Search

In the fields of oceanography, hydrodynamics, and marine engineering, many mathematicians and physicists are interested in Burgers-type equations to show the different dynamics of nonlinear wave phenomena, one of which is a (3+1)-dimensional Burgers system that is currently being studied. In this paper, we apply two different analytical methods, namely the generalized Kudryashov (GK) method, and the generalized exponential rational function method, to derive abundant novel analytic exact solitary wave solutions, including multi-wave solitons, multi-wave peakon solitons, kink-wave profiles, stripe solitons, wave-wave interaction profiles, and periodic oscillating wave profiles for a (3+1)-dimensional Burgers system with the assistance of symbolic computation. By employing the generalized Kudryashov method, we obtain some new families of exact solitary wave solutions for the Burgers system. Further, we applied the generalized exponential rational function method to obtain a large number of soliton solutions in the forms of trigonometric and hyperbolic function solutions, exponential rational function solutions, periodic breather-wave soliton solutions, dark and bright solitons, singular periodic oscillating wave soliton solutions, and complex multi-wave solutions under various family cases. Based on soft computing via Wolfram Mathematica, all the newly established solutions are verified by back substituting them into the considered Burgers system. Eventually, the dynamical behaviors of some established results are exhibited graphically through three - and two-dimensional wave profiles via numerical simulation.

This paper applies a powerful scheme, namely Bernoulli sub-equation function method, to some partial differential equations with high non-linearity. Many new travelling wave solutions, such as mixed dark-bright soliton, exponential and complex domain, are reported. Under a suitable choice of the values of parameters, wave behaviours of the results obtained in the paper – in terms of 2D, 3D and contour surfaces – are observed.

This paper deals with, in the framework of absolute stability, boundary stabilization for a nonlinear axially moving beam under boundary velocity feedback controls. The nonlinear boundary control that satisfies a slope-sector condition covering many types of nonlinear control schemes is a negative feedback of the transverse velocity at the right eyelet of the moving beam. Under the nonlinear control scheme, the well-posedness of the nonlinear partial differential equation which depends continuously on the initial value is investigated by means of the Faedo-Galerkin approximation and priori estimates. By exploiting the integral-type multiplier method, the exponential stability of the closed-loop system is established, where a novel energy like function is constructed. The numerical simulation examples using the finite element method are presented to illustrate the effectiveness of the established criterion of the controller.

In this paper, we prove the existence of nontrivial solutions and ground state solutions for the following planar Schrödinger–Poisson system with zero mass − Δ u + ϕ u = ( I α ∗ F ( u ) ) f ( u ) , x ∈ R 2 , Δ ϕ = u 2 , x ∈ R 2 , where α ∈ ( 0 , 2 ), I α : R 2 → R is the Riesz potential, f ∈ C ( R , R ) is of subcritical exponential growth in the sense of Trudinger–Moser. In particular, some new ideas and analytic technique are used to overcome the double difficulties caused by the zero mass case and logarithmic convolution potential.

Investigated in the present paper is a fifth-order nonlinear evolution (FONLE) equation, known as a nonlinear water wave (NLWW) equation, with applications in the applied sciences. More precisely, a traveling wave hypothesis is firstly applied that reduces the FONLE equation to a 1D domain. The Kudryashov methods (KMs) are then adopted as leading techniques to construct specific wave structures of the governing model which are classified as W -shaped and other solitons. In the end, the effect of changing the coefficients of nonlinear terms on the dynamical features of W -shaped and other solitons is investigated in detail for diverse groups of the involved parameters.

A novel approximate analytical solution to the linear damped Kawahara equation using a suitable hypothesis is reported for the first time. Based on the exact solutions (such as solitary waves, cnoidal waves, etc.) of the undamped Kawahara equation, the dissipative nonlinear structures like dissipative solitons and cnoidal waves are investigated. The obtained solution is considered a general solution, i.e., it can be applied for studying the properties of all dissipative traveling waves described by the linear damped Kawahara equation. Our technique is not limited to solve the linear damped Kawahara equation only, but it can be used for solving a large number of non-integrable evolution equations related to the realistic natural phenomena. Moreover, the maximum global residual error in the whole space-time domain is estimated for checking the accuracy of the obtained solutions. The obtained solutions can help many researchers in explaining the ambiguities about the mechanisms of propagation of nonlinear waves in complex systems such as seas, oceans, plasma physics, and much more.

The (2+1)-dimensional interaction of a Riemann wave propagating along the y-axis with a long wave along the x-axis is described by the space-time fractional Calogero-Degasperis (CD) and fractional potential Kadomstev-Petviashvili (PKP) equation. It can be modeled according to the Hamiltonian structure, the lax pair with the non-isospectral problem, and the pain level property. The proposed equations are widely used in beachfront ocean and coastal engineering to describe the propagation of shallow-water waves, demonstrate the propagation of waves in dissipative and nonlinear media, and reveal the propagation of waves in dissipative and nonlinear media. In this paper, we have established further exact solutions to the nonlinear fractional partial differential equation (NLFPDEs), namely the space-time fractional CD and fractional PKP equations using the modified Rieman-Liouville fractional derivative of Jumarie through the two variable ( G ′ / G , 1 / G )-expansion method. As far as trigonometric, hyperbolic, and rational function solutions containing parameters are concerned, solutions are acquired when unique characteristics are assigned to the parameters. Subsequently, the solitary wave solutions are generated from the solutions of the traveling wave. It is important to observe that this method is a realistic, convenient, well-organized, and ground-breaking strategy for solving various types of NLFPDEs.

This work investigates three nonlinear equations that describe waves on the oceans which are the Kadomtsev Petviashvili-modified equal width (KP-MEW) equation, the coupled Drinfel’d-Sokolov-Wilson (DSW) equation, and the Benjamin-Ono (BO) equation using the modified ( G ′ G 2 ) -expansion approach. The solutions of proposed equations by modified ( G ′ G 2 ) -expansion approach can be trigonometric, hyperbolic, or rational solutions. As a result, some new exact solutions are obtained and plotted.

This paper proves that the motion of small-slope vorticity fronts in the two-dimensional incompressible Euler equations is approximated on cubically nonlinear timescales by a Burgers–Hilbert equation derived by Biello and Hunter (2010) using formal asymptotic expansions. The proof uses a modified energy method to show that the contour dynamics equations for vorticity fronts in the Euler equations and the Burgers–Hilbert equation are both approximated by the same cubically nonlinear asymptotic equation. The contour dynamics equations for Euler vorticity fronts are also derived.

The large-amplitude free vibrations of an inextensible tapered beam with inertia and static nonlinearities were investigated using the continuous piecewise linearization method (CPLM). The CPLM solution was found to be in excellent agreement with exact numerical solutions and more accurate than other published approximate solutions. The study focuses on the large-amplitude vibrations where the nonlinear effects are strong and the response is characterized by anharmonicity. The present investigation showed that the CPLM solution predicted the anharmonic response due to strong nonlinearity whereas the other published solutions showed large errors. The effects of inertia and static nonlinearities were investigated and it was observed that an increase in the static nonlinear stiffness increases the vibration frequency whereas an increase in inertia nonlinearity reduces vibration frequency. Also, the combined effect of the inertia and static nonlinearities on the large-amplitude vibration response was investigated. It was observed that an increase in the ratio of the static to inertia nonlinearity increases the vibration frequency and vice versa.

In this study, we investigate the conditions for the existence of at least one and three positive solutions to nonlinear higher order multi-point fractional boundary value problems using Krasnosel'skii fixed point theorem and the five functionals fixed point theorem, respectively. © 2021 NSP Natural Sciences Publishing Cor. All Rights Reserved.

For any $\alpha \in (0,1)$, we construct an example of a solution to a parabolic equation with measurable coefficients in two space dimensions which has an isolated singularity and is not better that $C^\alpha$. We prove that there exists no solution to a fully nonlinear uniformly parabolic equation, in any dimension, which has an isolated singularity where it is not $C^2$ while it is analytic elsewhere, and it is homogeneous in $x$ at the time of the singularity. We build an example of a non homogeneous solution to a fully nonlinear uniformly parabolic equation with an isolated singularity, which we verify with the aid of a numerical computation.

Many applications and contexts including materials science, metallurgy, and solid-state physics, are concerned with the study of the behavior of viscoelastic materials. In addition, any composite or complex construction containing embedded polymers exhibits viscoelastic behavior under static and dynamic stress conditions. Wide types of linear/nonlinear constitutive models have been proposed to define the viscoelastic deformation process of viscoelastic materials in order to explore their mechanical behavior. However, it has been shown that the constitutive relationship in the integer-order of stress-strain available in conventional viscoelastic models may fail in some types of situations and do not match well with empirical evidence. In this paper, a novel mathematical model is provided that uses Caputo-Fabrizio fractional-order derivatives to describe the viscoelastic phenomena and is consistent with thermodynamic principles. The Caputo-Fabrizio kernel has many features, such as nonlocality and non-singularity in addition to the exponential form. The suggested model is used to study the dynamic reactions of an unbounded body with a spherical cavity made of viscoelastic material subjected to time-varying heat. Using the Laplace transform technique, numerical calculations of many physical fields are obtained and explored in depth.

This article studies novel soliton wave solutions of the nonlinear fractional ill-posed Boussinesq (NLFIPB) dynamic wave equation by applying the extended Riccati-expansion (ERE) method. Jacques Hadamard has formulated the investigated model to figure out the dynamic characterizations of waves in shallow water under gravity. The obtained solutions are explained through some sketches in 2D and 3D and contour plots. At the same time, the results’ accuracy is checked by comparing the obtained solutions with semi-analytical solutions through the well–known Adomian decomposition (AD) method. The superiority of the ERE method over the original method is explained. All constructed solutions are checked by submitting them back into the original model through Mathematica 12 software.

The (3+1)-dimensional Kadomtsev-Petviashvili and the modified KdV-Zakharov-Kuznetsov equations have a significant impact in modern science for their widespread applications in the theory of long-wave propagation, dynamics of shallow water wave, plasma fluid model, chemical kinematics, chemical engineering, geochemistry, and many other topics. In this article, we have assessed the effects of wave speed and physical parameters on the wave contours and confirmed that waveform changes with the variety of the free factors in it. As a result, wave solutions are extensively analyzed by using the balancing condition on the linear and nonlinear terms of the highest order and extracted different standard wave configurations, containing kink, breather soliton, bell-shaped soliton, and periodic waves. To extract the soliton solutions of the high-dimensional nonlinear evolution equations, a recently developed approach of the sine-Gordon expansion method is used to derive the wave solutions directly. The sine-Gordon expansion approach is a potent and strategic mathematical tool for instituting ample of new traveling wave solutions of nonlinear equations. This study established the efficiency of the described method in solving evolution equations which are nonlinear and with higher dimension (HNEEs). Closed-form solutions are carefully illustrated and discussed through diagrams.

Based on three-wave method, abundant nonautonomous solutions with different wave structures of a 3D variable-coefficient generalized shallow water wave equation are presented. In this situation the basic configurations (characteristic, amplitude, and speed) of the nonautonomous waves mutate with time due to the effect of nonlinearity and dispersion. Discussions indicate the interaction between lump wave and a pair of resonance stripe solitons solutions, the interaction between lump wave and periodic wave solutions, and the breather-type periodic soliton solutions. The dynamic properties of the obtained solutions are demonstrated and analyzed graphically for different choices of the arbitrary functions in these solutions.

This work aims to present nonlinear models that arise in ocean engineering. There are many models of ocean waves that are present in nature. In shallow water, the linearization of the equations requires critical conditions on wave capacity than it make in deep water, and the strong nonlinear belongings are spotted. We use Lie symmetry analysis to obtain different types of soliton solutions like one, two, and three-soliton solutions in a (2+1) dimensional variable-coefficient Bogoyavlensky Konopelchenko (VCBK) equation that describes the interaction of a Riemann wave reproducing along the y-axis and a long wave reproducing along the x-axis in engineering and science. We use the Lie symmetry analysis then the integrating factor method to obtain new solutions of the VCBK equation. To demonstrate the physical meaning of the solutions obtained by the presented techniques, the graphical performance has been demonstrated with some values. The presented equation has fewer dimensions and is reduced to ordinary differential equations using the Lie symmetry technique.

This paper investigates the air-gap field distribution of the permanent-magnet vernier machine (PMVM) using nonlinear exact conformal model (NECM) to account for slotting effect, flux modulation effect and iron nonlinearity. The exact conformal model based on the region of one-slot and one-flux-modulation-pole (OSECM) are introduced to show the effectiveness of linear analytical model for PMVM. It can keep high calculation accuracy and significantly reduce the computational burden. Then, the NECM is developed from OSECM by introducing the equivalent saturation current into the air region and coil region. The lumped parameter magnetic circuit model (LPMCM) model is used to obtain the magnetic potential of iron region and therefore calculate the equivalent saturation current. The NECM which combines LPMCM and OSECM can essentially improve the accuracy of linear analytical model. The harmonic analysis of air-gap field is performed to theoretically explain the component of electromagnetic torque. Both finite element model (FEM) simulation and test results are presented to validate the NECM.

This article presents the investigation of nonlinear vibration analysis of tapered porous functionally graded skew (TPFGS) plate considering the effects of geometrical non-uniformities to optimize the thickness in the structural design. The TPFGS plate is analyzed considering linearly, bi-linearly, and exponentially varying thicknesses. The plate's effective material properties are tailor-made using a modified power-law distribution in which gradation varies along the thickness direction of the TPFGS plate. Incorporating the non-linear finite element formulation to develop the kinematic equation's displacement model for the TPFGS plate is based on the first-order shear deformation theory (FSDT) in conjunction with von Karman's nonlinearity. The nonlinear governing equations are established by Hamilton's principle. The direct iterative method is adopted to solve the nonlinear mathematical relations to obtain the nonlinear frequencies. The influence of the porosity distributions and porosity parameter indices on the nonlinear frequency responses of the TPFGS plate for different skew angles and variable thicknesses are studied for various geometrical parameters. The influence of taper ratio, variable thickness, skewness, porosity distributions, gradation, and boundary conditions on the plate's nonlinear vibration is demonstrated. The nonlinear frequency analysis reveals that the geometrical non-uniformities and porosities significantly influence the porous functionally graded plates with varying thickness than the uniform thickness. Besides, exponentially and linearly variable thicknesses can be considered for the thickness optimizations of TPFGS plates in the structural design.

This study investigates the boundary stresses prescribed by the traction boundary condition on an incompressible magnetohydrodynamic Jeffery-Hamel flow of viscous fluid. The governing partial differential equations are transformed to a nonlinear ordinary differential equation (ODE) by the transformation obtained from continuity equation. We have modeled traction boundary conditions to solve the problem. We have not imposed flow symmetry condition and applied prescribed flow rate condition. We have computed analytical and numerical solutions of the problem. We have developed modified Adomian decomposition method based on Daun-Rach Approach (DRA) for general nonlinear third order ODE subject to Robin’s and integral boundary conditions. The governing equation is solved using the developed scheme. The Mathematica routine, NDSolve, is used to obtain numerical solution by first approximating the constant flow rate condition by Gauss-Lobatto integration formula of four and five points as well as by trapezoidal rule. Analytical and numerical results are compared and found in good agreement. The slip and no-slip scenarios are observed both for inertial and non-inertial flows. The back flow occurs both for convergent and divergent channel geometries. Flow bifurcations and boundary layer get strengthened due to presence of boundary stresses. The symmetric and asymmetric natures of flow are also captured due to presence of boundary stresses.

The tippedisk is a mechanical-mathematical archetype for friction-induced instability phenomena that exhibits an interesting inversion phenomenon when spun rapidly. The inversion phenomenon of the tippedisk can be modeled by a rigid eccentric disk in permanent contact with a flat support, and the dynamics of the system can therefore be formulated as a set of ordinary differential equations. The qualitative behavior of the nonlinear system can be analyzed, leading to slow-fast dynamics. Since even a freely rotating rigid body with six degrees of freedom already leads to highly nonlinear system equations, a general analysis for the full system equations is not feasible. In a first step the full system equations are linearized around the inverted spinning solution with the aim to obtain a local stability analysis. However, it turns out that the linear dynamics of the full system cannot properly describe the qualitative behavior of the tippedisk. Therefore, we simplify the equations of motion of the tippedisk in such a way that the qualitative dynamics are preserved in order to obtain a reduced model that will serve as the basis for a following nonlinear stability analysis. The reduced equations are presented here in full detail and are compared numerically with the full model. Furthermore, using the reduced equations we give approximate closed form results for the critical spinning speed of the tippedisk., Projekt DEAL

We study fluid-saturated porous materials that undergo poro-elastic deformations in thin domains. The mechanics in such materials are described using a biphasic model based on the theory of porous media (TPM) and consisting of a system of differential equations for material’s displacement and fluid’s pressure. These equations are in general strongly coupled and nonlinear, such that exact solutions are hard to obtain and numerical solutions are computationally expensive. This paper reduces the complexity of the biphasic model in thin domains with a scale separation between domain’s width and length. Based on standard asymptotic analysis, we derive a reduced model that combines two sub-models. Firstly, a limit model consists of averaged equations that describe the fluid pore pressure and displacement in the longitudinal direction of the domain. Secondly, a corrector model re-captures the mechanics in the transverse direction. The validity of the reduced model is finally tested using a set of numerical examples. These demonstrate the computational efficiency of the reduced model, while maintaining reliable solutions in comparison with original biphasic TPM model in thin domain., Deutsche Forschungsgemeinschaft, Projekt DEAL

Problems with variable exponents have attracted a great deal of attention lately and various existence, nonexistence and stability results have been established. The importance of such problems has manifested due to the recent advancement of science and technology and to the wide application in areas such as electrorheological fluids (smart fluids) which have the property that the viscosity changes drastically when exposed to heat or electrical fields. To tackle and understand these models, new sophisticated mathematical functional spaces have been introduced, such as the Lebesgue and Sobolev spaces with variable exponents. In this work, we are concerned with a system of wave equations with variable-exponent nonlinearities. This system can be regarded as a model for interaction between two fields describing the motion of two “smart” materials. We, first, establish the existence of global solutions then show that solutions of enough regularities stabilize to the rest state ( 0 , 0 ) either exponentially or polynomially depending on the range of the variable exponents. We also present some numerical tests to illustrate our theoretical findings.

Empirical distributions in a range of fields are often substantially non-Gaussian but smooth enough to suggest that the underlying population distribution has at least several derivatives. Linear combinations of many random variables often have smooth densities even if the random variables are not independent. The main result here generalizes the elementary result that a convolution of densities has as many derivatives as the sum of the number of derivatives of each component to certain nonlinear functions of random vectors whose components may not be independent. Linearity is replaced by an assumption that the nonlinear function has positive first partial derivatives bounded away from 0 along at least some coordinates and independence is replaced by an assumption that the joint density has certain mixed partial derivatives. This approach to justifying the smoothness of empirical distributions is contrasted with other possible approaches, including solutions of stochastic partial differential equations and ergodic distributions for chaotic dynamical systems.

In this paper, a generalized (3+1)-dimensional Kadomtsev–Petviashvili Benjamin–Bona–Mahony equation which describes the fluid flow in the case of an offshore structure, is investigated. Here, making use of the bilinear form and symbolic computation, we construct four kinds of rogue wave solutions consisting of independent breathers. Among these solutions, the fourth order rogue wave solution is rarely considered in nonlinear system. Exact locations of the highest and lowest peaks as well as the extreme values of the wave heights are systematically analyzed. The obtained rogue waves observe certain “circularity structure”, the highest or lowest peaks both sit at the same circular. Moreover, we show that the rogue waves are stable during the propagation.

Beam-like compliant elements have found wide-ranging application in many fields of engineering and science often where 3D large deflections can be of concern such as soft robotics, DNA mechanics and helicopter/wind turbine rotor blades. The homotopy analysis method (HAM) is used for the first time to obtain a novel analytical solution in converged series form for the arbitrary large deflection of geometrically exact beams subject to both conservative and follower loading scenarios. The homotopy analysis method, which offers desirable characteristics such as being free from small or large parameters, coupled with auxiliary parameters controlling convergence, is applied directly to the intrinsic governing equations of a geometrically exact beam theory. The system of first-order differential governing equations of geometrically exact beams with intrinsic formulation is free from rotation and displacement variables, and offers a low degree of nonlinearity (quadratic at most) and compact mathematical form, making it suitable for analytical solutions. Due to the relatively poor convergence of the original HAM algorithm, the iterative HAM technique is employed which is known to accelerate convergence and to improve the computational efficiency of the homotopy analysis method. The obtained homotopy series offers a number of novel features in the context of the analytical solutions for the large deflection of beams, including (a) the direct calculation of internal forces and moments which is significant for engineering design purposes, (b) being able to capture 3D deflections, (c) considering transverse shear effects which can be important for thicker beams or when the Young’s modulus to shear modulus ratio is significant (such as composite materials) and (d) considering conservative and follower tip and distributed loads, in a unified framework. In order to investigate the efficacy, applicability and accuracy of the proposed method, a number of numerical examples are considered in which a cantilever beam subject to tip or distributed loads undergoes large deflection. Large deflection results for both conservative and follower loads are compared against those of less comprehensive analytical solutions as well as against numerical methods including finite element and Chebyshev collocation methods where good agreement is observed. These results demonstrate the applicability and effectiveness of HAM for the large deflection analysis of geometrically exact beams.

The uniformly non-oscillatory (UNO) limiters with second-order accuracy are updated to handle nonuniform cell obtained from wavelet-based cell adaptation. These limiters are then used in the cell-adaptive second-order Kurganov-Tadmor (KT) central high resolution scheme. Over the adapted cells, the performance of the UNO limiters are compared with the generalized MINMOD (GMM) limiter with the total variation diminishing (TVD) feature, by measuring error of reconstruction of some functions and also by numerical solution of some benchmark partial differential equations (PDEs). Then, unified forms of generalized fully-coupled saturated thermo-poro-elastic systems are presented as first-order hyperbolic-parabolic systems in the 2D Cartesian and the asymmetric spherical coordinates. For these systems, corresponding flux, source (load), diffusion and nonlinear terms are derived. It is shown that in the case of the asymmetric spherical coordinate, due to the geometry, considerable complexities are introduced in the load (source) and flux terms. Finally, the coupled 1D and 2D systems are simulated by the adaptive central KT scheme with the GMM and UNO limiters.

This article critically examines the influences of Newtonian heating and slip effects on unsteady magnetohydrodynamic (MHD) flow of an Oldroyd-B fluid near an infinitely long plate. The nonlinear thermal radiation influence is considered in normal direction to the vertical plate. A modern definition of Caputo-Fabrizio fractional derivative is applied to generalize the constitutive mass and energy equations. The Laplace transformation technique and Durbin’s numerical algorithm are served to develop the solutions of proposed problem. The solutions are determined for both fractional and ordinary cases and Stehfest’s and Zakian’s numerical algorithms are employed to secure the authenticity of these solutions. The relations for Nusselt number and skin friction coefficient are efficiently computed to precisely estimate the shear stress and rate of heat transfer at boundary. The respective numerical computations are accessible through tables. To comprehensively analyze the dynamics of the proposed problem, physical influence of various parameters is studied and repercussions are graphically highlighted and discussed. Furthermore, two limiting models named as second grade model and Maxwell model are deduced to compare the pertinent flow characteristics. A comparative analysis between fractional and classical models depicts that the Caputo-Fabrizio fractional model explains the memory effects more adequately.

Analytical modeling (AM) of the magnetic field for axial-flux switched reluctance motors (AFSRMs), considering three-dimensional (3-D) features and nonlinear saturation effects, remains to be a knotty problem. This paper proposes a quasi-3-D nonlinear AM method to predict magnetic field distributions and electromagnetic performances of an AFSRM. Firstly, the harmonic modeling technique is applied to obtain the spatiotemporal distribution characteristics of the axial and tangential magnetic flux density at the mean radius. Secondly, the nonlinearity of the soft magnetic material in teeth is considered by an adaptive convergence rate algorithm. Finally, a radial dependence function is formulated to extend the solution of the annular section to any point in the 3-D space of the AFSRM. To confirm the accuracy of the proposed method, the analytical solutions of magnetic flux density and electromagnetic torque have been compared with FEM simulation results. Meanwhile, an additional validation has been done for the stability of this nonlinear AM method under severe saturation conditions. At last, the reasonable selection of the harmonic truncation order number has been discussed to reach a good balance between the accuracy and time cost. The quasi-3-D nonlinear AM method can consider the curvature and edge effect as well as the saturation effect effectively, which is suitable for performance predictions and parametric studies of AFSRMs.

In this paper, we study the long-time behavior of the following plate equation ε ( t ) u t t + g ( u t ) + Δ 2 u + λ u + f ( u ) = h , where the coefficient ε depends explicitly on time, the nonlinear damping and the nonlinearity both have critical growths.

An approximation by diffusion of a nonlinear Boltzmann equation modeling a Fermi-Dirac statistics is analyzed for an unbounded velocity space and Poisson coupling. A careful analysis of the entropy and entropy-dissipation allows to control the distribution function and to pass to the limit using duality method.

This paper presents a new active controller technique effect of functionally graded (FG) cylindrical shell structure model within joint harmonic and parametric excitations. The new active controller involves Nonlinear Proportional-Derivative (NPD) plus Negative Cubic Velocity Feedback (NCVF) as a new nonlinear control method (NPDCVF). The motivation of this study is the well-known observation that NPDCVF control method can offer a means to improve the performance of plant systems. The coupled nonlinear differential equations with two modes have been derived applying the von Karman nonlinear theory, Galerkin’s process, and the static condensation technique. Static condensation is the process of decreasing the number of free displacements or degrees of freedom. We have added other three different controller methods over the structure to select the best controller. The three applied controller methods to the considered structure are: Integral Resonant Control (IRC), Positive Position Feedback (PPF), and Nonlinear Integral Positive Position Feedback (NIPPF) which are applied to the considered structure. We establish that the best one for dipping the high vibration amplitudes is the NPDCVF as a new controller. The perturbation technique is useful to solve the present model including quadratic and cubic nonlinearities subjected to mixed harmonic and parametric forces within the simultaneous primary resonance case ( Ω = ω 1 , Ω = ω 2 ) as the worst resonance case of the system. All numerical outcomes have been completed with the help of MATLAB Ra 18.0 program software over the investigated model. The obtained results have proved that this new controller is excellent for improving the structure by reducing the risky vibrations caused during primary resonances. Frequency response equations have helped in observing the stability examination of the obtained numerical solutions. The actions of several factors performed on the controlled model have been examined and described. Some comparisons have also been prepared with the available recent published papers of a similar model.

A dispersive decay estimate is proven for the linear propagator of the three-scale rotating stratified Boussinesq equations in the stratification-dominant regime. Because the phase function of that propagator is both singular and degenerate, obtaining the estimate requires novel techniques involving cutting phase-space shells into several pieces and using different methods in each region. Using the decay estimate, we prove the long-time existence of solutions to the initial-value problem for the nonlinear equations and obtain a rate of convergence to the limit system and intermediate asymptotics.

In this paper, we consider the nonlinear stability of solutions to one-dimensional compressible micropolar equations. If the Riemann problem of corresponding Euler equations admits a solution which is composed of 1-rarefaction wave and 3-rarefaction wave, we proved that the solution to micropolar equations is nonlinear stable to the composite waves as time goes to infinity. Compared to the previous studies, we established the result under large initial perturbation. The key point in our analysis is how to deduce the positive lower and upper bounds of the specific volume and the absolute temperature.

In the present endeavour, the shifted Legendre collocation method is extended to obtain the solution of nonlinear fractional order (2+1)-dimensional advection–reaction–diffusion solute transport equation. The variations of solute concentration of the model for different fractional order space and time derivatives are presented graphically for various particular cases. The main feature of the present contribution is the graphical exhibitions of the effects of advection term, reaction term and fractional-order parameters on the solution profile. To authenticate the effectiveness of the method, a drive has been taken to compare the obtained results with the existing analytical results of the integer-order form of the considered model through error analysis which are displayed in tabular and pictorial forms.

This study presents a semi-analytical formulation for the nonlinear vibration and dynamic instability of a randomly distributed carbon nanotube-reinforced composite (RD-CNTRC) plate. Three cases of localized in-plane periodic loadings are studied. The analytical stress fields within the RD-CNTRC plate for all the in-plane stress components (σij, (i, j = x, y)) are developed by solving the in-plane elastic problem using Airy's stress approach. The effective mechanical properties of the RD-CNTRC plate are evaluated by the Eshelby-Mori-Tanaka technique. The plate is modeled based on higher-order shear deformation theory (HSDT) in conjunction with the von-Karman nonlinearity. Using Hamilton's principle, the governing partial differential equations (PDEs) are derived, whose approximate solution is sought, referring to the Galerkin method. The resulting nonlinear ODEs are solved using the Incremental Harmonic Balance (IHB) Method to compute the nonlinear vibration response of the RD-CNTRC plate. Further dropping the nonlinear terms, these ODEs are solved by Bolotin's method to trace the instability region. The proposed semi-analytical method is an effective strategy for studying the influence of different parameters such as agglomeration models, CNT mass fraction, pre-loading, and boundary conditions on the nonlinear vibration and dynamic instability characteristics of the RD-CNTRC plates. The reduced computational effort allows the design phase to be supported in selecting parameters when designing RD-CNTRC plates with stability and vibration requirements.

In this paper, a damped p-Laplacian type wave equation with logarithmic nonlinearity is considered and finite time blow-up of solutions is proved with initial data at different initial energy levels. More precisely, when the initial energy is subcritical, a sufficient condition for the solutions to blow up in finite time is derived, by combining the Nehari manifold with concavity argument. When the initial energy is supercritical, some new skills are invented to establish another finite time blow-up criterion for this problem. Meanwhile, for each of the above two cases, the lifespan of the weak solutions is estimated from above. Moreover, by making full use of the strongly damping term and choosing appropriate parameters, a lower bound for the blow-up time is given.

Forced oscillation is challenging power system stability. Most previous research was based on linear models, so analysis of nonlinear characteristics for forced oscillation in power system is missing. Thus, this letter aims to theoretically reveal the effect of the quadratic nonlinearity on forced oscillation. A multi-scale technique is employed, by which the amplitude-frequency function of the steady response and the approximated first-order analytical expression of the dynamic response are derived for nonlinear forced oscillation. Interesting phenomena of frequency deviation and jumping caused by nonlinearity are found and explained. Above expressions and phenomena are verified by numerical simulation. Although revealing all characteristics of nonlinear forced oscillation in power system is difficult, the analysis results in this letter could offer insights.

We present a monotone finite volume element scheme for the diffusion problem on triangular grids, which stems from some idea about the two-point flux in the literature [15] . A new nonlinear two-point flux formulation is obtained for some part of one control volume edge, and this straight edge crosses through some common edge of two neighboring triangular elements where two inner points from these neighboring elements are chosen and linked so that it does not cross through other triangle elements even if the meshes are severely distorted. The approximation of κ∇u in some quadrilateral region including this straight control volume edge, are obtained by some weighted average with the gradient of some finite element functions, and the weighted coefficients are determined by the nonlinear two-point flux idea. Furthermore, we prove that the new scheme is monotone under some conditions that one modified barycenter dual partition Ω h ⁎ is chosen for some given partition Ω h and the exact solution u ∈ C 1 ( Ω ) , and the corresponding coefficient matrix is of M-matrix. Finally, we carry on some typical experiments. Numerical results verify the monotonicity of this new scheme, and also confirm the accuracy and stability.

The analysis of motion equations of flexible-link manipulators indicates that nonlinear terms and behavioral instability exist in their outcomes. In this system, when the differential equations possess structural time-dependent parameters, there is a higher chance for system instability, especially chaos in comparison to time-invariant ones. It is a point for abnormal behavior that is unpredictable and always accompanied by damages. Therefore, the physical parameters and system inputs that cause chaotic responses should be detected and prevented as much as possible. To this aim, the dynamic equations of flexible link manipulator with revolute-prismatic joints as a case of time-variant dynamic system are obtained by Hamilton’s principle with the assumption of nonlinear strains. The discretized motion equations are solved and the results are presented in the forms of bifurcation diagrams (for variation of torque/force amplitude), Poincare maps, phase-plane portraits, and the largest Lyapunov exponent. Finally, the obtained results are validated with the aid of the experimental setup. The results indicate that although the system response in low torque/force amplitudes is subharmonic (amplitude of 0.01) and quasi-periodic (amplitudes of 0.02-0.03), it becomes quasi-periodic/chaotic with a mild slope with respect to time in high amplitude values. Moreover, since the behavior of modal generalized coordinate is different from the rigid ones, the quasi-periodic and chaotic vibrations do not hurt the joints.

In this paper, the stability of $ (\omega, c) $-periodic solutions of non-instantaneous impulses differential equations is studied. The exponential stability of homogeneous linear non-instantaneous impulsive problems is studied by using Cauchy matrix, and some sufficient conditions for exponential stability are obtained. Further, by using Gronwall inequality, sufficient conditions for exponential stability of $ (\omega, c) $-periodic solutions of nonlinear noninstantaneous impulsive problems are established. Finally, some examples are given to illustrate the correctness of the conclusion.