Nonlinear Forcing Term Research Articles

Blowing‐up solutions for the Moorea–Gibson–Thompson equation with a viscoelastic memory and an external force

The Moore–Gibson–Thompson equation with a viscoelastic memory and a forcing term is considered. The existence and uniqueness of a local solution are obtained via Faedo–Galerkin's method. Furthermore, it is shown that solutions with or without a positive initial energy experience blowing‐up due to the nonlinear forcing term.

Bernoulli factories and duality in Wright–Fisher and Allen–Cahn models of population genetics

Mathematical models of genetic evolution often come in pairs, connected by a so-called duality relation. The most seminal example are the Wright–Fisher diffusion and the Kingman coalescent, where the former describes the stochastic evolution of neutral allele frequencies in a large population forwards in time, and the latter describes the genetic ancestry of randomly sampled individuals from the population backwards in time. As well as providing a richer description than either model in isolation, duality often yields equations satisfied by quantities of interest. We employ the so-called Bernoulli factory – a celebrated tool in simulation-based computing – to derive duality relations for broad classes of genetics models. As concrete examples, we present Wright–Fisher diffusions with general drift functions, and Allen–Cahn equations with general, nonlinear forcing terms. The drift and forcing functions can be interpreted as the action of frequency-dependent selection. To our knowledge, this work is the first time a connection has been drawn between Bernoulli factories and duality in models of population genetics.

Mechanical response of tunnels crossing active fault zones under normal or reverse faulting: A refined nonlinear analytical approximate solution

Tunnels crossing active fault zones will be severely damaged under normal or reverse faulting. The tunnel-stratum interaction is nonlinear, and the fault zone width significantly affects the tunnel response. Nevertheless, the existing analytical models neglect the effect of nonlinear tunnel-stratum interaction or fault zone width and adopt unrealistic assumptions. This work proposes a refined nonlinear solution, considering the nonlinear tunnel-stratum interaction and fault zone width, and some widely employed assumptions are removed. Specifically, a tunnel under normal or reverse faulting is simplified to an elastic beam acting on a nonlinear foundation, the nonlinear tunnel-stratum interaction is considered by setting a series of nonlinear axial and vertical springs, and the fault zone width is considered by reducing the spring stiffness. Additionally, the nonlinear axial force and geometrical nonlinearity terms are considered in the governing equations. The comparison results indicate that the results obtained from the proposed model agree well with those from model tests and numerical models and have a better accuracy than the existing analytical models. Finally, the effects of fault, stratum, and lining parameters on the response of tunnels crossing normal or reverse faults are discussed regarding stress distribution and failure scope.

On the energy decay of a viscoelastic piezoelectric beam model with nonlinear internal forcing terms and a nonlinear feedback

In this paper, we consider a piezoelectric beam model with nonlinear source terms and investigate the interaction between a viscoelastic damping and a nonlinear frictional damping. Under general assumptions on the relaxation function and the nonlinear feedback, we establish explicit formulae for the energy decay rates of this system and prove that the energy decays at a rate dictated by the weaker damping. Our results substantially improve some earlier related results in the literature.

On a nonlinear system of plate equations with variable exponent nonlinearity and logarithmic source terms: Existence and stability results

<abstract><p>In this paper, we consider a coupling non-linear system of two plate equations with logarithmic source terms. First, we study the local existence of solutions of the system using the Faedo-Galerkin method and Banach fixed point theorem. Second, we prove the global existence of solutions of the system by using the potential wells. Finally, using the multiplier method, we establish an exponential decay result for the energy of solutions of the system. Some conditions on the variable exponents that appear in the coupling functions and the involved constants that appear in the source terms are determined to ensure the existence and stability of solutions of the system. A series of lemmas and theorems have been proved and used to overcome the difficulties caused by the variable exponent and the logarithmic nonlinearities. Our result generalizes some earlier related results in the literature from the case of only constant exponent of the nonlinear internal forcing terms to the case of variable exponent and logarithmic source terms, which is more useful from the physical point of view and needed in several applications.</p></abstract>

Two Extratropical Pathways to Forcing Tropical Convective Disturbances

Abstract Observational evidence of two extratropical pathways to forcing tropical convective disturbances is documented through a statistical analysis of satellite-derived OLR and ERA5 reanalysis. The forcing mechanism and the resulting disturbances are found to strongly depend on the structure of the background zonal wind. Although Rossby wave propagation is prohibited in easterlies, modeling studies have shown that extratropical forcing can still excite equatorial waves through resonance between the tropics and extratropics. Here this “remote” forcing pathway is investigated for the first time in the context of convectively coupled Kelvin waves over the tropical Pacific during northern summer. The extratropical forcing is manifested by eddy momentum flux convergence that arises when extratropical eddies propagate into the subtropics and encounter their critical line. This nonlinear forcing has similar wavenumbers and frequencies with Kelvin waves and excites them by projecting onto their meridional eigenstructure in zonal wind, as a form of resonance. This resonance is also evidenced by a momentum budget analysis, which reveals the nonlinear forcing term is essential for maintenance of the waves, while the remaining linear terms are essential for propagation. In contrast, the “local” pathway of extratropical forcing entails the presence of a westerly duct during northern winter that permits Rossby waves to propagate into the equatorial east Pacific, while precluding any sort of resonance with Kelvin waves due to Doppler shifting effects. The intruding disturbances primarily excite tropical “cloud plumes” through quasigeostrophic forcing, while maintaining their extratropical nature. This study demonstrates the multiple roles of the extratropics in forcing in tropical circulations and illuminates how tropical–extratropical interactions and extratropical basic states can provide be a source of predictability at the S2S time scale. Significance Statement This study seeks to understand how circulations in the midlatitudes excite the weather systems in the tropics. Results show that the mechanisms, as well as the types of tropical weather systems excited, are strongly dependent on the mean large-scale wind structure. In particular, when the large-scale wind blows from east to west, a special type of eastward-moving tropical weather system, the Kelvin wave, is excited owing to its resonance with remote eastward-moving weather systems in the extratropics. On the contrary, when the average wind blows from west to east, midlatitude systems are observed to intrude into the lower latitudes and directly force tropical convection, the cloud plumes, while maintaining their extratropical nature. These results speak to how the midlatitudes can excite distinct types of tropical weather systems under different climatological wind regimes. Understanding these tropical weather systems and their interactions with the midlatitudes may ultimately help to improve predictions of weather beyond 2 weeks.

Solutions to aliasing in time-resolved flow data

Avoiding aliasing in time-resolved flow data obtained through high fidelity simulations while keeping the computational and storage costs at acceptable levels is often a challenge. Well-established solutions such as increasing the sampling rate or low-pass filtering to reduce aliasing can be prohibitively expensive for large data sets. This paper provides a set of alternative strategies for identifying and mitigating aliasing that are applicable even to large data sets. We show how time-derivative data, which can be obtained directly from the governing equations, can be used to detect aliasing and to turn the ill-posed problem of removing aliasing from data into a well-posed problem, yielding a prediction of the true spectrum. Similarly, we show how spatial filtering can be used to remove aliasing for convective systems. We also propose strategies to prevent aliasing when generating a database, including a method tailored for computing nonlinear forcing terms that arise within the resolvent framework. These methods are demonstrated using a non-linear Ginzburg-Landau model and large-eddy simulation (LES) data for a subsonic turbulent jet.

A neural-corrector method for prediction of the vertical motions of a high-speed craft

A machine learning method is developed which is capable of making predictions of nonlinear ship motions in a range of wave conditions when trained on response data from only a single seaway. The method is formulated around the equations of motion in the time domain, but the equations are augmented with data-driven terms that act to correct the force-balance in the equations. The data-driven nonlinear forcing terms are modeled using Long Short-Term Memory (LSTM) recurrent neural networks. The resulting hybrid governing equations are solved numerically. Predictions from the method are compared to nonlinear test data of 2-DOF motion of a Generic Prismatic Planing Hull (GPPH) at forward speed in head seas, with time histories given for both regular and irregular waves. The training data requirements to classify a specific seaway are investigated and quantified. Predictions over a range of significant wave heights and peak periods are performed using training data from only a single seaway to show the effectiveness of the method in generalizing across different environmental conditions.

A preservative splitting approximation of the solution of a variable coefficient quenching problem

This paper studies the numerical solution of a two-dimensional quenching type nonlinear reaction-diffusion problem via dimensional splitting. The variable coefficient differential equation considered possesses a nonlinear forcing term, and may lead to strong quenching singularities that have profound multiphysics and engineering applications to the energy industry. Our investigations focus on the construction and preservative properties of a semi-adaptive Peaceman-Rachford procedure for solving the aforementioned problem. In the study, the multidimensional differential equation is decomposed into single dimensional subequations so that the computational efficiency can be effectively raised. Temporal adaptation is implemented through arc-length estimations of the rate-of-change of the numerical solution. The positivity, monotonicity and localized linear stability of the variable step splitting scheme are analyzed. These ensure key preservations of underlying physical features of the computed approximations for applications. Computational experiments are presented to illustrate our results as well as to demonstrate the viability and accuracy of the splitting method for solving singular quenching-combustion problems.

Stepanov pseudo‐almost periodic functions and applications

In this work, we present basic results and applications of Stepanov pseudo‐almost periodic functions with measure. Using only the continuity assumption, we prove a new composition result of μ‐pseudo‐almost periodic functions in Stepanov sense. Moreover, we present different applications to semilinear differential equations and inclusions with weak regular forcing terms in Banach spaces. We prove the existence and uniqueness of μ‐pseudo‐almost periodic solutions (in the strong sense) to a class of semilinear fractional inclusions and semilinear evolution equations respectively, provided that the nonlinear forcing terms are only Stepanov μ‐pseudo‐almost periodic in the first variable and not a uniformly strict contraction with respect to the second argument. Our results are obtained using the Meir–Keeler principle and the Banach fixed point principle respectively. Some examples of fractional and nonautonomous partial differential equations illustrating our theoretical results are also presented.

Optimal eddy viscosity for resolvent-based models of coherent structures in turbulent jets

Abstract

Existence and uniqueness of weak solution to a three-dimensional stochastic modified-Leray-alpha model of fluid turbulence

In this paper, we study the stochastic three-dimensional modified Leray-alpha model arising from the turbulent flows of fluids. We prove the existence of the probabilistic weak solution under the non-Lipschitz condition for the nonlinear forcing terms. We also discuss its uniqueness.

Mild solutions of a class of semilinear fractional integro-differential equations subjected to noncompact nonlocal initial conditions

In this paper, we prove the existence of mild solutions of a class of fractional semilinear integro-differential equations of order \(\beta\in(1,2]\) subjected to noncompact initial nonlocal conditions. We assume that the linear part generates an arbitrarily strongly continuous \(\beta\)-order fractional cosine family, while the nonlinear forcing term is of Caratheodory type and satisfies some fairly general growth conditions. Our approach combines the Monch fixed point theorem with some recent results regarding the measure of noncompactness of integral operators. Our conclusions improve and generalize many earlier related works. An example is provided to illustrate the main results.

The colour of forcing statistics in resolvent analyses of turbulent channel flows

Abstract

Spectral proper orthogonal decomposition and resolvent analysis of near-wall coherent structures in turbulent pipe flows

Abstract

On the numerical structure preservation of nonlinear damped stochastic oscillators

The paper is focused on analyzing the conservation issues of stochastic 𝜃-methods when applied to nonlinear damped stochastic oscillators. In particular, we are interested in reproducing the long-term properties of the continuous problem over its discretization through stochastic 𝜃-methods, by preserving the correlation matrix. This evidence is equivalent to accurately maintaining the stationary density of the position and the velocity of a particle driven by a nonlinear deterministic forcing term and an additive noise as a stochastic forcing term. The provided analysis relies on a linearization of the nonlinear problem, whose effectiveness is proved theoretically and numerically confirmed.

Implicit-explicit time integration of nonlinear fractional differential equations

Efficient long-time integration of nonlinear fractional differential equations is significantly challenging due to the integro-differential nature of the fractional operators. In addition, the inherent non-smoothness introduced by the inverse power-law kernels deteriorates the accuracy and efficiency of many existing numerical methods. We develop two efficient first- and second-order implicit-explicit (IMEX) methods for accurate time-integration of stiff/nonlinear fractional differential equations with fractional order α∈(0,1] and prove their convergence and linear stability properties. The developed methods are based on a linear multi-step fractional Adams-Moulton method (FAMM), followed by the extrapolation of the nonlinear force terms. In order to handle the singularities nearby the initial time, we employ Lubich-like corrections to the resulting fractional operators. The obtained linear stability regions of the developed IMEX methods are larger than existing IMEX methods in the literature. Furthermore, the size of the stability regions increase with the decrease of fractional order values, which is suitable for stiff problems. We also rewrite the resulting IMEX methods in the language of nonlinear Toeplitz systems, where we employ a fast inversion scheme to achieve a computational complexity of O(Nlog⁡N), where N denotes the number of time-steps. Our computational results demonstrate that the developed schemes can achieve global first- and second-order accuracy for highly-oscillatory stiff/nonlinear problems with singularities.

Stability estimates and a Lagrange-Galerkin scheme for a Navier-Stokes type model of flow in non-homogeneous porous media

The purposes of this work are to study the $L^{2}$-stability of a Navier-Stokes type model for non-stationary flow in porous media proposed by Hsu and Cheng in 1989 and to develop a Lagrange-Galerkin scheme with the Adams-Bashforth method to solve that model numerically.The stability estimate is obtained thanks to the presence of a nonlinear drag force term in the model which corresponds to the Forchheimer term. We derive the Lagrange-Galerkin scheme by extending the idea of the method of characteristics to overcome the difficulty which comes from the non-homogeneous porosity. Numerical experiments are conducted to investigate the experimental order of convergence of the scheme. For both simple and complex designs of porosities, our numerical simulations exhibit natural flow profiles which well describe the flow in non-homogeneous porous media.

Existence of local stable manifolds for some nondensely defined nonautonomous partial functional differential equations

In this paper, we establish the existence of local stable manifolds for a semi-linear differential equation, where the linear part is a Hille–Yosida operator on a Banach space and the nonlinear forcing term [Formula: see text] satisfies the [Formula: see text]-Lipschitz conditions, where [Formula: see text] belongs to certain classes of admissible function spaces. The approach being used is the fixed point arguments and the characterization of the exponential dichotomy of evolution equations in admissible spaces of functions defined on the positive half-line.

A blow-up result for nonlocal thin-film equation with positive initial energy

In this note, we consider a thin-film equation including a diffusion term, a fourth order term and a nonlocal source term under the periodic boundary conditions. In particular, a finite time blow-up result is established for the case of positive initial energy provided that \[ \frac{\pi^2}{a^2}\leq \frac{2}{p-1},\] where $a$ is the length of the interval and $p>1$ is the power of nonlinear force term. Also upper and lower blow-up times are estimated.