either to a persistence of sticking or to a reverse sliding. To distinguish between these different modes, Routh's incremental method is employed. A critical coefficient of friction, dependent solely on the system configuration, is identified and used to determine whether the contact mode is sliding or sticking. Three definitions of coefficient of restitution are introduced to model the plasticity of the impact. Analytical algebraic solutions and a computational strategy are offered to identify the mode of impact and to evaluate the impact variables. An example is thoroughly examined to demonstrate the effectiveness of the formulation, validate the algebraic solutions, and evaluate the outcomes of the three definitions of the coefficient of restitution. [ABSTRACT FROM AUTHOR]

We use potential analysis to study the properties of positive solutions of a discrete Wolff-type equation $$ \begin{align*} w(i)=W_{\beta,\gamma}(w^q)(i), \quad i \in \mathbb{Z}^n. \end{align*} $$ Here, $n \geq 1$ , $\min \{q,\beta \}>0$ , $1 and $\beta \gamma. Such an equation can be used to study nonlinear problems on graphs appearing in the study of crystal lattices, neural networks and other discrete models. We use the method of regularity lifting to obtain an optimal summability of positive solutions of the equation. From this result, we obtain the decay rate of $w(i)$ when $|i| \to \infty $. [ABSTRACT FROM AUTHOR]

The solution of one-dimensional nonlinear problem of stationary heat conduction in a dielectric layer with volumetric energy release at a constant difference in electric potentials on the curvilinear heat transfer surfaces, bounding the layer, is presented in a closed analytical form. The analytical form of the solution, being a set of integral relations, allows to conduct a quantitative analysis of the influence of governing parameters on temperature state of the layer, including the one preceding the occurrence of thermal dielectric breakdown. Such form of solution can be used to test the results of numerical solution of this problem. An example of quantitative analysis of the presented integral relations is given. Approximate formulas that make it possible to find estimates of the parameters of temperature state of dielectric layer and the maximum difference in electrical potentials, preceding the thermal breakdown of the layer, are obtained. [ABSTRACT FROM AUTHOR]

It has been observed in experiments that, for some materials, when a sheet is subjected to increasing equal biaxial tensile forces on its edges, its deformation can change from homogeneous equal biaxial extension to homogeneous unequal biaxial extension. Such response has been analyzed in the literature as a bifurcation from the base deformation to a new deformation. The analyses have used a constitutive equation that expresses the stress tensor as a function of a deformation tensor. The present work is concerned with analyzing this bifurcation using a new class of constitutive equations in which the deformation tensor is expressed as a function of the stress tensor. Using such a model for an incompressible isotropic nonlinear elastic material, a condition is derived for determining when during an equal biaxial extension of a sheet, there is a bifurcation into an unequal biaxial extension. An example is provided using a constitutive equation that has been fit to experimental data. [ABSTRACT FROM AUTHOR]

In this paper, we shall deal with the long‐time dynamics of a nonlinear system of wave equation with a fractional damping and a small delay. Inspired by the works of Chueshov and Lasiecka on the property of quasi‐stability of dynamic systems, we show first the quasi‐stability property holds for the problem considered here and then prove the existence of global and exponential attractors. We shall also prove the upper semicontinuities of global attractors with respect to the fractional exponents. Finally, we shall show the continuities of global attractors with respect to a small parameter multiplying the delay term in a residual dense set and their upper semicontinuities in a complete metric space. [ABSTRACT FROM AUTHOR]

In this paper, a 2D multiterm nonlinear problem of the form ∑m=1lqmDtαmu−Δu=f(u)$$ {\sum}_{m=1}^l{q}_m{D}_t^{\alpha_m}u-\Delta u=f(u) $$ is considered, where each Caputo fractional derivative is of order αm∈(0,1)$$ {\alpha}_m\in \left(0,1\right) $$. We use the Grünwald–Letnikov(GL) scheme on uniform mesh to discretize the multiterm Caputo fractional derivative and finite difference scheme is used for spatial discretization, and then we construct a fully discrete GL‐ADI scheme. A discrete Gronwall inequality is introduced for getting the sharp pointwise‐in‐time error estimate on uniform mesh. Numerical examples are provided to verify the sharpness of our error estimate. [ABSTRACT FROM AUTHOR]

This work investigates the non‐global existence of solutions for the inhomogeneous nonlinear Schrödinger problem without gauge invariance. Indeed, initially, we establish the non‐global existence of certain mass‐subcritical solutions. Subsequently, we demonstrate the non‐global existence of energy‐subcritical solutions characterized by small initial data. These results are are acknowledged to be unsuccessful for the classical gauge invariant inhomogeneous nonlinear Schrödinger equation (INLS). [ABSTRACT FROM AUTHOR]

In this study, we utilize a specific combination of orthogonal Gegenbauer polynomials as basis functions that satisfy homogeneous boundary conditions in the spatial variable and homogeneous initial conditions in the temporal variable. We derive an expression for the fractional derivative of the temporal basis and another expression for the second‐order derivative of the spatial basis. Subsequently, we discretize the nonlinear Fisher problem using the spectral collocation method to transform the problem into the solution of a nonlinear system of algebraic equations. We then solve this resulting system using Newton's method with an initial guess approaching zero. Furthermore, we conduct an error analysis of the proposed method and assess its applicability through three test problems, providing comparisons for validation. [ABSTRACT FROM AUTHOR]

This paper considers the viscous/inviscid Moore–Gibson–Thompson (MGT) equations with memory of type I in the whole space ℝn$$ {\mathrm{\mathbb{R}}}^n $$. For one thing, associating with a new condition on initial data, we derive the optimal L2$$ {L}^2 $$ estimates and the optimal leading term of the acoustic velocity potential for large time, where we analyze different contributions from viscous, thermally relaxing, as well as hereditary fluids on large time asymptotic behavior for the acoustic waves models. For another, by using the multiscale analysis and energy methods in the Fourier space, we demonstrate the L∞$$ {L}^{\infty } $$ inviscid limits (i.e., as the diffusivity of sound tends to zero), which match our Wentzel‐Kramers‐Brillouin (WKB) expansion of the solution. Finally, we give a further application of our results on large time behavior for the nonlinear Jordan‐MGT equation in viscous hereditary fluids. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of weak solutions for the nonlinear parabolic problem associated with the following fourth‐order equation, under Navier boundary conditions: ∂u∂t+Δp(x)2u+g(x,t,u,∇u)=f(x,t)inQ:=Ω×(0,T),$$ \frac{\partial u}{\partial t}+{\Delta}_{p(x)}^2u+g\left(x,t,u,\nabla u\right)=f\left(x,t\right)\kern3.0235pt \mathrm{in}\kern0.30em \mathcal{Q}:= \Omega \times \left(0,T\right), $$where Ω⊂ℝn(N≥2)$$ \Omega \subset {\mathrm{\mathbb{R}}}^n\kern3.0235pt \left(N\ge 2\right) $$ is a bounded domain. Under some appropriate assumptions, we obtain the existence of a weak solution for the above problem by using the topological degree theory for operators of the type L+S+C$$ \mathcal{L}+\mathcal{S}+\mathcal{C} $$, in the framework space Lp−(0,T;V)$$ {L}^{p-}\left(0,T;\mathcal{V}\right) $$. [ABSTRACT FROM AUTHOR]

The aim of this paper is to derive sufficient conditions for the existence, uniqueness, and Hyers–Ulam stability of solutions to a new nonlinear fractional integro‐differential equation with functional boundary conditions, using several fixed‐point theorems, the multivariate Mittag‐Leffler function and Babenko's approach. A few examples are also presented to illustrate the applications of our results based on approximate values of a couple of Mittag‐Leffler functions calculated by Python codes. Furthermore, the approaches used have a wide range of applications to various fractional differential equations with initial or boundary conditions or integral equations in complete spaces. [ABSTRACT FROM AUTHOR]

We present some existence and localization results for periodic solutions of impulsive first‐order coupled nonlinear systems of two equations, without requiring periodicity for the nonlinearities. The arguments are based on Schauder's fixed point theorem together with the upper and lower solution methods, where the upper and lower solutions are not necessarily well ordered. In addition, results on equi‐regulated functions are required for the impulsive analysis. An application to a Wilson‐Cowan system of two strongly coupled neurons illustrates one of the main results. [ABSTRACT FROM AUTHOR]

In this paper, feedback laws for a class of infinite horizon control problems under state constraints are investigated. We provide a two-player game representation for such control problems assuming time-dependent dynamics and Lagrangian and the set constraints merely compact. Using viability results recently investigated for state constrained problems in an infinite horizon setting, we extend some known results for the linear-quadratic regulator problem to a class of control problems with nonlinear dynamics in the state and affine in the control. Feedback laws are obtained under suitable controllability assumptions. [ABSTRACT FROM AUTHOR]

We consider the aeroelastic simulation of flexible mechanical structures submerged in subsonic fluid flows at low Mach numbers. The nonlinear kinematics of flexible bodies are described in the total Lagrangian formulation and discretized by finite elements. The aerodynamic loads are computed using the unsteady vortex-lattice method wherein a free wake is tracked over time. Each implicit time step in the dynamic simulation then requires solving a nonlinear equation system in the structural variables with additional aerodynamic load terms. Our focus here is on the efficient numerical solution of this system by accelerating the Newton algorithm. The particular structure of the aeroelastic nonlinear system suggests the structural derivative as an approximation to the full derivative in the linear Newton system. We investigate and compare two promising algorithms based on this approximation, a quasi-Newton type algorithm and a novel inexact Newton algorithm. Numerical experiments are performed on a flexible plate and on a wind turbine. Our computational results show that the approximation can indeed accelerate the Newton algorithm substantially. Surprisingly, the theoretically preferable inexact Newton algorithm is much slower than the quasi-Newton algorithm, which motivates further research to speed up derivative evaluations. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of generalized saddle points for a function defined on a product of topological spaces. The generalized saddle points are defined as minima of a special function, that is the supremum of Nikaido-Isoda function. We prove elementary properties of the generalized saddle points and study their dependence on a parameter. We show that under natural assumptions if a function has the only generalized saddle point, then this point is stable. Moreover, for functions defined over a sets of solutions to nonlinear equations with a parameter in finite-dimensional spaces, we provide sufficient conditions of the generalized saddle point stability in terms of λ-truncations. [ABSTRACT FROM AUTHOR]

The present research paper deals with the study of magnetized dissipative Soret of steady viscous incompressible two-dimensional Maxwell fluid flow in a porous medium over a stretching sheet with chemical reaction and Joule heating. Maxwell nanofluid flow model using to differentiate it from the Newtonian fluids. Nonlinear coupled partial differential equations model has been not applicable to any of the direct methods. A Von-Karman similarity transformations are used to convert the nonlinear system of differential equations and are solved numerically via the bvp4c MATLAB function. The simulated results are shown for various values of physical parameters in the flow regime. In addition to, the numerical results of magnetized dissipative Soret, chemical reaction and Joule heating are calculated and tabularized. The effects of various parameters like Maxwell fluid, magnetic parameter, chemical reaction parameter, wall temperature, etc. on normal velocity profile, axial velocity profile, temperature profile and concentration profile. Comparison of current similarity solutions with available results indicates the accuracy and guarantee of the present numerical results and the used method. The results indicate that the effects of the Maxwell fluid, magnetic parameter, chemical reaction parameter, wall temperature are very pronounced. [ABSTRACT FROM AUTHOR]

The provided research deals with a mathematical formulation to achieve nonlinear statics for the sandwich annular structure's thermal post-buckling behavior. The structure's core is filled by graphene nanoplatelets (GPLs), while its face sheets have been made of SMA/graphite/epoxy. The first-order shear deformation (FOSD) model has been employed for modeling face sheets, and quasi-3D hybrid and quadratic kind of higher-order shear deformation (Q-3D-HSD) model is supposed for core's in-plane transverse and displacements, respectively. The boundary conditions (BCs) and nonlinear governing equations have been modeled by employing the Hamiltonian, and they have been solved applying the generalized differential quadrature (GDQ) approach. The direct-iterative method has been provided for solving the set of equations, including highly nonlinear factors. Ultimately, the outcomes reveal that the radius ratio of outer to the inner, SMA fiber, GPLs' configurational factors, and large amplitude would play a vital role in the annular sandwich structure's thermal post-buckling response. Moreover, the system with GPL-O distribution pattern has maximum thermal post-buckling load while GPL-X type of distribution would result in the minimum thermal post-buckling to thermal buckling ratio for the annular structure. [ABSTRACT FROM AUTHOR]

This paper investigates the nonlinear vibrational characteristics of the two-dimensional temperature-dependent, and imperfect functionally graded (2D-FG) truncated conical nanotubes based on the nonlocal strain gradient theory coupled with Von-Kármán nonlinear theory using a novel high-order tube theory along with the first-order beam theory. The Hamilton principle is used to derive the nonlinear and nonlocal governing equations and associated boundary conditions. A numerical approach, including a couple of iteration techniques and the generalized differential quadrature method (GDQM), is used to solve linear and nonlinear equations for different clamped and simply-supported boundary conditions. A composition makes the functionally graded temperature/- and porosity-dependent material of ceramic and metal phases in the radius direction that the tube radius also changes along the tube length, which means the properties of the material are varied in both axial and radial direction that means it is two-dimensional functionally graded material (2D-FGM). The obtained results are discussed in detail to investigate the effect of the nonlocal and strain gradient parameter, temperature change, nonlinear amplitude, porosity parameter, radius aspect ratio, etc., on the vibrational behavior of nanotubes. [ABSTRACT FROM AUTHOR]

The goal of this article is to examine the heat and mass transportation of a Maxwell fluid flow in the rotating frame with double stratification and porous medium through a linear stretching surface. The heat transfer scrutiny has been presented in the existence of Cattaneo-Christov theory and variable thermal conductivity. The suitable transformation is considered to modify the flow model into the nonlinear system of ordinary differential equations. The numerical computation for the nonlinear system of equations is built-in by the Bvp4c MATLAB approach. It is found that both rotation and stretching have a noteworthy impact on temperature and the velocity profile. Thermal stratification enhances the heat transfer rate as well as higher estimations of rotation parameter condense the heat flux. The novel idea of the present problem is to analyze the heat transfer and Maxwell fluid flow into the rotating frame along Cattaneo-Christov theory induced by the stretching sheet. A relative analysis of current computations with prevailing literature revealed an exceptional agreement. [ABSTRACT FROM AUTHOR]

Solitons and waves with memory effects are two examples of nonlinear wave phenomena that can be studied mathematically using the fractional nonlinear sine-Gordon equation. It is a modification of the traditional sine-Gordon equation that takes memory effects and nonlocal interactions into account by adding fractional derivatives. A more complex explanation of particle dynamics and interactions within relativistic quantum mechanics is made possible by the fractional Klein–Gordon model, a theoretical framework that expands the standard equation to include fractional derivatives. The study uses shifted Legendre–Gauss–Lobatto and shifted Legendre–Gauss–Radau collocation techniques to solve numerically two-dimensional sine-Gordon and Klein–Gordon models. The study handles two-dimensional sine-Gordon and Klein–Gordon models by extending a collocation approach using basis functions. It suggests a collocation technique that uses a suggested basis to automatically satisfy the conditions. The suggested methods' spectral accuracy and efficiency are validated by numerical results. [ABSTRACT FROM AUTHOR]

This article studies the cluster consensus problem for the nonlinear multi‐agent systems subject to input saturation under the context of event triggered control. Taking ordinary consensus object as a special case, this article considers the problem of cluster consensus. The event triggered mechanism is designed and the Zeno phenomenon is ruled out. The sufficient conditions of control protocol and range estimation of domain of attraction are formulated by the linear matrix inequalities for the constraints of input saturation and nonlinear terms existed in the system dynamics. Finally, the proposed results are validated by one example in the background of spring damping system. [ABSTRACT FROM AUTHOR]

Efficient mineral hydraulic collection is a key issue in the mining of deep-sea minerals, where the collection efficiency is related to the lifted fluid velocity and the mineral particle velocity. In this paper, a robust control contraction metric (RCCM)-based tube model predictive control (Tube-MPC) for efficient mineral hydraulic collection is proposed. Firstly, a simplified control model is designed with empirical formula of hydrodynamic forces to avoid the computational burden in solution of the fluid equations. Secondly, a nonlinear programing problem (NLP) for efficiency optimization is formulated to calculate the nominal control law and the optimal state. To address external disturbance and model mismatch from the simplified model, a robust control contraction metric is utilized to calculate the feedback control law to track the optimal state. Finally, numerical simulations and experiments are conducted to verify the mineral hydraulic collection performance of the control strategy. [ABSTRACT FROM AUTHOR]

This article describes a projection method based on a combination of rational and polynomial approximations for efficiently solving large nonlinear eigenvalue problems. In a first stage the nonlinear matrix function T(λ)$$ T\left(\lambda \right) $$ under consideration is approximated by a matrix polynomial in λ$$ \lambda $$. The error resulting from this polynomial approximation is in turn approximated by rational functions with the help of the Cauchy integral formula. The two approximations are combined and a linearization is performed. A key ingredient of the proposed approach is a projection method that uses subspaces spanned by vectors of the same dimension as that of the original problem instead of that of the linearized problem. A procedure is also presented to automatically select shifts and to partition the region of interest into a few subregions. This allows to subdivide the problem into smaller subproblems that are solved independently. The accuracy of the proposed method is theoretically analyzed and its performance is illustrated with a few test problems that have been discussed in the literature. [ABSTRACT FROM AUTHOR]

Earlier, an approximation technique was presented for solving strong nonlinear oscillator equations. Due to arising algebraic complexities, the method fails to determine suitable solutions of some nonlinear oscillators such as quadratic oscillators, the cubical Duffing oscillator of softening springs, and pendulum equations. Then, rearranging an algebraic equation related to amplitude and frequency, the method covers the noted problems. In this article, the latter technique is applied to handling anti-symmetric constant force oscillators and anti-symmetric quadratic nonlinear oscillators. [ABSTRACT FROM AUTHOR]

Parametric timed automata (PTA) extend timed automata (TA) with parameters instead of fixed timing constraints, providing the flexibility to accommodate uncertainties during the design phase. Once a parametric model is obtained, the next step is finding the optimal parameters such that the resulting TA satisfies the specifications. This paper introduces a new algorithm for determining parameters from safety specifications for PTA with bounded integer parameters and no nested cycles. The algorithm searches for unsafe paths through a depth-first search and generates parameter constraints. In particular, the realizability of simple and cyclic paths are encoded via mixed integer linear programming and non-linear programming problems. Then, the parameter constraints rendering the path unrealizable are derived via quantifier elimination. The accumulated constraints through the depth-first search guarantee that a parameter valuation satisfying these constraints solves the synthesis problem. The results are illustrated over benchmarks. [ABSTRACT FROM AUTHOR]

The model-implied simulation-based power estimation (MSPE) approach is a new general method for power estimation (Irmer et al., 2024). MSPE was developed especially for power estimation of non-linear structural equation models (SEM), but it also can be applied to linear SEM and manifest models using the R package powerNLSEM. After first providing some information about MSPE and the new adaptive algorithm that automatically selects sample sizes for the best prediction of power using simulation, a tutorial on how to conduct the MSPE for quadratic and interaction SEM (QISEM) using the powerNLSEM package is provided. Power estimation is demonstrated for four methods, latent moderated structural equations (LMS), the unconstrained product indicator (UPI), a simple factor score regression (FSR), and a scale regression (SR) approach to QISEM. In two simulation studies, we highlight the performance of the MSPE for all four methods applied to two QISEM with varying complexity and reliability. Further, we justify the settings of the newly developed adaptive search algorithm via performance evaluations using simulation. Overall, the MSPE using the adaptive approach performs well in terms of bias and Type I error rates. [ABSTRACT FROM AUTHOR]

The geometries constructed using the standard CAD software are always boundary-represented (B-Rep). Due to this, analysis-suitable volumetric parameterization of the geometries constructed using the standard CAD software is necessary for three-dimensional isogeometric analysis (IGA). This paper presents a user-friendly, robust volumetric parametrization tool capable of processing the B-Rep of the geometries constructed in one of the most popular CAD software, i.e. Rhinoceros®. The developed tool traverses multiple stages, starting with reading the standard 3DM CAD files of an arbitrarily shaped geometry. It then extracts surface control points, organizes them to establish an outer structural framework, and employs Coons volume parameterization to generate internal control points based on user-defined specifications. We use the NURBS-Python library for reading 3DM files and the NURBS toolbox to visualize and evaluate NURBS functions in the constructed geometries. We construct several baseline and intricate shape geometries to showcase the robustness and generality of the developed tool. Additionally, to demonstrate the effectiveness and applicability of our tool for IGA of non-linear three-dimensional problems, we simulate frictionless contact between two deformable geometries constructed using the developed tool. [ABSTRACT FROM AUTHOR]

Fractional calculus is an important mathematical tool that is widely used in control systems. It is established in the literature that fractional order models are more accurate and more effective in system modelling. In this study, an alternative and novel technique is proposed to identify the fractional order time-delayed model of an unknown system. The method is based on obtaining the approximate stability boundary locus (SBL) curve of the unknown system by applying three different experimental tests. Three points on the SBL curve are determined by the experimental tests and then the parameters of the fractional order time-delayed model are computed by solving the nonlinear systems of equation. The system model with double fractional order element plus a time delay is obtained using the proposed method. The proposed method is explained through simulations on a twin rotor system. The proposed method is also used in model order reduction calculation of the higher order transfer functions. [ABSTRACT FROM AUTHOR]