Time-dependent Nonlinear Differential Equations Research Articles

Quantitative Aeroelastic Stability Prediction of Wings Exhibiting Nonlinear Restoring Forces

In engineering practice, eigen-solution is used to assess the stability of linear dynamical systems. However, the linearity assumption in dynamical systems sometimes implies simplifications, particularly when strong nonlinearities exist. In this case, eigen-analysis requires linerisation of the problem and hence fails to provide a direct stability estimation. For this reason, a more reliable tool should be implemented to predict nonlinear phenomena such as chaos or limit cycle oscillations. One method to overcome this difficulty is the Lyapunov Characteristic Exponents (LCEs), which provide quantitative indications of the stability characteristics of dynamical systems governed by nonlinear time-dependent differential equations. Stability prediction using Lyapunov Characteristic Exponents is compatible with the eigen-solution when the problem is linear. Moreover, LCE estimations do not need a steady or equilibrium solution and they can be calculated as the system response evolves in time. Hence, they provide a generalization of traditional stability analysis using eigenvalues. These properties of Lyapunov Exponents are very useful in aeroelastic problems possessing nonlinear characteristics, which may significantly alter the aeroelastic characteristics, and result in chaotic and limit cycle behaviour. A very common nonlinearity in flexible systems is the nonlinear restoring force such as cubic stiffness, which would substantially benefit from using LCEs in stability assessment. This work presents the quantitative evaluation of aeroelastic stability indicators in the presence of nonlinear restoring force. The method is demonstrated on a two-dimensional aeroelastic problem by comparing the system behaviour and estimated Lyapunov Exponents.

Modeling of Virus Survival Time in Respiratory Droplets on Surfaces: A New Rational Approach for Antivirus Strategies

The modeling of the viability decay of viruses in sessile droplets is addressed considering a droplet sitting on a smooth surface characterized by a specific contact angle. To investigate, at prescribed temperature, how surface energy of the material and ambient humidity cooperate to determine the virus viability, we propose a model which involves the minimum number of thermodynamically relevant parameters. In particular, by considering a saline water droplet (one salt) as the simplest approximation of real solutions (medium and natural/artificial saliva), the evaporation is described by a first-order time-dependent nonlinear differential equation properly rearranged to obtain the contact angle evolution as the sole unknown function. The analyses were performed for several contact angles and two typical droplet sizes of interest in real situations by assuming constant ambient temperature and relative humidity in the range 0–100%. The results of the simulations, given in terms of time evolution of salt concentration, vapor pressure, and droplet volume, elucidate some previously not yet well-understood dynamics, demonstrating how three main regimes—directly implicated in nontrivial trends of virus viability and to date only highlighted experimentally—can be recognized as the function of relative humidity. By recalling the concept of cumulative dose of salts (CD), to account for the effect of the exposition of viruses to salt concentration on virus viability, we show how the proposed approach could suggest a chart of a virus fate by predicting its survival time at a given temperature as a function of the relative humidity and contact angle. We found a good agreement with experimental data for various enveloped viruses and predicted in particular for the Phi6 virus, a surrogate of coronavirus, the characteristic U-shaped dependence of viability on relative humidity. Given the generality of the model and once experimental data are available that link the vulnerability of a certain virus, such as SARS-CoV-2, to the concentrations of salts or other substances in terms of CD, it is felt that this approach could be employed for antivirus strategies and protocols for the prediction/reduction of human health risks associated with SARS-CoV-2 and other viruses.

Coupled effects of magnetic field, number of walls, geometric imperfection, temperature change, and boundary conditions on nonlocal nonlinear vibration of carbon nanotubes resting on elastic foundations

The present study focusses on the investigations of the combined effects of magnetic field, temperature, nonlocal parameter, number of walls, initial geometric imperfection and end supports on the nonlinear transverse vibrations of carbon nanotubes resting on linear and nonlinear elastic foundations. With the aids of Erigen's nonlocal elasticity, Euler-Bernoulli beam theories and van der Waal forces equation, systems of nonlinear partial differential equations governing the dynamics responses of slightly curved multi-walled carbon nanotubes resting on Winkler and Pasternak foundations in a thermal-magnetic environment are developed. The developed equations are decomposed into spatial and temporal equations using Galerkin separation approach. The time-dependent nonlinear differential equations are solved by homotopy perturbation method. Detailed studies are conducted to investigate the effects of the models parameters on the nonlinear vibrations of the structures. The parametric studies reveal that the frequency ratio decreases as the number of magnetic field strength, number of nanotube walls, temperature, spring constants and the ratio of the radius of curvature to the length of the slightly curved nanotubes increase. It is established that these trends are the same for all the boundary conditions considered. The study also reveals that the clamped-simple supported multi-walled nanotubes have the highest frequency ratio while the clamped-clamped supported multi-walled nanotubes have the lowest frequency ratio. Further investigations show that quadruple-walled carbon nanotubes can be taken as pure linear vibration even at any value of linear Winkler and Pasternak foundations constants. Such result can be used to control the nonlinear vibration of the carbon nanotubes and also restrains the chaos vibration in the objective structure. The findings in this work will help in the design of multi-walled carbon nanotubes for various structural, electrical, mechanical and biological applications in a thermal and magnetic environment.

Time to Approach Similarity

Summary In a recent article, Ball and Huppert (J. Fluid Mech., 874, 2019) introduced a novel method for ascertaining the characteristic timescale over which the similarity solution to a given time-dependent nonlinear differential equation converges to the actual solution, obtained by numerical integration, starting from given initial conditions. In this article, we apply this method to a range of different partial differential equations describing propagating gravity currents of fixed volume as well as modifying the techniques to apply to situations for which convergence to the numerical solution is oscillatory, as appropriate for gravity currents propagating at large Reynolds numbers. We investigate properties of convergence in all of these cases, including how different initial geometries affect the rate at which the two solutions agree. It is noted that geometries where the flow is no longer unidirectional take longer to converge. A method of time-shifting the similarity solution is introduced to improve the accuracy of the approximation given by the similarity solution, and also provide an upper bound on the percentage disagreement over all time.

Analysis of transient heat transfer in radial moving fins with temperature-dependent thermal properties

In this article, thermal analysis of radial moving fins of hyperbolic and rectangular profiles is performed. A time-dependent nonlinear differential equation modeling heat transfer in a radial moving fin is considered. The thermal conductivity and heat transfer coefficient are modeled as linear and power-law functions of temperature, respectively. The analytical solutions are generated using the differential transform method which is an analytical solution technique that can be applied to various types of differential equations. The accuracy of the analytical solutions is validated by benchmarking against the numerical solutions obtained by applying the inbuilt numerical solver in Maple (pdsolve). A good agreement is observed between the analytical and numerical solutions. The effects of embedding parameters on the dimensionless temperature and heat transfer rate are analyzed. The results show that the fin rapidly dissipates heat to the surrounding fluid with an increase in the speed of the moving fin, that is, an increase in the values of the Peclet number. The results also show that the heat transfer through a rectangular fin is more efficient when compared to a fin of hyperbolic profile.

Learning Across Scales---Multiscale Methods for Convolution Neural Networks

In this work, we establish the relation between optimal control and training deep Convolution Neural Networks (CNNs). We show that the forward propagation in CNNs can be interpreted as a time-dependent nonlinear differential equation and learning can be seen as controlling the parameters of the differential equation such that the network approximates the data-label relation for given training data. Using this continuous interpretation, we derive two new methods to scale CNNs with respect to two different dimensions. The first class of multiscale methods connects low-resolution and high-resolution data using prolongation and restriction of CNN parameters inspired by algebraic multigrid techniques. We demonstrate that our method enables classifying high-resolution images using CNNs trained with low-resolution images and vice versa and warm-starting the learning process. The second class of multiscale methods connects shallow and deep networks and leads to new training strategies that gradually increase the depths of the CNN while re-using parameters for initializations.

Functionality of reduced graphene oxide flakes at the growth of conducting zone in polyaniline-graphene composite films

When reduced graphene oxide (RGO) flakes were included in polyaniline (PANI), the PANI-RGO composite film exhibited faster redox conversion than the PANI film did. A reason of the enhancement of the conversion rate was searched by examining the growth rates of the conducting zone which appeared at the boundary between the oxidized and the reduced PANI in the film when the reduced composite was oxidized electrochemically from one end of the film. The film was prepared by drying PANI-RGO colloidal suspensions, which were formed by coating flaky RGO particles with PANI. Higher anodic potentials and higher ratio of RGO fractions enhanced the growth rates. There are several possible reasons for the increase in the rates; electro-catalytic properties for the oxidation, enhancement of the conductivity by RGO, the electric percolation, and enhancement of specific surface area of RGO. The growth rate was modeled with the oxidation at the Tafel-typed rate, which was restrained by the IR-drop between the conducting front and the electrode. The growth length was expressed by a time-dependent non-linear differential equation. The numerical solution allowed us to analyze the experimental data of the time variation of the growth length. RGO functions as an increase in the rough surface area of the oxidation rather than electric percolation, enhancement of conductivity and catalytic effects.

Predictive Modeling Applied to a Spent Fuel Dissolver Model—I: Adjoint Sensitivity Analysis

This work presents a comprehensive sensitivity analysis of a paradigm dissolver model that has been selected because of its applicability to material separations and its potential role in diversion activities associated with proliferation and international safeguards. This dissolver model consists of eight active compartments in which the time-dependent nonlinear differential equations modeling the physical and chemical processes comprise 16 time-dependent spatially dependent state functions and 635 model parameters related to the model’s equation of state and inflow conditions. The most important response for the dissolver model is the computed nitric acid in the compartment farthest away from the inlet, where measurements are available for comparisons. The sensitivities to all model parameters of the acid concentrations at each of these instances in time are computed exactly and efficiently using the adjoint sensitivity analysis method for nonlinear systems. The relative importance of the sensitivities in contributing to the uncertainties in the computed model responses is quantified numerically and analyzed in the dissolver’s physics context. The sensitivities computed in this work will be used in a companion paper for uncertainty analysis and predictive modeling, which aims at validating the paradigm dissolver model using the available experimental data and subsequently obtaining best-estimate predicted nominal values for the acid concentrations, with reduced predicted uncertainties, for the longer-term purpose of coupling this dissolver model to other nuclear facilities of interest to nonproliferation objectives.

Non-linear dynamic analysis of symmetric and antisymmetric cross-ply laminated orthotropic thin shells

In this paper, the governing equations for non-linear free vibration of truncated, thin, laminated, orthotropic conical shells using the theory of large deformations with the Karman-Donnell-type of kinematic nonlinearity are derived. Applying superposition principle and Galerkin’s method, these equations are reduced to a time dependent non-linear differential equation. The frequency-amplitude relationship for the laminated orthotropic thin truncated conical shell is obtained using the method of weighted residuals. In the particular case, we can obtain the similar relationships for the single-layer and laminated orthotropic cylindrical shells, also. The influence played by geometrical parameters of the conical shell and physical parameters of the laminate (i.e. material properties, staking sequences and number of layers) on the non-linear vibration behavior of the conical shell is examined. It is noticed that the non-linear vibration of shells is highly dependent on laminate characteristics and, from these observations, it is concluded that specific configurations of laminates should be designed for each kind of application. Present results are compared with available data for special cases.

A Wavelet Method for Solving Nonlinear Time-Dependent Partial Differential Equations

A wavelet method is proposed for solving a class of nonlinear time-dependent partial differential equations. Following this method, the nonlinear equations are first transfoulled into a system of ordinary differential equations by using the modified wavelet Galerkin method recently developed by the authors. Then, the classical fourth-order explicit Runge-Kutta method is employed to solve the resulting system of ordinary differential equations. To justify the present method, the coupled viscous Burgers' equations are solved as examples, results demonstrate that the proposed wavelet algorithm have a much better accuracy and efficiency than many existing numerical methods, and the order of convergence of such a wavelet method can even reach about 5.

Large amplitude vibration of orthotropic sandwich elliptic plates

The present paper deals with non-linear vibration of sandwich elliptic plates of uniform thickness made of orthotropic material. The governing partial differential equations are deduced by the method of ‘constant deflection contour lines’ proposed by Mazumder. The time dependent non-linear differential equation has finally been solved. The graphs corresponding to the ratio of non-linear to linear frequencies versus non-dimensional amplitude are drawn for movable as well as for immovable edge conditions and for different aspect ratios and compared with the existing results.

Large Amplitude Vibration of Thin Homogeneous Heated Orthotropic Sandwich Elliptic Plates

ABSTRACT This paper deals with nonlinear vibration of homogeneous heated sandwich elliptic plates of uniform thickness made of orthotropic material under thermal loading. The governing partial differential equations are deduced by the method of constant deflection contour lines proposed by Mazumder and Jones. The time-dependent nonlinear differential equation has finally been solved. Graphs corresponding to the ratio of nonlinear to linear frequencies versus nondimensional amplitude are drawn for movable as well as for immovable edge conditions and for different aspect ratios and compared with the existing results.

Non-linear free vibration analysis of laminated non-homogeneous orthotropic cylindrical shells

The free vibrations of laminated non-homogeneous orthotropic thin cylindrical shells are studied, with geometric non-linearity taken into account. The basic relations and equations of motion, based on the Donnel-Mushtari shell equations considering finite deformations and the Airy stress function, are obtained for laminated thin cylindrical shells, the elasticity modulus and density of which vary piecewise continuously in the through-thickness direction. Applying the Galerkin method to the foregoing equations, a non-linear time dependent differential equation is obtained for the displacement amplitude. The frequency is obtained from this equation as a function of the shell displacement amplitude and then compared with the results in the literature. Finally, the effect of non-linearity, non-homogeneity and the number and ordering of layers on the frequency is found for different mode numbers, presented graphically and compared with other work.

Non-ideal Brownian motion, generalized Langevin Equation and its application to the security market

Brownian motion has been extensively applied in the field of mathematical finance in modeling the stochastic processes of returns on securities. In this paper basic and generalized Langevin Equations with memory are used to augment Brownian motion to capture the well stylized facts of the financial market that frictions and imperfect information exist. The operator method of Fourier-Laplace transform with an appropriate kernel (influence function) is used to circumvent the difficulty associated with solving a time dependent nonlinear differential Equation, and a practical computational method is proposed.

Large-amplitude vibration of thin homogeneous orthotropic elastic plates under uniform heating—revisited

The paper deals with the non-linear vibration of homogeneous elastic plates of uniform thickness made of orthotropic material under uniform heating. The governing partial differential equations are deduced by the method of constant deflection contour lines, proposed by Mazumdar and Jones and subsequently modified by Sinha Ray and Banerjee. The time-dependent non-linear differential equation has finally been solved by the simple and elegant method of Adomian.

Thermosolutal convection during directional solidification

During solidification of a binary alloy at constant velocity vertically upward, thermosolutal convection can occur if the solute rejected at the crystal-melt interface decreases the density of the melt. We assume that the crystal-melt interface remains planar and that the flow field is periodic in the horizontal direction. The time-dependent nonlinear differential equations for fluid flow, concentration, and temperature are solved numerically in two spatial dimensions for small Prandtl numbers and moderately large Schmidt numbers. For slow solidification velocities, the thermal field has an important stabilizing influence: near the onset of instability the flow is confined to the vicinity of the crystal-melt interface. Further, for slow velocities, as the concentration increases, the horizontal wavelength of the flow decreases rapidly — a phenomenon also indicated by linear stability analysis. The lateral in-homogeneity in solute concentration due to convection is obtained from the calculations. For a narrow range of solutal Rayleigh numbers and wavelengths, the flow is periodic in time.

Some Conditions for the Stability of Nonlinear Time-Dependent Differential Equations

Some Conditions for the Stability of Nonlinear Time-Dependent Differential Equations