Coupled Evolution Equations Research Articles

Existence and Hyers‐Ulam stability of fractional nonlinear impulsive switched coupled evolution equations

In this paper, we study a class of fractional nonlinear impulsive switched coupled evolution equations. Existence and uniqueness of solutions as well as Hyers‐Ulam stability results are presented. An example is provided for the verification of our results.

Advance Exp (-Φ(ξ)) Expansion Method and Its Application to Find the Exact Solutions for Some Important Coupled Nonlinear Physical Models

The theoretical investigations of resonance physical phenomena by nonlinear coupled evolution equations are become important in currently. Hence, the purpose of this paper is to represent an advance exp (-Φ(ξ))-expansion method with nonlinear ordinary differential equation for finding exact solutions of some nonlinear coupled physical models. The present method is capable of evaluating all branches of solutions simultaneously and this difficult to distinguish with numerical technique. To verify its computational efficiency, the coupled classical Boussineq equation and (2+1)-dimensional Boussinesq and Kadomtsev-Petviashili equation are considered. The obtained solutions in this paper reveal that the method is a very effective and easily applicable of formulating the exact traveling wave solutions of the nonlinear coupled evolution equations arising in mathematical physics and engineering.

Nonlinear stability in three-layer channel flows

The nonlinear stability of viscous, immiscible multilayer flows in plane channels driven both by a pressure gradient and gravity is studied. Three fluid phases are present with two interfaces. Weakly nonlinear models of coupled evolution equations for the interfacial positions are derived and studied for inertialess, stably stratified flows in channels at small inclination angles. Interfacial tension is demoted and high-wavenumber stabilisation enters due to density stratification through second-order dissipation terms rather than the fourth-order ones found for strong interfacial tension. An asymptotic analysis is carried out to demonstrate how these models arise. The governing equations are $2\times 2$ systems of second-order semi-linear parabolic partial differential equations (PDEs) that can exhibit inertialess instabilities due to interaction between the interfaces. Mathematically this takes place due to a transition of the nonlinear flux function from hyperbolic to elliptic behaviour. The concept of hyperbolic invariant regions, found in nonlinear parabolic systems, is used to analyse this inertialess mechanism and to derive a transition criterion to predict the large-time nonlinear state of the system. The criterion is shown to predict nonlinear stability or instability of flows that are stable initially, i.e. the initial nonlinear fluxes are hyperbolic. Stability requires the hyperbolicity to persist at large times, whereas instability sets in when ellipticity is encountered as the system evolves. In the former case the solution decays asymptotically to its uniform base state, while in the latter case nonlinear travelling waves can emerge that could not be predicted by a linear stability analysis. The nonlinear analysis predicts threshold initial disturbances above which instability emerges.

Asymptotic analysis of the evaporation dynamics of partially wetting droplets

We consider the dynamics of an axisymmetric, partially wetting droplet of a one-component volatile liquid. The droplet is supported on a smooth superheated substrate and evaporates into a pure vapour atmosphere. In this process, we take the liquid properties to be constant and assume that the vapour phase has poor thermal conductivity and small dynamic viscosity so that we may decouple its dynamics from the dynamics of the liquid phase. This leads to a so-called ‘one-sided’ lubrication-type model for the evolution of the droplet thickness, which accounts for the effects of evaporation, capillarity, gravity, slip and kinetic resistance to evaporation. By asymptotically matching the flow near the contact line region and the bulk of the droplet in the limit of a small slip length and commensurably small evaporation and kinetic resistance effects, we obtain coupled evolution equations for the droplet radius and volume. The predictions of our asymptotic analysis, which also include an estimate of the evaporation time, are found to be in excellent agreement with numerical simulations of the governing lubrication model for a broad range of parameter regimes.

Best rates of decay for coupled waves with different propagation speeds

We consider an abstract system of two coupled evolution equations. One of these equations has an internal damping, and the other is simply elastic. When both equations have the same propagation speed, Alabau et al. (J Evol Equ 2:127–150, 2002) showed that the semigroup of this system decays polynomially in time with the rate $$t^{-1/2}$$ . In this work, we consider this coupled system when the propagation speeds of the equations are different, and we study the asymptotic behavior of the semigroup. For this case, we show that the semigroup still decays polynomially with a slower rate as $$t^{-1/4}$$ . Moreover, we prove that this rate of decay is the best.

A new model for dynamic recrystallization under hot working conditions based on critical dislocation gradients

Abstract Dynamic recrystallization (DRX) occurs during hot working of metals and alloys with a sufficiently low stacking-fault energy. It is characterized by the nucleation and growth of new grains, which takes place concurrently to plastic deformation. DRX processes are used in industrial hot forming processes to control the microstructure and properties of the workpiece. Various models have been developed that consider the coupled evolution of microstructure and flow stress during hot deformation processes. It is widely accepted that the onset of DRX is characterized by an inflection point of the strain hardening rate as a function of stress. This so-called second-derivative criterion was derived by Poliak and Jonas in the 1990s using results from the thermodynamics of irreversible processes. Recent results show that models which use classical Avrami kinetics to describe the evolution of the recrystallized volume fraction violate the criterion if the Avrami exponent assumes a value of 3 or less. This inconsistency has its root in the assumptions made for the nucleation and growth of DRX nuclei and was shown to affect the accuracy of models used to design and control metal forming processes. This paper presents a new, consistent model for the nucleation of DRX. Typical nucleation models assume a constant dislocation density throughout the parent grains to derive a critical dislocation density for nucleation. However, it is clear from crystal plasticity that the dislocation density is higher at the grain boundaries than inside the grains. In order to involve dislocation gradients, coupled evolution equations for mobile and immobile dislocation densities are derived and solved under hot working conditions in this work. A new nucleation criterion is developed based upon the strain-dependent dislocation density gradient in parent grains. Once the critical gradient is reached, nucleation starts. The results of the newly developed criterion are compared with conventional models and experimental data. The comparison shows that the new criterion predicts more realistic critical dislocation densities in contrast to conventional models which over-estimate the critical dislocation density. As a consequence, more accurate predictions in the design and control of metal forming processes may be achieved with the new model.

A nonlinear small-deformation theory for transient droplet electrohydrodynamics

The deformation of a viscous liquid droplet suspended in another liquid and subject to an applied electric field is a classic multiphase flow problem best described by the Melcher–Taylor leaky dielectric model. The main assumption of the model is that any net charge in the system is concentrated on the interface between the two liquids as a result of the jump in Ohmic currents from the bulk. Upon application of the field, the drop can either attain a steady prolate or oblate shape with toroidal circulating flows both inside and outside arising from tangential stresses on the interface due to action of the field on the surface charge distribution. Since the pioneering work of Taylor (Proc. R. Soc. Lond. A, vol. 291, 1966, pp. 159–166), there have been numerous computational and theoretical studies to predict the deformations measured in experiments. Most existing theoretical models, however, have either neglected transient charge relaxation or nonlinear charge convection by the interfacial flow. In this work, we develop a novel small-deformation theory accurate to second order in electric capillary number $O(Ca_{E}^{2})$ for the complete Melcher–Taylor model that includes transient charge relaxation, charge convection by the flow, as well as transient shape deformation. The main result of the paper is the derivation of coupled evolution equations for the induced electric multipoles and for the shape functions describing the deformations on the basis of spherical harmonics. Our results, which are consistent with previous models in the appropriate limits, show excellent agreement with fully nonlinear numerical simulations based on an axisymmetric boundary element formulation and with existing experimental data in the small-deformation regime.

Contributions to theUY(1)Chern-Simons term and the evolution of fermionic asymmetries and hypermagnetic fields

We study simultaneous evolution of electron, neutrino and quark asymmetries, and large scale hypermagnetic fields in the symmetric phase of the electroweak plasma in the temperature range $100$GeV$\leq T\leq 10$TeV, taking into account the chirality flip processes via inverse Higgs decays and fermion number violation due to Abelian anomalies. We present a derivation of the coefficient of the Chern-Simons term for the hypercharge gauge field, showing that the left-handed and right-handed components of each fermion species contribute with opposite sign. This is in contrast to the results presented in some of the previous works. The $\textrm{U}_{\textrm{Y}}(1)$ Chern-Simons term affects the resulting anomalous magnetohydrodynamic (AMHD) equations. We solve the resulting coupled evolution equations for the lepton and baryon asymmetries, as well as the hypermagnetic field to obtain their time evolution along with their values at the electroweak phase transition ($T_{EW} \sim 100$GeV) for a variety of critical ranges for their initial values at $T=10$TeV. We first investigate the results of this sign change, by directly comparing our results with those obtained in one of the previous works and find that matter asymmetry generation increases considerably in the presence of a strong hypermagnetic field. Furthermore, we find that a strong hypermagnetic field can generate matter asymmetry starting from absolutely zero asymmetry, while matter asymmetry can generate a hypermagnetic field provided the initial value of the latter is nonzero.

Application of Tanh Method to Complex Coupled Nonlinear Evolution Equations

M.A. Abdelkawy, A.H. Bhrawy, E. Zerrad and A. Biswas Department of Mathematics and Statistics, College of Science, Al-Imam Mohammad Ibn Saud Islamic University, Riyadh-11566, Saudi Arabia Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt Department of Physics and Engineering, Delaware State University, Dover, DE 19901-2277, USA Department of Mathematical Sciences, Delaware State University, Dover, DE 19901-2277, USA Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah-21589, Saudi Arabia

Combined Effects of Thermal Diffusion and Diffusion-Thermo Effects on Transient MHD Natural Convection and Mass Transfer Flow in a Vertical Channel with Thermal Radiation

We study the nonlinear coupled evolution equations which model the transient MHD natural convection and mass transfer flow of viscous, incompressible and electrically conducting fluid between two infinite vertical plates in the presence of the transversal magnetic field, thermal radiation, thermal diffusion and diffusion-thermo effects. Both analytical and numerical methods are used for this study.

A micromechanical model for the transformation induced plasticity in polycrystalline steels

AbstractDue to the effect of transformation induced plasticity (TRIP) , TRIP‐steels are very promising materials, e.g. for the automobile industry. The material behavior is characterized by very complex inner processes, namely phase transformation coupled with plastic deformation and kinematic hardening. We establish a micromechanical model which uses the volume fractions of the single phases, the plastic strain and the hardening parameter in every grain of the polycrystalline material as internal variables. Furthermore, we apply the Principle of the Minimum of the Dissipation Potential to derive the associated evolution equations. The use of a coupled dissipation functional and a combined Voigt/Reuss bound directly results in coupled evolution equations for the internal variables and in one combined yield function. Additionally, we show numerical results which prove our model's ability to give a first prediction of the TRIP‐steels' complex material behavior. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Higher-order generalized hydrodynamics: Foundations within a nonequilibrium statistical ensemble formalism.

Construction, in the framework of a nonequilibrium statistical ensemble formalism, of a higher-order generalized hydrodynamics, also referred to as mesoscopic hydrothermodynamics, that is, covering phenomena involving motion of fluids displaying variations short in space and fast in time-unrestricted values of Knudsen numbers, is presented. In that way, an approach is provided enabling the coupling and simultaneous treatment of the kinetics and hydrodynamic levels of descriptions. It is based on a complete thermostatistical approach in terms of the densities of matter and energy and their fluxes of all orders covering systems arbitrarily driven away from equilibrium. The set of coupled nonlinear integrodifferential hydrodynamic equations is derived. They are the evolution equations of the Gradlike moments of all orders, derived from a generalized kinetic equation built in the framework of the nonequilibrium statistical ensemble formalism. For illustration, the case of a system of particles embedded in a fluid acting as a thermal bath is fully described. The resulting enormous set of coupled evolution equations is of unmanageable proportions, thus requiring in practice to introduce an appropriate description using the smallest possible number of variables. We have obtained a hierarchy of Maxwell times, associated to the set of all the higher-order fluxes, which have a particular relevance in the process of providing criteria for establishing the contraction of description.

Interfacial instabilities of two-layer plane Poiseuille flows of viscoelastic fluids

The linear stability of a two-layer flow of viscoelastic fluids on an inclined channel is examined. A simplified model is derived by using a weighted residual method combined with a long-wavelength expansion. A set of two coupled evolution equations for two fields, the film thickness h(x,t) and the flow rate q(x,t), is obtained. Then, the correct threshold of instability of the interface is characterized analytically. The effect of the elasticity stratification on the stability of the interface is considered within the long-wavelength assumption.

Numerical Solutions of Coupled Nonlinear Evolution Equations via El-gendi Nodal Galerkin Method

In this research the solution of coupl ed modified Korteweg -de Vries equation (mKdV) and the generalized Hirota–Satsuma coupled KdV equation by using El-gendi nodal Galerkin (EGG) approaches are presented. El-gendi nodal Galerkin (EGG) (EGG) approaches consist of two approaches, the first is El-gendi Chebyshev nodal Galerkin (ECG) and the second approach is called El-gendi Legendre nodal Galerkin (ELG). In these new approaches spaces of the solution and the weak form to the system are presented. The resulted systems of ODES are solved by the fourth order Runge-Kutta solver. The convergence and the stability of these new methods are analyzed numerically. Numerical results are presented and compared with the results obtained by pseudo-spectral method. Original Research Article

Bimodality and re-entrant behaviour in the hierarchical self-assembly of polymeric nanoparticles.

We show that a film of a suspension of polymer grafted nanoparticles on a liquid substrate can be employed to create two-dimensional nanostructures with a remarkable variation in the pattern length scales. The presented experiments also reveal the emergence of concentration-dependent bimodal patterns as well as re-entrant behaviour that involves length scales due to dewetting and compositional instabilities. The experimental observations are explained through a gradient dynamics model consisting of coupled evolution equations for the height of the suspension film and the concentration of polymer. Using a Flory-Huggins free energy functional for the polymer solution, we show in a linear stability analysis that the thin film undergoes dewetting and/or compositional instabilities depending on the concentration of the polymer in the solution. We argue that the formation via 'hierarchical self-assembly' of various functional nanostructures observed in different systems can be explained as resulting from such an interplay of instabilities.

Spatio-temporal dynamics induced by competing instabilities in two asymmetrically coupled nonlinear evolution equations.

Pattern formation often occurs in spatially extended physical, biological, and chemical systems due to an instability of the homogeneous steady state. The type of the instability usually prescribes the resulting spatio-temporal patterns and their characteristic length scales. However, patterns resulting from the simultaneous occurrence of instabilities cannot be expected to be simple superposition of the patterns associated with the considered instabilities. To address this issue, we design two simple models composed by two asymmetrically coupled equations of non-conserved (Swift-Hohenberg equations) or conserved (Cahn-Hilliard equations) order parameters with different characteristic wave lengths. The patterns arising in these systems range from coexisting static patterns of different wavelengths to traveling waves. A linear stability analysis allows to derive a two parameter phase diagram for the studied models, in particular, revealing for the Swift-Hohenberg equations, a co-dimension two bifurcation point of Turing and wave instability and a region of coexistence of stationary and traveling patterns. The nonlinear dynamics of the coupled evolution equations is investigated by performing accurate numerical simulations. These reveal more complex patterns, ranging from traveling waves with embedded Turing patterns domains to spatio-temporal chaos, and a wide hysteretic region, where waves or Turing patterns coexist. For the coupled Cahn-Hilliard equations the presence of a weak coupling is sufficient to arrest the coarsening process and to lead to the emergence of purely periodic patterns. The final states are characterized by domains with a characteristic length, which diverges logarithmically with the coupling amplitude.

Parametric subharmonic instability of internal waves: locally confined beams versus monochromatic wavetrains

AbstractInternal gravity wavetrains in continuously stratified fluids are generally unstable as a result of resonant triad interactions which, in the inviscid limit, amplify short-scale perturbations with frequency equal to one half of that of the underlying wave. This so-called parametric subharmonic instability (PSI) has been studied extensively for spatially and temporally monochromatic waves. Here, an asymptotic analysis of PSI for time-harmonic plane waves with locally confined spatial profile is made, in an effort to understand how such wave beams differ, in regard to PSI, from monochromatic plane waves. The discussion centres upon a system of coupled evolution equations that govern the interaction of a small-amplitude wave beam with short-scale subharmonic wavepackets in a nearly inviscid uniformly stratified Boussinesq fluid. For beams with general localized profile, it is found that triad interactions are not strong enough to bring about instability in the limited time that subharmonic perturbations overlap with the beam. On the other hand, for quasi-monochromatic wave beams whose profile comprises a sinusoidal carrier modulated by a locally confined envelope, PSI is possible if the beam is wide enough. In this instance, a stability criterion is proposed which, under given flow conditions, provides the minimum number of carrier wavelengths a beam of small amplitude must comprise for instability to arise.

Nonlinear Evolution Equations for Broader Bandwidth Wave Packets in Crossing Sea States

Two coupled nonlinear equations are derived describing the evolution of two broader bandwidth surface gravity wave packets propagating in two different directions in deep water. The equations, being derived for broader bandwidth wave packets, are applicable to more realistic ocean wave spectra in crossing sea states. The two coupled evolution equations derived here have been used to investigate the instability of two uniform wave trains propagating in two different directions. We have shown in figures the behaviour of the growth rate of instability of these uniform wave trains for unidirectional as well as for bidirectional perturbations. The figures drawn here confirm the fact that modulational instability in crossing sea states with broader bandwidth wave packets can lead to the formation of freak waves.

Engineering magnetic polariton system with distributed coefficients: Applications to soliton management

In the wake of the recent design of a powerful method for generating higher-dimensional evolution systems with distributed coefficients Kuetche (2014) [15] illustrated on the dynamics of the current-fed membrane of zero Young’s modulus, we construct the general Lax-representation of a new higher-dimensional coupled evolution equations with varying coefficients. Discussing the physical meanings of these equations, we show that the coupled system above describes the propagation of magnetic polaritons within saturated ferrites, resulting structurally from the fast-near adiabatic magnetization dynamics combined to the Maxwell’s equations. Accordingly, we address some practical issues of the nonautonomous soliton managements underlying in the fast remagnetization process of data inputs within magnetic memory devices.

Coupled second order evolution equations with fading memory: Optimal energy decay rate

Of concern is the Cauchy problem for coupled second order semilinear evolution equations in a Hilbert space, with indirect memory-damping. We find an approach to obtain successfully an optimal rate of uniform decay for the system energy, only under basic conditions on the memory kernels. Simultaneously, the same rate is also obtained (with less difficulty) for the corresponding single memory-dissipative second order evolution equations. As can be seen, our results essentially improve the previously related ones in the literature. The abstract results are then applied to several concrete problems in the real world.