Set Of Differential Equations Research Articles

Fast and scalable computation of shape-morphing nonlinear solutions with application to evolutional neural networks

We develop fast and scalable methods for computing reduced-order nonlinear solutions (RONS). RONS was recently proposed as a framework for reduced-order modeling of time-dependent partial differential equations (PDEs), where the modes depend nonlinearly on a set of time-varying parameters. RONS uses a set of ordinary differential equations (ODEs) for the parameters to optimally evolve the shape of the modes to adapt to the PDE's solution. This method has already proven extremely effective in tackling challenging problems such as advection-dominated flows and high-dimensional PDEs. However, as the number of parameters grow, integrating the RONS equation and even its formation become computationally prohibitive. Here, we develop three separate methods to address these computational bottlenecks: symbolic RONS, collocation RONS and regularized RONS. We demonstrate the efficacy of these methods on two examples: Fokker–Planck equation in high dimensions and the Kuramoto–Sivashinsky equation. In both cases, we observe that the proposed methods lead to several orders of magnitude in speedup and accuracy. Our proposed methods extend the applicability of RONS beyond reduced-order modeling by making it possible to use RONS for accurate numerical solution of linear and nonlinear PDEs. Finally, as a special case of RONS, we discuss its application to problems where the PDE's solution is approximated by a neural network, with the time-dependent parameters being the weights and biases of the network. The RONS equations dictate the optimal evolution of the network's parameters without requiring any training.

Impact of thermal radiations, heat generation/absorption and porosity on MHD nanofluid flow towards an inclined stretching surface: Non‐similar analysis

AbstractObjective of this investigation is to provide a novel study of the flow of a magnetohydrodynamics (MHD) nanofluid over an inclined stretching surface in a permeable medium in a 2‐dimensional boundary layer. The convection model further incorporates heat radiation and heat sources/sinks. Nanomaterials include magnesium oxide and copper oxide . Using non‐similarity transformations, the set of governing equations is transformed into a set of dimensionless partial differential equations (PDEs). Using the local non‐similarity technique, the resulting system of dimensionless PDEs is treated as a set of ordinary differential equations (ODEs), and then solved with the help of the bvp4c MATLAB tools. This comprehensive research involves a thorough investigation of the significant starting physical characteristics and their impacts on the velocity and nanofluid temperature profiles. These profiles are shown using a set of graphical representations and numerical tables. It is worth noting that changes in the magnetic and porosity parameters result in a decrease in the velocity profile, but the Grashof number exhibits a reciprocal correlation. When we account for the effects of the Eckert number and the radiation parameters in a given flow regime, we also see an increase in the nanofluid's temperature profile. It has been shown that when the magnetic number and the Eckert number are augmented, the heat transmission rate decreases. The skin friction coefficient grows on a logarithmic scale when magnetic parameters and nanoparticle concentration are increased in estimation. The numerical calculations are checked for accuracy by comparing them to previously published findings, which guarantees an exact match in a constrained setting. Scientists investigating nanomaterials and their applications in manufacturing and thermal engineering may find this research instructive.

Analytical tangents for arbitrary material laws derived from rheological models at large deformations

AbstractThe development of suitable material laws for various material classes is an essential preliminary task for conducting realistic simulations. Within the framework of large deformations, one recognized approach is the utilization of rheological connections allowing the construction of arbitrary models. A common method to calculate the stress response of such a material model is to formulate a set of algebraic and ordinary differential equations and to solve them numerically. However, in this work, only stress relations between different rheological elements are formulated and directly solved by a numeric algorithm without the need to derive the typical system of algebraic/differential equations. The required derivatives for the solution of these equations for this algorithm and the stiffness of the material model are calculated analytically following the same general principle as the algorithm calculating the stress response. This improves stability and computation effort compared to a forward difference scheme.

Impact of Second-Order Slip on Radiative Magnetohydrodynamics Rotating Flow in Channel

In our current investigation, we have discussed the effects of second-order slip and radiation on the flow of a generalized rotating fluid inside a channel in the presence of a magnetic field. The flow equations are transformed into a set of ordinary differential equations with the help of suitable similarity transformations, and the resultant coupled nonlinear ordinary differential equations are solved analytically. Using the obtained results, the effects of the involved physical parameters on the flow dynamics are examined and described physically. In addition, Pearson’s coefficient of correlation has been utilized to measure the linear association between different variables. The impact of emerging dimensionless parameters such as Hartman number, rotation parameter, radiation parameter, viscosity parameter, first- and second-order slip parameter, Schmidt number, and suction parameter on the Nusselt number and Sherwood number have all been investigated using a slope linear regression approach across the whole data points. The t-test approach has been adopted to explore the relationship between various involved physical parameters.

Dislocation-density-related constitutive model and its applicability to cyclic deformation and damage behavior of 316LN SS at 823 K

In the present investigation, an in-depth understanding of the relationship between dislocation kinetics and cyclic plasticity, along with the description of yield asymmetry behavior during loading and unloading in nuclear grade 316LN austenitic stainless steel at 823 K, was examined by employing a constitutive framework using the evolution of physically-based internal-state variables. The formulation encompassed a set of first-order simultaneous differential equations that governed the evolution of dislocation density, mean free path of dislocations, and back stress. These variables were interdependent with the power law which described the kinetics of plastic strain rate. The evolution equations were numerically integrated, and the simulated data were fitted to the experimental data up to stabilized cycle at a temperature of 823 K and total strain amplitudes of ±0.004, ±0.006, ±0.008 and ± 0.010. The predicted cyclic stress response as a function of the cyclic number displayed the continuous cyclic hardening from the first cycle to the initiation of stabilized cycle. Further, good agreement between the predicted and experimental hysteresis loops was found at different cycle numbers. The predicted dislocation density and tensile peak values of back/effective stress increased with cyclic hardening and approached the value of saturation at the stabilized cycle. On the other hand, a rapid decrease in mean free path with cycle number was noticed. An increase in total strain amplitude resulted in a systematic increase in dislocation density, peak stress, back stress and effective stress. At the total strain amplitudes above ±0.006, the predicted higher rates of dislocation accumulation and annihilation, and the faster evolution rate of back stress towards saturation led to the development of stable dislocation substructure configurations with a clear distinction between dislocation high/low dense regions. Furthermore, the applicability of the model was extended by including an empirical relationship between the damage variable and cycle number within the power law, thereby enabling predictions of the cyclic stress-strain response beyond the stabilized cycle.

Combined reaction and diffusion across the interface in reactive nonuniform liquid systems

A recently introduced theoretical framework is generalized to multicomponent reactive nonuniform systems. The modified and generalized Cahn-Hilliard equation is combined with a chemical kinetics model. Kinetic rate equations, which are formulated with activities rather than concentrations, are combined with non-Fickian diffusion driven by gradients in chemical potential. Based on the incompressible density gradient theory, which provides an expression for the Gibbs energy of a nonuniform system, a generalized method is developed to model the combined reaction and diffusion associated with reactive liquid-liquid systems. A set of partial differential equations describing the temporal evolution of the mole fractions of all the involved components in the bulk phases and the spatial and temporal evolution of the mole fractions in the interface has been derived. The obtained equations were used to investigate the dynamics of the interfacial properties of a ternary mixture with a simple chemical reaction. Special attention is given to reacting mixtures with reaction rates that are much smaller than diffusion rates. In particular, it was found that the mixture will remain in phase equilibrium upon reaction and that the interfacial chemical reaction will not affect the dynamics of the overall system, the latter being only affected by the bulk phase kinetics.

Adaptive constrained tracking control for dynamical actuator driven linear 2 × 2 hyperbolic PDE systems with nonlinear uncertainties

The paper develops an adaptive constrained tracking control technique for a class of 2 × 2 hyperbolic partial differential equation (PDE) systems with boundary actuator dynamics, which are described by a set of ordinary differential equations (ODEs) in the presence of unknown parametric nonlinearities. Since the control input only appears in the uncertain ODE subsystem rather than directly on the boundary of PDE subsystem, the control task becomes quite difficult and the existing direct boundary control approaches are ineffective. Moreover, in this paper, a more challenging problem is considered such that the controlled output and the states of ODE actuators are constrained. To this end, by utilizing finite and infinite dimensional backstepping techniques, barrier Lyapunov functions (BLFs) and adaptive methods, a novel adaptive tracking control approach is proposed. It is the first time that such a constrained tracking control problem is addressed for the PDE-ODE coupled systems considered in this paper. On the basis of the presented method, the rigorous theoretical proof is provided to show that the PDE controlled output and all the states of the ODE actuator stay within the predefined compact sets. Finally, the results are illustrated via a comparative numerical simulation.

Coupled Price–Volume Equity Models with Auto-Induced Regime Switching

In this work, we present a rigorous development of a model for the Price–Volume relationship of transactions introduced in 2009. For this development, we rely on the precise formulation of diffusion auto-induced regime-switching models presented in our previous work of 2020. The auto-induced regime-switching models referred to may be based on a finite set of stochastic differential equations (SDE)—all defined on the same bounded time interval—and a sequence of interlacing stopping times defined by the hitting threshold times of the trajectories of the solutions of the SDE. The coupling between price and volume—which we take as a proxy of liquidity—is assumed to be the following: the regime switching in the price variable occurs at the stopping times for which there is a change of region—in the product state space of price and liquidity—for the liquidity variable (and vice versa). The regimes may be defined parametrically—that is, the SDE coefficients keep the same functional form but with varying parameters—or the functional form of the SDE coefficients may change with each regime. By using the same noise source for both the price and the liquidity regime-switching models—volume (liquidity), which, in general, is not a tradable asset—we ensure that despite incorporating information on liquidity, the price part of the coupled model can be assumed to be arbitrage free and complete, allowing the pricing and hedging of derivatives in a simple way.

Computational analysis of entropy generation in three-dimensional mixed convection flow with thermal dissipation and variable thermal conductivity

Entropy generation in steady three-dimensional thermal flow is investigated under the coupled influence of viscous dissipation and mixed convection. The stretching sheet with an exponential velocity profile induces the flow. In this examination, we presume that the fluid is incompressible and possesses a Newtonian behavior and temperature-dependent thermal conductivity. Using the appropriate classical similarity variables reduces the dimensional nonlinear dynamical system of governing equations to a dimensionless nonlinear dynamical set of ordinary differential equations. It’s important to note that the selection of an exponentially varying surface temperature is deliberate, as it enables precise similarity transformations to be established in situations involving both viscous dissipation and buoyancy forces. The simplified finite difference approach (SFDM) is used to solve the acquired nonlinear dynamical set of ordinary differential equations. The tridiagonal matrix approach is used to solve the resulting system of algebraic equations. In a particular situation, the present numerical simulation is numerically checked with the previously available results and observed a good agreement. The effects of physical parameters on velocity, temperature, surface stresses and heat tranfer rate, entropy generation, and the irreversibility parameter are investigated. The entropy generation rate exhibits a decreasing trend with respect to the mixed convection parameter in the vicinity of the stretching surface. Entropy reduction can be achieved by employing a working fluid with a low Prandtl number and by minimizing the fluid friction.

Thermal and entropy behavior of sustainable solar energy in water solar collectors due to non-Newtonian power-law hybrid nanofluids

Introduction: Nanofluids, hybrid nanofluid possesses thermophysical features that boost the fluid performance. This research work is motivated by the utilization of water solar collectors that incorporate non-Newtonian, power-law hybrid nanofluid in a three-dimensional model, considering the two-phase model.Method: The primary objective of this study is to transform the governing equations of the flow model into a set of ordinary differential equations by employing the three-parameters group technique. Based on the innovative discoveries, two models incorporating new associated functions have been successfully developed for two distinct scenarios characterized by the power-law index, n. The impact of physical factors on the velocity profile, temperature distribution, concentration field, and entropy output of the system is clearly illustrated through a variety of graphs.Results: The results indicated that the inclination angle of 20° had the best thermal characteristics compared to other inclinations. The entropy generation reached its maximum value at temperature difference of 13 K due to irreversibility of the system, which indicates that the system is more efficient.Discussion: Furthermore, the increasing percentage in Nusselt number is predicted to be 28.18% when the Prandtl number is taken a range. The Sherwood number enhanced up to 18.61% with a range of Brownian motion. A quantitative comparison is conducted between the present results and the literature in order to validate the superior efficiency of the used method.

A differential-algebraic extended Kalman filter with exact constraint satisfaction

This work proposes an Augmented Manifold Differential-Algebraic Extended Kalman Filter for state and input estimation for multibody systems described by a set of differential algebraic equations. The proposed Kalman Filter allows for exact constraint satisfaction and does not require reformulation of the multibody system equations. The approach consists of two main steps: (i) elimination of the Lagrange multipliers from the system state via a nullspace projection and (ii) an additional correction step that constrains the a-posteriori estimated system state to lie on the constraint manifold. The latter is achieved by solving a constrained optimization problem using a Euclidean descent approach and projection on the constraint manifold, i.e. manifold optimization. The proposed approach is numerically validated on both open and closed kinematic chain rigid multibody models and is shown to be robust and indifferent to the type of kinematic chain being considered. Furthermore, an experimental validation is carried out where the forces and torques acting on the wheel center of a McPherson car suspension test setup are successfully estimated using a flexible multibody model and the proposed Kalman Filter.

An Open‐Source, Physics‐Based, Tropical Cyclone Downscaling Model With Intensity‐Dependent Steering

AbstractAn open‐source, physics‐based tropical cyclone (TC) downscaling model is developed, in order to generate a large climatology of TCs. The model is composed of three primary components: (a) a random seeding process that determines genesis, (b) an intensity‐dependent beta‐advection model that determines the track, and (c) a non‐linear differential equation set that determines the intensification rate. The model is entirely forced by the large‐scale environment. Downscaling ERA5 reanalysis data shows that the model is generally able to reproduce observed TC climatology, such as the global seasonal cycle, genesis locations, track density, and lifetime maximum intensity distributions. Inter‐annual variability in TC count and power‐dissipation is also well captured, on both basin‐wide and global scales. Regional TC hazard estimated by this model is also analyzed using return period maps and curves. In particular, the model is able to reasonably capture the observed return period curves of landfall intensity in various sub‐basins around the globe. The incorporation of an intensity‐dependent steering flow is shown to lead to regionally dependent changes in power dissipation and return periods. Advantages and disadvantages of this model, compared to other downscaling models, are also discussed.

Effects of magnetization and heat transfer on electrically conducting magnetohydrodynamics flow with mixed convection flow over porous duct

The effects of magnetic field and heat transfer problem on incompressible magnetohydrodynamics (MHD) mixed unsteady convective flow of an electrically conducting viscous fluid in a duct filled with a porous medium are investigated. The fluid flow is subjected to an applied uniform magnetic field vertical to the duct and uniform free stream of constant velocity and temperature. The system of nonlinear differential equations governing flow and heat transfer phenomena are derived and reduced into a set of ordinary differential equations using the auxiliary variable method. The reduced equations are solved numerically using Runge-Kutta method associated with slip boundary conditions. The graphical results for velocity, temperature and heat transfer rate are obtained using MATLAB software. It is observed that the velocity profiles increase with an increase of similarity parameter (fI) while temperature gradient decreases rapidly with an increase of similarity parameter (θI) for different values of Grosh number, Magnetization and Prandtl number respectively.

MHD boundary layer flow due to an exponentially stretching surface through porous medium with radiation effect

The purpose of this article is to study the boundary layer flow and heat transfer of the MHD second-grade fluid. By utilizing similarity transformations, the governing equations are transformed into a set of non-linear ordinary differential equations. To get semi-analytical formulations of velocity, temperature, and other variables, we use the homotopy analysis technique (HAM). Then, we employ the Wolfram Language function NSolve to get the solutions. The main finding of the present work is that the flow variables have been influenced by the magnetic field parameter, the porous parameter, and the radiation parameter.

Spectro-spatial evolution of the CMB. Part I. Discretisation of the thermalisation Green's function

Spectral distortions of the cosmic microwave background (CMB) have been recognized as an important future probe of the early Universe. Existing theoretical studies primarily focused on describing the evolution and creation of average distortions, ignoring spatial perturbations in the plasma. One of the main reasons for this choice is that a treatment of the spectro-spatial evolution of the photon field deep into the primordial Universe requires solving a radiative transfer problem for the distortion signals, which in full detail is computationally challenging. Here we provide the first crucial step towards tackling this problem by formulating a new spectral discretisation of the underlying average thermalisation Green's function. Our approach allows us to convert the high-dimensional partial differential equation system (≃ 103–104 equations) into and set of ordinary differential equations of much lower dimension (≃ 10 equations). We demonstrate the precision of the approach and highlight how it may be further improved in the future. We also clarify the link of the observable spectral distortion parameters (e.g., μ and y) to the computational spectral basis that we use in our frequency discretisation. This reveals how several basis-dependent ambiguities can be interpreted in future CMB analysis. Even if not exact, the new Green's function discretisation can be used to formulate a generalised photon Boltzmann-hierarchy, which can then be solved with methods that are familiar from theoretical studies of the CMB temperature and polarisation anisotropies. We will carry this program out in a series of companion papers, thereby opening the path to full spectro-spatial exploration of the CMB with future CMB imagers and spectrometers.

Levenberg–Marquardt neural network-based intelligent computation for the non-Newtonian polymer during forward roll coating

Scientists and researchers widely recognize the effectiveness of artificial intelligence (AI)-based machine learning and intelligent computing solvers, demonstrating qualities such as resilience, robustness, stability, and rapid convergence. One particularly significant and rapidly growing field within AI is artificial neural networks. This research uses a supervised neural network model based on Levenberg–Marquardt backpropagation (LMB-SNNs) to examine the Sisko fluid model for the forward roll coating process (SFM-FRCP). A suitable transformation is applied to the partial differential equations based SFM-FRCP mathematical model, resulting in a set of nonlinear ordinary differential equations. The perturbation method has been used to find the analytical solutions for the velocity profile, pressure gradient, and pressure profile. A dataset for varying the pertinent parameters is generated, and the LMB-SNNs technique has been used to estimate the velocity profile, pressure gradient, and pressure profile behavior during FRCP for numerous scenarios. The numerical solution for SFM-FRCP in different scenarios, such as the validation, training, and testing procedures of LMB-SNNs, is carried out. Moreover, the state transition index, fitness outline, mean square error, histogram error, and regression presentation also endorse the strength and reliability of the solver LMB-SNNs for SFM-FRCP. The comparative analyses and performance studies through outputs of regression drawings, absolute error, and error histograms validate the effectiveness of the suggested solver LMB-SNNs. The method's precision is verified by the closest numerical outputs of both built and dataset values with similar levels 10−11–10−14. Furthermore, it has been observed that as the non-Newtonian parameter increases, the fluid velocity decreases. The research work carried out in this paper is original and fills a gap in the existing research by showing the rheological properties of the Sisko fluid model and the implementation of the LMB-SNNs during the FRCP.

Brunovsky Decomposition for Dynamic Interval Localization

This paper proposes a new set-membership method for estimating the trajectories of dynamical systems, when the states are completely unknown and only non-linear observations are available. The first part of the proposed method is symbolic and follows the decomposition of Brunovsky, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i.e.</i> , it decomposes the set of differential equations describing the dynamical system into two blocks of constraints: one block gathers non-linear analytical equations that do not involve differential operators, while the other block is composed of linear chains of integrators. The second part of the method, that relies on the symbolic decomposition, is numerical and based on a contractor approach. It involves a specific optimal operator for narrowing the sets of feasible solutions. This approach is shown to be efficient on a difficult problem of dynamic localization of a mobile robot, without any prior knowledge about its states.

Beyond the Drake Equation: A Time-dependent Inventory of Habitable Planets and Life-bearing Worlds in the Solar Neighborhood

We introduce a mathematical framework for statistical exoplanet population and astrobiology studies that may help direct future observational efforts and experiments. The approach is based on a set of differential equations and provides a time-dependent mapping between star formation, metal enrichment, and the occurrence of exoplanets and potentially life-harboring worlds over the chemo-population history of the solar neighborhood. Our results are summarized as follows: (1) the formation of exoplanets in the solar vicinity was episodic, starting with the emergence of the thick disk about 11 Gyr ago; (2) within 100 pc from the Sun, there are as many as 11,000(η ⊕/0.24) Earth-size planets in the habitable zone (“temperate terrestrial planets” or TTPs) of K-type stars. The solar system is younger than the median TTP, and was created in a star formation surge that peaked 5.5 Gyr ago and was triggered by an external agent; (3) the metallicity modulation of the giant planet occurrence rate results in a later typical formation time, with TTPs outnumbering giant planets at early times; and (4) the closest, life-harboring Earth-like planet would be ≲20 pc away if microbial life arose as soon as it did on Earth in ≳1% of the TTPs around K stars. If simple life is abundant (fast abiogenesis), it is also old, as it would have emerged more than 8 Gyr ago in about one-third of all life-bearing planets today. Older Earth analogs are more likely to have developed sufficiently complex life capable of altering their environment and producing detectable oxygenic biosignatures.

Numerical and intelligent neuro-computational modelling with Fourier’s energy and Fick’s mass flux theory of 3D fluid flow through a stretchable surface

Current work explores the intricacies of magnetohydrodynamic and mix convectional boundary-layer flow concerning couple stress Casson nanofluid (CSCNF) dynamics via a 3D stretchable surface. The addition of active and passive control mechanisms for nanoscales creates an innovative dimension to the exploration. Remarkably, the analysis incorporates the influence of non-Fourier and non-Fickian heat and mass flux, alongside the effects of thermophoresis and Brownian diffusion, to systematically investigate the heat and mass transportation phenomena. The governing equations (PDEs) describing the MHD-3DCSCNF model are converted into a set of ordinary differential equations to facilitate the ANN analysis. Employing the bvp4c technique, a dataset is systematically generated for the back propagation artificial neural network with Levenberg–Marquardt Algorithm (BANN-LMA) through four distinct scenarios. Via accurate testing, validation, and training, the BANN-LMA produces estimated results for the MHD-3DCSCNF problem. The performance validation of BANN-LMA is executed through several metrics, involving the mean squared error, error histogram and regression analysis. The training process, characterized by minimizing the MSE through a gradient descent methodology with optimized weights, exhibits a compelling correlation R = 1, between the target and network output. Furthermore, the consistent convergence observed highlights the method robustness and reliability.

Weak disturbance decoupling of high‐order fully actuated nonlinear systems

SummaryIt is well known that all practical control systems are subject to external disturbances. Therefore, a feedback controller should be designed so that the closed‐loop system has some desired properties such as stability, good tracking performance, at the same time, the impact of external disturbances on the performance of the closed‐loop system is attenuated to some degree. On the other hand, almost all models for real fully actuated systems, which are derived based on the physical laws, are described by a set of high order differential equations. The traditional method for controlling these fully actuated systems is to convert the high order differential equations to first order differential equations, which are called the state equations, and then to design feedback controllers by using the state‐space design methods. Such conversion causes some inconveniences. To overcome such inconveniences, a high‐order fully actuated (HOFA) design method has been proposed to control HOFA systems. The main goal of this paper is to design a weak disturbance decoupling controller for HOFA nonlinear systems to guarantee the stability and desired output tracking performance of the closed‐loop system and to attenuate the impact of disturbances.