Complex Ordinary Differential Equations Research Articles

Modeling and Path-Following Control for Dubins Dynamics in Complex State Space

A path-following control design in the complex state space for the Dubins vehicle is proposed. The vehicle dynamics are modeled as a bilinear system of two complex ordinary differential equations with a scalar complex control input. The developed control solution represents a feedback linearizing proportional–integral–derivative servomechanism, with real valued control gains applied to complex valued tracking error components. The design is performed within the model reference control framework. It is applied to solving path-following and waypoint routing problems for curves with bounded curvature and vehicles with bounded speeds. The path-following controller regulates the system turn rate and the reference model speed. The vehicle speed dynamics are independently maintained as desired. In that regard, the developed control policy forces the system position to track a virtual target point defined by the reference model. Verifiable sufficient conditions are formulated to enable steering the Dubins vehicle along the designated routes in the presence of unknown bounded environmental disturbances, such as wind and gust. Application examples of the developed path-following control solution to waypoint routing guidance for unmanned aerial platforms are discussed.

On the feed-forward neural network for analyzing pantograph equations

Ordinary differential equations (ODEs) are fundamental tools for modeling and understanding a wide range of chemistry, physics, and biological phenomena. However, solving complex ODEs often presents significant challenges, necessitating advanced numerical approaches beyond traditional analytical techniques. Thus, a novel machine learning (ML)-based method for solving and analyzing ODEs is proposed in the current investigation. In this study, we utilize a feed-forward neural network (FNN) with five fully connected layers trained on data samples generated from the exact solutions of specific ODEs. To show the efficacy of our suggested method, we will conduct a thorough evaluation by comparing the anticipated solutions of the FNN with the exact solutions for some ODEs. Furthermore, we analyze the absolute error and present the loss functions for some ODE examples, providing valuable insights into the model’s performance and potential areas for further development.

Domain walls and vector solitons in the coupled nonlinear Schrödinger equation

We outline a program to classify domain walls (DWs) and vector solitons in the 1D two-component coupled nonlinear Schrödinger (CNLS) equation without restricting the signs or magnitudes of any coefficients. The CNLS equation is reduced first to a complex ordinary differential equation (ODE), and then to a real ODE after imposing a restriction. In the real ODE, we identify four possible equilibria including ZZ, ZN, NZ, and NN, with Z(N) denoting a zero (nonzero) value in a component, and analyze their spatial stability. We identify two types of DWs including asymmetric DWs between ZZ and NN and symmetric DWs between ZN and NZ. We identify three codimension-1 mechanisms for generating vector solitons in the real ODE including heteroclinic cycles, local bifurcations, and exact solutions. Heteroclinic cycles are formed by assembling two DWs back-to-back and generate extended bright-bright (BB), dark-dark (DD), and dark-bright (DB) solitons. Local bifurcations include the Turing (Hamiltonian–Hopf) bifurcation that generates Turing solitons with oscillatory tails and the pitchfork bifurcation that generates DB, bright-antidark, DD, and dark-antidark solitons with monotonic tails. Exact solutions include scalar bright and dark solitons with vector amplitudes. Any codimension-1 real vector soliton can be numerically continued into a codimension-0 family. Complex vector solitons have two more parameters: a dark or antidark component can be numerically continued in the wavenumber, while a bright component can be multiplied by a constant phase factor. We introduce a numerical continuation method to find real and complex vector solitons and show that DWs and DB solitons in the immiscible regime can be related by varying bifurcation parameters. We show that collisions between two DB solitons with a nonzero phase difference in their bright components typically feature a mass exchange that changes the frequencies and phases of the two bright components and the two soliton velocities.

Optimizing ODE solutions: application of Nelder-Mead algorithm for solving mixed boundary value problems

This abstract introduces a novel application of the Nelder-Mead algorithm in optimizing solutions to Ordinary Differential Equations (ODEs) with derivative boundary conditions. The study presents a refined methodology that leverages the Nelder-Mead algorithm’s adaptability to tackle ODEs characterized by intricate derivative constraints. With the aim of enhancing the toolkit for solving complex ODEs, the objective of this research is to showcase the algorithm’s efficacy in optimizing solutions under derivative boundary conditions. The methodology involves adapting the Nelder-Mead algorithm to navigate the param-eter space while satisfying both the ODEs and their derivative constraints. Experimental results demonstrate the algorithm’s capability to identify solutions that meet these stringent requirements, marking a significant advancement in addressing ODEs with derivative boundary conditions. The study concludes by emphasizing the algorithm’s potential for advancing ODE-solving techniques, particularly in scenarios where gradient-based methods struggle, thus widening the scope of applications across various scientific and engineering domains.

Insight into acoustic wave-driven gas bubble dynamics in Williamson's fluid

This study's motivation on the behavior of a gas bubble and surrounding fluid during the collapse phase of an acoustic wave leads to the emission of light, which can be modelled using the Rayleigh-Plesset equation. The significant contribution to the major addition to the understanding of gas bubble dynamics in viscoelastic fluids. The results of the study have the potential to impact a wide range of applications, from sonar and underwater acoustic propagation to biomedical applications and industrial operations. Understanding the dynamics of acoustic wave-driven gas bubble oscillations in non-Newtonian fluids is crucial for various applications in medical imaging, therapy, and microfluidics. The aim of this study is to discover the conditions under which these bubbles can undergo sonoluminescence, the emission of light due to the collapse of bubbles under severe sonic pressure. The complex ordinary differential equation is solved numerically, and graphs are generated to evaluates various parameters such as velocity, radius, and pressure. A comparison with the viscous case is also presented to assess the influence of Newtonian and non-Newtonian fluids. A program designed by Mathematica software (13.0) “parametric NDSolve package” is used to solve the equations. The numerical solution supports the boundary conditions, indicating stability and reliability. Notably, the study identifies distinct fluctuations in the phase and stable characteristics associated with non-Newtonian effects. Interestingly, certain fluid parameters lead to discrete group modulation of radial excursions, revealing a novel and previously unknown phenomenon. It is concluded that Higher Reynolds numbers result in turbulent flow, which causes the bubble to enlarge and increase its volume. Furthermore the pressure gradient is affected by magnetic drag force, and as size of bubble increases, required pressure gradient to counteract the drag force also increases. As the magnetic field intensity increases, the velocity of the bubble decreases. Also, considering the implications of these results, the study explores their relevance to medical ultrasonography applications.

Analysis of bidirectional flow of Williamson micropolar fluid in porous medium with activation energy and thermal radiation

The prime objective of the proposed model is to explore the impacts of viscous dissipation and Ohmic heating on the bidirectional flow of micropolar and Williamson fluids in porous medium over a stretching surface. The velocity slips and heat convection are assessed at the surface as well. The concernment of chemical reaction, thermophoresis, and Brownian motion repercussions is also considered. More complex nonlinear ordinary differential equations arise when the variables are combined in a suitable manner to represent the mathematical model of the problem. The Galerkin weighted residual approach is used to solve non - linear differential equations’ system that has been developed. Flow patterns are presented by utilizing multiple plots and assessing the influence of various emerging factors. Tables are frequently utilized to characterize the rates of mass and heat transfer. The current findings have yielded several important crucial findings, including the fact that skin friction improves with improvements in the porosity, Williamson fluid, and micropolar fluid parameters. The Nusselt number decays with increments in Eckert numbers and heat source parameters. The velocity field speeds up in the x-direction as the slip parameter enhances, but it lessens in both directions with rise in the porosity parameter and the Williamson fluid parameter. The microrotation velocity profiles in both directions experience a significant reduction as the microrotation slip parameter increases. Also, it has been noted that the nano-liquid concentration profile becomes lesser while the temperature distribution becomes higher for higher values of the Eckert numbers. The thermal profile accelerates as the heat source parameter increases.

On the Analysis of a Fractional Tuberculosis Model with the Effect of an Imperfect Vaccine and Exogenous Factors under the Mittag–Leffler Kernel

This research study aims to investigate the effects of vaccination on reducing disease burden by analyzing a complex nonlinear ordinary differential equation system. The study focuses on five distinct sub-classes within the system to comprehensively explore the impact of vaccination. Specifically, the mathematical model employed in this investigation is a fractional representation of tuberculosis, utilizing the Atangana–Baleanu fractional derivative in the Caputo sense. The validity of the proposed model is established through a rigorous qualitative analysis. The existence and uniqueness of the solution are rigorously determined by applying the fundamental theorems of the fixed point approach. The stability analysis of the model is conducted using the Ulam–Hyers approach. Additionally, the study employs the widely recognized iterative Adams–Bashforth technique to obtain an approximate solution for the suggested model. The numerical simulation of the tuberculosis model is comprehensively discussed, with a particular focus on the assumptions made regarding vaccination. The model assumes that only a limited portion of the population is vaccinated at a steady rate, and the efficacy of the vaccine is a critical factor in reducing disease burden. The findings of the study indicate that the proposed model can effectively assess the impact of vaccination on mitigating the spread of tuberculosis. Furthermore, the numerical simulation underscores the significance of vaccination as an effective control measure against tuberculosis.

Direct 0D-3D coupling of a lattice Boltzmann methodology for fluid-structure aortic flow simulations.

This work introduces a numerical approach and implementation for the direct coupling of arbitrary complex ordinary differential equation- (ODE-)governed zero-dimensional (0D) boundary conditions to three-dimensional (3D) lattice Boltzmann-based fluid-structure systems for hemodynamics studies. In particular, a most complex configuration is treated by considering a dynamic left ventricle- (LV-)elastance heart model which is governed by (and applied as) a nonlinear, non-stationary hybrid ODE-Dirichlet system. Other ODE-based boundary conditions, such as lumped parameter Windkessel models for truncated vasculature, are also considered. Performance studies of the complete 0D-3D solver, including its treatment of the lattice Boltzmann fluid equations and elastodynamics equations as well as their interactions, is conducted through a variety of benchmark and convergence studies that demonstrate the ability of the coupled 0D-3D methodology in generating physiological pressure and flow waveforms-ultimately enabling the exploration of various physical and physiological parameters for hemodynamics studies of the coupled LV-arterial system. The methods proposed in this paper can be easily applied to other ODE-based boundary conditions as well as to other fluid problems that are modeled by 3D lattice Boltzmann equations and that require direct coupling of dynamic 0D boundary conditions.

Liouville operators over the Hardy space

The role of Liouville operators in the study of dynamical systems through the use of occupation measures has been an active area of research in control theory over the past decade. This manuscript investigates Liouville operators over the Hardy space, which encode complex ordinary differential equations in an operator over a reproducing kernel Hilbert space.

A new hierarchical approach to multi-level model abstraction for simplifying ODE models of biological networks and a case study: The G1/S Checkpoint/DNA damage signalling pathways of mammalian cell cycle

Model reduction is an important topic in studies of biological systems. By reducing the complexity of large models through multi-level models while keeping the essence (biological meaning) of the model, model reduction can help answer many important questions about these systems. In this paper, we present a new reduction method based on hierarchical representation and a lumping approach. We used G1/S checkpoint pathway represented in Ordinary Differential Equations (ODE) in Iwamoto et al. (2011) as a case study to present this reduction method. The approach consists of two parts; the first part represents the biological network as a hierarchy (multiple levels) based on protein binding relations, which allowed us to model the biological network at different levels of abstraction. The second part applies different levels (level 1, 2 and 3) of lumping the species together depending on the level of the hierarchy, resulting in a reduced and transformed model for each level. The model at each level is a representation of the whole system and can address questions pertinent to that level. We develop and simulate reduced models for levels-1, 2 and 3 of lumping for the G1/S checkpoint pathway and evaluate the biological plausibility of the proposed method by comparing the results with the original ODE model of Iwamoto et al. (2011). The results for continuous dynamics of the G1/S checkpoint pathway with or without DNA-damage for reduced models of level- 1, 2 and 3 of lumping are in very good agreement and consistent with the original model results and with biological findings. Therefore, the reduced models (level-1, 2 and 3) can be used to study cell cycle progression in G1 and the effects of DNA damage on it. It is suitable for reducing complex ODE biological network models while retaining accurate continuous dynamics of the system. The 3 levels of the reduced models respectively achieved 20%, 26% and 31% reduction of the base model. Moreover, the reduced model is more efficient to run (30%, 44% and 52% time reduction for the three levels) and generate solutions than the original ODE model. Simplification of complex mathematical models is possible and the proposed reduction method has the potential to make an impact across many fields of biomedical research.

General Solutions of Some Complex Third-order Differential Equations

According to the Nevanlinna theory, many researches have undertaken the behaviors of meromorphic solutions of complex ordinary differential equations (ODEs). Most of these researches have concentrated on the value distribution and growth of meromorphic solutions of ODEs. However, the existence of a meromorphic general solution is often used as a way to identify equations that are integrable. Especially, the existence of global meromorphic solutions of differential equation with entire coefficient can be settled, resulting in the characterization of Schwarzian derivatives. This is concerning with the linearly independent solutions of linear differential equations . The purpose of this present paper is to find explicit solutions of differential equation in terms of finite combinations of known functions, that is, we use local series methods and reduction of order to solve all linearly independent solutions of some third-order ODEs with entire coefficient in the neighborhood of z0.

Determination of the blow up point for complex nonautonomous ODE with cubic nonlinearity

We present a method for determination of the blow up point for a complex nonautonomous ordinary differential equation with a cubic nonlinearity. This equation describes stationary nonlinear modes for two important NLS-based models: (i) the Gross–Pitaevskii equation with complex potential and repulsive nonlinearity and (ii) the Lugiato–Lefever equation in the case of normal dispersion. We derive and justify an asymptotic expansion in the vicinity of the blow up point. This expansion is employed for the construction of a numeric procedure for the computation of the coordinate of the blow up point. We illustrate applications of the proposed procedure by two examples, where exact analytical solutions are available. It is shown that it allows one to find the blow up point with high accuracy. The method may be efficiently used for the search of soliton solutions for vector and scalar NLS-type equations within the strategy of ’filtering out’ of blow up solutions (Alfimov et al., Physica D, 394, 39 (2019)).

Magnetohydrodynamic (MHD) boundary layer heat and mass transfer characteristics of nanofluid over a vertical cone under convective boundary condition

This paper investigates the boundary layer flow, heat and mass transfer characteristics over a vertical cone filled with nanofluid saturated porous medium with the influence of magnetic field, thermal radiation and first order chemical reaction subject to the convective boundary condition. Similarity transformation technique is used for the purpose of converting non-linear partial differential equations into the system of complex ordinary differential equations. The computational Finite element method has been employed to solve the flow, heat and mass transfer equations together with boundary conditions. The impact of various pertinent parameters on hydrodynamic, thermal and solutal boundary layers is investigated and the results are displayed graphically. Furthermore, the values of local skin-friction coefficient, rate of temperature and rate of concentration is also calculated and the results are presented graphically. The comparisons with previously published work is made and found good agreement. The thickness of thermal boundary layer is increased with increase in the values of Brownian motion parameter (Nb) and thermophoresis parameter (Nt).

A dynamic Bayesian Markov model for health economic evaluations of interventions in infectious disease

BackgroundHealth economic evaluations of interventions in infectious disease are commonly based on the predictions of ordinary differential equation (ODE) systems or Markov models (MMs). Standard MMs are static, whereas ODE systems are usually dynamic and account for herd immunity which is crucial to prevent overestimation of infection prevalence. Complex ODE systems including distributions on model parameters are computationally intensive. Thus, mainly ODE-based models including fixed parameter values are presented in the literature. These do not account for parameter uncertainty. As a consequence, probabilistic sensitivity analysis (PSA), a crucial component of health economic evaluations, cannot be conducted straightforwardly.MethodsWe present a dynamic MM under a Bayesian framework. We extend a static MM by incorporating the force of infection into the state allocation algorithm. The corresponding output is based on dynamic changes in prevalence and thus accounts for herd immunity. In contrast to deterministic ODE-based models, PSA can be conducted straightforwardly. We introduce a case study of a fictional sexually transmitted infection and compare our dynamic Bayesian MM to a deterministic and a Bayesian ODE system. The models are calibrated to simulated time series data.ResultsBy means of the case study, we show that our methodology produces outcome which is comparable to the “gold standard” of the Bayesian ODE system.ConclusionsIn contrast to ODE systems in the literature, the dynamic MM includes distributions on all model parameters at manageable computational effort (including calibration). The run time of the Bayesian ODE system is 15 times longer.

Combination of singularly perturbed vector field method and method of directly defining the inverse mapping applied to complex ODE system prostate cancer model

ABSTRACTWe propose a new method to solve a system of complex ordinary differential equations (ODEs) with hidden hierarchy. Given a complex system of the ODE, the hierarchy of the system is generally hidden. Once we reveal the hierarchy of the system, the system can be reduced into subsystems called slow and fast subsystems. This division of slow and fast subsystems reduces the analysis and hence reduces the computation time, which can be expensive. In our new method, we first apply the singularly perturbed vector field method that is the global quasi-linearization method. This method exposes the hierarchy of a given complex system. Subsequently, we apply a version of the homotopy analysis method called the method of directly defining the inverse mapping. We applied our new method to the immunotherapy of advanced prostate cancer.

EXCEL BASED SETTLEMENT OF BEAMS ON ELASTIC FOUNDATIONS WITH FREE-ENDS AND ARBITRARY LOADING

The deflection of long footings placed on homogeneous and isotropic soils involves soil-structure differential equations models whose solution may not be possible for most practical problems. The analytical solution of beams on elastic foundation problem involves soil modulus of subgrade reaction and simplifying assumptions relative to applied loading. The exact solutions are available in relatively simple cases of loading, uniform cross sectional properties of the footing and constant soil modulus of subgrade reaction. Therefore, the Finite Difference Method (FDM) or Finite Element Method (FEM), are typically used to compute the deformation of beams with variable loading and geometry resting on elastic foundations with variable modulus of subgrade reaction. The finite differences method was used to solve this problem for long beams with arbitrary loading and constant cross-sections using an Excel Workbook to compute beam deflections providing both numerical and graphical output. The foundation is modeled as a long beam with free ends and a constant modulus of subgrade reaction. The proposed solution presents an efficient method involving a complex ordinary differential equation model for beams on elastic foundations encountered in engineering practice.

Simplified variational iteration method for solving ordinary differential equations and eigenvalue problems

A simplified variational iteration method is proposed to solve high-order homogeneous or nonhomogeneous linear ordinary differential equation and ordinary differential equation eigenvalue problems more efficiently and conveniently. The simplification includes two aspects: (1) explicitly deducing the general form of the differential equation for the identification of the general Lagrange multiplier while avoiding the complexity of variational calculations during identification and (2) simplifying the iterative expressions to reduce the computational work of each iteration. Three ordinary differential equations in mechanics are solved by this simplified variational iteration method, which proves that it is valid and more concise than traditional methods. To make the method more practical, it is suggested that some complicated analytical derivations be executed numerically, thereby achieving a simplified semi-analytical variational iteration method that can be easily implemented by computer programs. The method is then used to numerically solve two complex ordinary differential equation problems derived from the continuum analysis of tall building structures: a sixth-order nonhomogeneous ordinary differential equation with complex boundary conditions and a sixth-order ordinary differential equation eigenvalue problem. Numerical computer programs are developed for these two problems, and corresponding examples are provided to verify the accuracy and efficiency of the simplified variational iteration method in solving complex ordinary differential equation problems.

The construction of a two-dimensional reproducing kernel function and its application in a biomedical model

There are two major classes of cardiac tissue models: the ionic model and the FitzHugh-Nagumo model. During computer simulation, each model entails solving a system of complex ordinary differential equations and a partial differential equation with non-flux boundary conditions. The reproducing kernel method possesses significant applications in solving partial differential equations. The derivative of the reproducing kernel function is a wavelet function, which has local properties and sensitivities to singularity. Therefore, study on the application of reproducing kernel would be advantageous. Applying new mathematical theory to the numerical solution of the ventricular muscle model so as to improve its precision in comparison with other methods at present. A two-dimensional reproducing kernel function inspace is constructed and applied in computing the solution of two-dimensional cardiac tissue model by means of the difference method through time and the reproducing kernel method through space. Compared with other methods, this method holds several advantages such as high accuracy in computing solutions, insensitivity to different time steps and a slow propagation speed of error. It is suitable for disorderly scattered node systems without meshing, and can arbitrarily change the location and density of the solution on different time layers. The reproducing kernel method has higher solution accuracy and stability in the solutions of the two-dimensional cardiac tissue model.

PeTTSy: a computational tool for perturbation analysis of complex systems biology models.

BackgroundOver the last decade sensitivity analysis techniques have been shown to be very useful to analyse complex and high dimensional Systems Biology models. However, many of the currently available toolboxes have either used parameter sampling, been focused on a restricted set of model observables of interest, studied optimisation of a objective function, or have not dealt with multiple simultaneous model parameter changes where the changes can be permanent or temporary.ResultsHere we introduce our new, freely downloadable toolbox, PeTTSy (Perturbation Theory Toolbox for Systems). PeTTSy is a package for MATLAB which implements a wide array of techniques for the perturbation theory and sensitivity analysis of large and complex ordinary differential equation (ODE) based models. PeTTSy is a comprehensive modelling framework that introduces a number of new approaches and that fully addresses analysis of oscillatory systems. It examines sensitivity analysis of the models to perturbations of parameters, where the perturbation timing, strength, length and overall shape can be controlled by the user. This can be done in a system-global setting, namely, the user can determine how many parameters to perturb, by how much and for how long. PeTTSy also offers the user the ability to explore the effect of the parameter perturbations on many different types of outputs: period, phase (timing of peak) and model solutions. PeTTSy can be employed on a wide range of mathematical models including free-running and forced oscillators and signalling systems.To enable experimental optimisation using the Fisher Information Matrix it efficiently allows one to combine multiple variants of a model (i.e. a model with multiple experimental conditions) in order to determine the value of new experiments. It is especially useful in the analysis of large and complex models involving many variables and parameters.ConclusionsPeTTSy is a comprehensive tool for analysing large and complex models of regulatory and signalling systems. It allows for simulation and analysis of models under a variety of environmental conditions and for experimental optimisation of complex combined experiments. With its unique set of tools it makes a valuable addition to the current library of sensitivity analysis toolboxes. We believe that this software will be of great use to the wider biological, systems biology and modelling communities.Electronic supplementary materialThe online version of this article (doi:10.1186/s12859-016-0972-2) contains supplementary material, which is available to authorized users.

Sound Synthesis Based on Ordinary Differential Equations

Ordinary differential equations (ODEs) have been studied for centuries as a means to model complex dynamical processes from the real world. Nevertheless, their application to sound synthesis has not yet been fully exploited. In this article we present a systematic approach to sound synthesis based on first-order complex and real ODEs. Using simple time-dependent and nonlinear terms, we illustrate the mapping between ODE coefficients and physically meaningful control parameters such as pitch, pitch bend, decay rate, and attack time. We reveal the connection between nonlinear coupling terms and frequency modulation, and we discuss the implications of this scheme in connection with nonlinear synthesis. The ability to excite a first-order complex ODE with an external input signal is also examined; stochastic or impulsive signals that are physically or synthetically produced can be presented as input to the system, offering additional synthesis possibilities, such as those found in excitation/filter synthesis and filter-based modal synthesis.