Single Ordinary Differential Equation Research Articles

Nussbaum-Based Adaptive Neural Networks Tracking Control for Nonlinear PDE-ODE Systems Subject to Deception Attacks.

In this article, the novel adaptive neural networks (NNs) tracking control scheme is presented for nonlinear partial differential equation (PDE)-ordinary differential equation (ODE) coupled systems subject to deception attacks. Because of the special infinite-dimensional characteristics of PDE subsystem and the strong coupling of PDE-ODE systems, it is more difficult to achieve the tracking control for coupled systems than single ODE system under the circumstance of deception attacks, which result in the states and outputs of both PDE and ODE subsystems unavailable by injecting false information into sensors and actuators. For efficient design of the controllers to realize the tracking performance, a new coordinate transformation is developed under the backstepping method, and the PDE subsystem is transformed into a new form. In addition, the effect of the unknown control gains and the uncertain nonlinearities caused by attacks are alleviated by introducing the Nussbaum technology and NNs. The proposed tracking control scheme can guarantee that all signals in the coupled systems are bounded and the good tracking performance can be achieved, despite both sensors and actuators of the studied systems suffering from attacks. Finally, a simulation example is given to verify the effectiveness of the proposed control method.

Application of Lie symmetry to find an analytical solution for reactor point kinetics equation with prompt jump approximation and power feedback

The reactor Point Kinetics Equations (PKE) are simpler zero-dimensional approximation to space dependent dynamical models of nuclear reactor core, that are accurate enough to describe transients in small to medium size fast reactor cores. Even these simplified equations can be solved only by numerical methods, except in a very few restrictive cases, where they are amenable to analytical solution. Symmetry methods using Lie's point symmetry have been shown to be a systematic and powerful tool to solve any given ordinary or partial differential equation. An approximation of the PKE, known as Prompt Jump Approximation (PJA) converts the coupled system of first order ODEs with one delayed-neutron precursor group and power feedback, into a single first order nonlinear ordinary differential equation. In this study, we demonstrate an application of Lie symmetry method for solving the point kinetics equation under PJA. The analytical solution obtained is compared with benchmark numerical solution of PKE with PJA.

Machine learning for the identification of phase transitions in interacting agent-based systems: A Desai-Zwanzig example.

Deriving closed-form analytical expressions for reduced-order models, and judiciously choosing the closures leading to them, has long been the strategy of choice for studying phase- and noise-induced transitions for agent-based models (ABMs). In this paper, we propose a data-driven framework that pinpoints phase transitions for an ABM-the Desai-Zwanzig model-in its mean-field limit, using a smaller number of variables than traditional closed-form models. To this end, we use the manifold learning algorithm Diffusion Maps to identify a parsimonious set of data-driven latent variables, and we show that they are in one-to-one correspondence with the expected theoretical order parameter of the ABM. We then utilize a deep learning framework to obtain a conformal reparametrization of the data-driven coordinates that facilitates, in our example, the identification of a single parameter-dependent ordinary differential equation(ODE) in these coordinates. We identify this ODE through a residual neural network inspired by a numerical integration scheme (forward Euler). We then use the identified ODE-enabled through an odd symmetry transformation-to construct the bifurcation diagram exhibiting the phase transition.

Mathematical modelling of impurity deposition during evaporation of dirty liquid in a porous material

When a contaminated liquid evaporates from within a porous material, the impurities or dirt accumulate and deposit within the pore space. This occurs during the cleaning of filters and fouling of textiles, and is related to the ‘coffee-ring’ problem. To investigate how and where dirt is deposited in the pore space, we present a model for the motion of an evaporation front through a porous material, and the related accumulation, transport, and deposition of dirt, assuming that the liquid remains stationary. For physically relevant parameters, vapour transport out of the porous material is quasi-steady and we derive a single ordinary differential equation describing the motion of the evaporation front in time. Model solutions exhibit spatially non-uniform profiles of the deposited dirt-layer thickness through the porous material. The dirt accumulation and evaporation problems are coupled: deposited dirt hinders vapour transport through the porous material, slowing the evaporation. We identify two scenarios in which the porous material becomes clogged with dirt. Accumulation of suspended dirt at the evaporating interface along with slow dirt diffusion results in the deposited dirt layers clogging the pores at the evaporating interface, halting the drying and trapping liquid in the porous material. Alternatively, slow dirt deposition results in the suspended dirt being pushed far into the porous material by the evaporation, eventually leaving only dirt (with no liquid) in the pore space. We investigate the dynamics of both clogging scenarios, characterising the parameter regimes for which each occurs. Both clogging scenarios must be avoided in practice since they may be detrimental to future filter efficacy or textile breathability.

Water Temperature Model: A Case Study in Alagoinhas-BA

In recent years, climate change has generated disturbances at both local and global scales, directly impacting the water temperature of rivers. Temperature fluctuations have become a common occurrence in river water, where after experiencing heating, the return to its natural temperature is gradual. Consequently, temperature changes affect various internal processes within rivers, including nutrient consumption, food availability, and dissolved oxygen concentration. These alterations subsequently impact the growth and distribution of aquatic organisms. This article focuses on the physical variables derived from a model characterized by limited computational complexity. The final structure of the model consists of a single ordinary differential equation, which is linearly dependent on air and water temperature, as well as flow rate. Additionally, to handle data sequences such as time series more efficiently than traditional Recurrent Neural Networks (RNNs), we employ a Long Short-Term Memory (LSTM) neural network.

On elementary four-wave interactions in dispersive media

The cubic interactions in a discrete system of four weakly nonlinear waves propagating in a conservative dispersive medium are studied. By reducing the problem to a single ordinary differential equation governing the motion of a classical particle in a quartic potential, the complete explicit branches of solutions are presented, either steady, periodic, breather or pump, thus recovering or generalizing some already published results in hydrodynamics, nonlinear optics and plasma physics, and presenting some new ones. Various stability criteria are also formulated for steady equilibria. Theory is applied to deep-water gravity waves for which models of isolated quartets are described, including bidirectional standing waves and quadri-directional travelling waves, steady or not, resonant or not.

An explicit expression of generating function for one-loop tensor reduction

This work introduces an explicit expression for the generating function for the reduction of an n-gon to an (n – k)-gon. A novel recursive relation of generating function is formulated based on Feynman Parametrization in projective space, involving a single ordinary differential equation. The explicit formulation of generating functions provides crucial insights into the complex analytic structure inherent in loop amplitudes.

On the existence and uniqueness of a positive solution to a boundary value problem for a nonlinear ordinary differential equation of 4n order

The paper considers a two-point boundary value problem with homogeneous boundary conditions for a single nonlinear ordinary differential equation of order 4n. Using the well-known Krasnoselsky theorem on the expansion (compression) of a cone, sufficient conditions for the existence of a positive solution to the problem under consideration are obtained. To prove the uniqueness of a positive solution, the principle of compressed operators was invoked. In conclusion, an example is given that illustrates the fulfillment of the obtained sufficient conditions for the unique solvability of the problem under study.

Experimental study on the impulsively started motion of a close-to-neutral buoyancy freely decelerating sphere

An experimental study is presented that addresses the problem of a freely decelerating sphere in a still water tank. The diameter of the sphere was 0.04 m. Three different solid-to-fluid density ratios were considered: 0.78, 0.88, and 0.94. The submerged sphere was impulsively started upon being rammed by an actuator-mass system. Six initial velocities were considered: 0.91, 2.03, 2.54, 2.94, 3.29, and 3.78 m/s. The Reynolds number of the initial velocities based on the sphere diameter was 3.6 × 104, 8.1 × 104, 1.01 × 105, 1.17 × 105, 1.31 × 105, and 1.51 × 105 (subcritical). It was observed that both sphere dynamics and associated flow topology (identified via an optical system and a particle image velocimetry system, respectively) differed significantly from the case of an accelerating sphere. In the present case, a large vortex ring structure (both torus diameters of the order of the sphere's diameter) formed and attached to the sphere surface. This vortex ring followed the sphere motion all the way down the falling trajectory. From the data reduction standpoint, it was found that a suitably defined dimensionless acceleration parameter allowed for collapsing the kinematics variables of the sphere trajectory, namely, position, velocity, and acceleration, into a single ordinary differential equation.

Lie-group method solutions for a viscous flow in a dilating-squeezing permeable channel with velocity slip

The investigation focuses on the effects of wall velocity slip on the solution of a viscous, laminar, incompressible channel flow subjected to small-scaled contraction and expansion of the weakly permeable walls. The study of such flow systems is often contextual for fluid transport in biological organisms. In the considered flow configuration, the vertically moving porous walls enable the fluid to enter or exit with a constant rate. The tangential slip velocity of the flow at the porous walls is modeled with the Navier slip boundary condition. The flow dynamics inside the channel is governed by the full Navier–Stokes equations. The Lie symmetry analysis and the invariant method are adopted to reduce the number of independent variables in the system of governing equations. Consequently, a single fourth-order ordinary differential equation is obtained, which is solved analytically by the double perturbation method and the variation of iteration method. The solutions are compared for different arrangements. Furthermore, the approximated analytical solutions are likened to the numerical solutions obtained from a fourth-order Runge–Kutta solver embedding the Shooting method to check the accuracy. It is observed that the boundary layers are formed, and the flow rapidly turns near the walls, when suction and wall contraction coexist. Alternatively, if injection and wall expansion are paired, the flow adjacent to the walls is delayed. The existence of wall velocity slip advances the near-wall velocity and cuts down the speed of centerline velocity. It results in a change in the volumetric flow rate and shear rate. The overall pressure is also varied by higher wall velocity slip. The results are explored for different values of the permeation Reynold number and the dimensionless wall dilation rate to capture all possible impacts of the flow parameters. The current analysis rectifies the existing errors in the work of Boutros et al. [“Lie-group method solution for two-dimensional viscous flow between slowly expanding or contracting walls with weak permeability,”Appl. Math. Model. 31(6), 1092–1108 (2007)] with the no-slip boundary condition and discusses the overall influences of slip boundary condition on the Lie symmetry solution of the flow system.

Phenomenological model of superconducting optoelectronic loop neurons

Superconducting optoelectronic loop neurons are a class of circuits potentially conducive to networks for large-scale artificial cognition. These circuits employ superconducting components including single-photon detectors, Josephson junctions, and transformers to achieve neuromorphic functions. To date, all simulations of loop neurons have used first-principles circuit analysis to model the behavior of synapses, dendrites, and neurons. These circuit models are computationally inefficient and leave opaque the relationship between loop neurons and other complex systems. Here we introduce a modeling framework that captures the behavior of the relevant synaptic, dendritic, and neuronal circuits at a phenomenological level without resorting to full circuit equations. Within this compact model, each dendrite is discovered to obey a single nonlinear leaky-integrator ordinary differential equation, while a neuron is modeled as a dendrite with a thresholding element and an additional feedback mechanism for establishing a refractory period. A synapse is modeled as a single-photon detector coupled to a dendrite, where the response of the single-photon detector follows a closed-form expression. We quantify the accuracy of the phenomenological model relative to circuit simulations and find that the approach reduces computational time by a factor of ten thousand while maintaining an accuracy of one part in ten thousand. We demonstrate the use of the model with several basic examples. The net increase in computational efficiency enables future simulation of large networks, while the formulation provides a connection to a large body of work in applied mathematics, computational neuroscience, and physical systems such as spin glasses.

The novel PINN/gPINN-based deep learning schemes for non-Fickian coupled diffusion-elastic wave propagation analysis

This paper presents a cutting-edge approach inspired by the physics-informed neural network (PINN) to solve the coupled non-Fickian diffusion-elasticity system. This is the first time that the PINN framework has been applied to non-Fick diffusion-elasticity in order to investigate the transient behavior of the concentration and displacement fields. This problem pertains to a strip subjected to a time-dependent shock loading. In light of recent developments in gradient-enhanced PINN (gPINN), there is now considerable concern regarding its applicability to systems of coupled partial differential equations (PDEs). As of yet, this methodology has only been applied to a single ordinary differential equation (ODE) or a single PDE. gPINN's performance is also examined for coupled PDEs in the current problem. Moreover, some analyses are performed to analyze the impact of parameter selection on the outputs. Successful implementations of the PINN approach without the need for mesh generation or the limitations of earlier numerical approaches confirm its capabilities in solving complex systems of coupled PDEs.

Simplifying population balance models to promote broader use in industry

The approach outlined in this paper takes advantage of the existence of stationary states and similarity solutions to reduce a one-dimensional population balance model with particle volume as the internal dimension from a partial integro-differential equation to a single ordinary differential equation for the evolution of the mean size. The mechanisms of accretional growth, collisional growth and breakage, either individually or in all possible combinations, are considered. Power-law rate kernels are employed and a self-similar fragment distribution is assumed for the breakage term. When expressed in terms of the evolution of the scaled number-mean particle volume, the model parameters consist of kernel orders and the characteristic times associated with each mechanism. Equations for the scaled moments of the stationary states and similarity solutions are exhibited for all possible mechanism combinations as are equations for the trajectories of scaled number-mean particle volume. Stability of the stationary states and similarity solutions is addressed in an appendix. It should be clear that when this level of simplification is possible, the subject of population balance modeling becomes considerably easier to teach while its use becomes much easier to promote in industry. The use of this simplified approach for extraction of model parameters from experiment is demonstrated in an example, followed by a brief discussion of using the results for scale up. To close out the paper, the same approach, but now in two internal dimensions (particle volume and surface area), is used to demonstrate the capability of the model to unify a set of operating conditions for carbon black manufacture across multiple grades and two different types of synthesis reactors. This is done by extracting the characteristic coalescence time at multiple temperatures and correlating the results in an Arrhenius plot. The model matches known values of primary particle size, and predicts a key indicator of the product’s fractal aggregate structure, namely the number of primary particles per aggregate. The model results are then pushed to yield insight into the nature of the possible collision and coalescence mechanisms.

Profile likelihood-based parameter and predictive interval analysis guides model choice for ecological population dynamics

Calibrating mathematical models to describe ecological data provides important insight via parameter estimation that is not possible from analysing data alone. When we undertake a mathematical modelling study of ecological or biological data, we must deal with the trade-off between data availability and model complexity. Dealing with the nexus between data availability and model complexity is an ongoing challenge in mathematical modelling, particularly in mathematical biology and mathematical ecology where data collection is often not standardised, and more broad questions about model selection remain relatively open. Therefore, choosing an appropriate model almost always requires case-by-case consideration. In this work we present a straightforward approach to quantitatively explore this trade-off using a case study exploring mathematical models of coral reef regrowth after some ecological disturbance, such as damage caused by a tropical cyclone. In particular, we compare a simple single species ordinary differential equation (ODE) model approach with a more complicated two-species coupled ODE model. Univariate profile likelihood analysis suggests that the both models are practically identifiable. To provide additional insight we construct and compare approximate prediction intervals using a new parameter-wise prediction approximation, confirming both the simple and complex models perform similarly with regard to making predictions. Our approximate parameter-wise prediction interval analysis provides explicit information about how each parameter affects the predictions of each model. Comparing our approximate prediction intervals with a more rigorous and computationally expensive evaluation of the full likelihood shows that the new approximations are reasonable in this case. All algorithms and software to support this work are freely available as jupyter notebooks on GitHub so that they can be adapted to deal with any other ODE-based models.

Analysis of Mixed Convection on Two-Phase Nanofluid Flow Past a Vertical Plate in Brinkman-Extended Darcy Porous Medium with Nield Conditions

The rapid advancement in technology in recent years has shown that nanofluids are very vital to further development in science and technology. Moreover, many industrial specifications cannot be met by allowing natural convection only, hence the need to incorporate forced convection and natural convection into a single flow regime. The research aims to quantify the mixed convective two-phase flow past a vertical permeable surface in a Brinkman-Extended Darcy porous medium (BEDPM) induced by nanofluid, with heat and mass transfer. In addition, the Nield condition is also incorporated. The model of the problem was initially constructed in the vital form of leading governing equations (LGEs). These LGEs are specifically called partial differential equations (PDEs) (because of two or more independent variables) which were later converted into a set of the single independent variable of ordinary differential equations (ODEs) by implementing the similarity transformations. The set of single independent ODEs was numerically solved via the boundary value problem of fourth-order (bvp4c) technique. The bvp4c is one of the most frequently recommended built-in MATLAB subroutines based on the three-stage Labatto formula. The impact of several physically embedded influential parameters on the fluid flow, along with mass and thermal properties of the nanofluid in a Brinkman-Extended Darcy porous medium for the cases of buoyancy assisting flow (BAF) and buoyancy opposing flow (BOF), were investigated and argued. The numerical outcomes clarify that the porosity parameter reduces the velocity, whereas the concentration and the temperature enhance in the case of the buoyancy assisting and buoyancy opposing flows. In addition, the wall drag force elevates for the larger value of the dimensionless permeability parameter K1 and the buoyancy ratio parameter N, while it declines for the modified porosity parameter ε1.

Discontinuous stationary solutions to certain reaction-diffusion systems

Systems consisting of a single ordinary differential equation coupled with one reaction-diffusion equation in a bounded domain and with the Neumann boundary conditions are studied in the case of particular nonlinearities from the Brusselator model, the Gray-Scott model, the Oregonator model and a certain predator-prey model. It is shown that the considered systems have the both smooth and discontinuous stationary solutions, however, only discontinuous ones can be stable.

Continuum multiscale modeling of absorption processes in micro- and nanocatalysts

In this paper, we propose a novel, semi-analytic approach for the two-scale, computational modeling of concentration transport in packed bed reactors. Within the reactor, catalytic pellets are stacked, which alter the concentration evolution. Firstly, the considered experimental setup is discussed and a naive one-scale approach is presented. This one-scale model motivates, due to unphysical fitted values, to enrich the computational procedure by another scale. The computations on the second scale, here referred to as microscale, are based on a proper investigation of the diffusion process in the catalytic pellets from which, after continuum-consistent considerations, a sink term for the macroscopic advection–diffusion–reaction process can be identified. For the special case of a spherical catalyst pellet, the parabolic partial differential equation at the microscale can be reduced to a single ordinary differential equation in time through a semi-analytic approach. After the presentation of our model, we show results for its calibration against the macroscopic response of a simple standard mass transport experiment. Based thereon, the effective diffusion parameters of the catalyst pellets can be identified.

Numerical method for determining the coefficient of hydraulic resistance two-phase flow in a gas lift well

The process of stationary movement of a gas-liquid mixture in the lifting pipe of a gas-lift well is considered. To describe this two-phase flow, a mathematical model is proposed that includes the equation of flow motion and the continuity equation for each phase. The presented model is transformed to a single nonlinear ordinary differential equation with respect to pressure. Within the framework of the obtained model, the task is set to determine the hydraulic resistance coefficient of a two-phase flow according to an additionally specified condition with respect to pressure. An additional condition, presented in the form of a nonlinear algebraic equation, is transformed into an ordinary differential equation with respect to an unknown coefficient of hydraulic resistance by applying the method of differentiation by parameter. The solution of the resulting Cauchy problem is determined by the finite difference method. Based on the proposed computational algorithm, numerical experiments were carried out for model data. Keywords: gas lift; two-phase flow; hydraulic resistance coefficient; parameter differentiation method; finite difference method.

Similarity solutions of Prandtl mixing length modelled two dimensional turbulent boundary layer equations

The exact similarity solutions of two dimensional laminar boundary layer were obtained by Blasius in 1908, however, for two dimensional turbulent boundary layers, no Blasius type similarity solutions (special exact solutions) have ever been found. In the light of Blasius’ pioneer works, we extend Blasius similarity transformation to the two dimensional turbulent boundary layers, and for a special case of flow modelled by Prandtl mixing-length, we successfully transform the two dimensional turbulent boundary layers partial differential equations into a single ordinary differential equation. The ordinary differential equation is numerically solved and some useful quantities are produced. For numerical calculations, a complete Maple code is provided.

A Damping-Tunable Snap System: From Dissipative Hyperchaos to Conservative Chaos.

A hyperjerk system described by a single fourth-order ordinary differential equation of the form has been referred to as a snap system. A damping-tunable snap system, capable of an adjustable attractor dimension () ranging from dissipative hyperchaos () to conservative chaos (), is presented for the first time, in particular not only in a snap system, but also in a four-dimensional (4D) system. Such an attractor dimension is adjustable by nonlinear damping of a relatively simple quadratic function of the form , easily tunable by a single parameter A. The proposed snap system is practically implemented and verified by the reconfigurable circuits of field programmable analog arrays (FPAAs).