System Of Nonlinear Ordinary Differential Equations Research Articles

The Hunter‐Saxton Equation: A Numerical Approach Using Collocation Method

In this study, we are going to present an overview on the Hunter‐Saxton equation which is a famous equation modelling waves in a massive director field of a nematic liquid crystal. The collocation finite element method is based on quintic B‐spline basis for obtaining numerical solutions of the equation. Using this method, after discretization, solution of the equation expressed as linear combination of shape functions and B‐spline basis. So, Hunter‐Saxton equation converted to nonlinear ordinary differential equation system. With the aid of the error norms and , some comparisons are presented between numeric and exact solutions for different step sizes. As a result, the authors observed that the method is a powerful, suitable and reliable numerical method for solving various kind of partial differential equations.

LASSIM-A network inference toolbox for genome-wide mechanistic modeling.

Recent technological advancements have made time-resolved, quantitative, multi-omics data available for many model systems, which could be integrated for systems pharmacokinetic use. Here, we present large-scale simulation modeling (LASSIM), which is a novel mathematical tool for performing large-scale inference using mechanistically defined ordinary differential equations (ODE) for gene regulatory networks (GRNs). LASSIM integrates structural knowledge about regulatory interactions and non-linear equations with multiple steady state and dynamic response expression datasets. The rationale behind LASSIM is that biological GRNs can be simplified using a limited subset of core genes that are assumed to regulate all other gene transcription events in the network. The LASSIM method is implemented as a general-purpose toolbox using the PyGMO Python package to make the most of multicore computers and high performance clusters, and is available at https://gitlab.com/Gustafsson-lab/lassim. As a method, LASSIM works in two steps, where it first infers a non-linear ODE system of the pre-specified core gene expression. Second, LASSIM in parallel optimizes the parameters that model the regulation of peripheral genes by core system genes. We showed the usefulness of this method by applying LASSIM to infer a large-scale non-linear model of naïve Th2 cell differentiation, made possible by integrating Th2 specific bindings, time-series together with six public and six novel siRNA-mediated knock-down experiments. ChIP-seq showed significant overlap for all tested transcription factors. Next, we performed novel time-series measurements of total T-cells during differentiation towards Th2 and verified that our LASSIM model could monitor those data significantly better than comparable models that used the same Th2 bindings. In summary, the LASSIM toolbox opens the door to a new type of model-based data analysis that combines the strengths of reliable mechanistic models with truly systems-level data. We demonstrate the power of this approach by inferring a mechanistically motivated, genome-wide model of the Th2 transcription regulatory system, which plays an important role in several immune related diseases.

Sampled‐data fuzzy control for non‐linear parabolic distributed parameter systems with control inputs missing

In this study, an H ∞ sampled-data fuzzy control problem is considered for non-linear parabolic partial differential equation (PPDE) systems with control inputs missing. It is assumed that the actuator to the plant is set to zero if the control inputs missing which occurs in a random way satisfies a Bernoulli distribution. With the aid of the modal decomposition technique, a non-linear ordinary differential equation (ODE) system is initially obtained to represent the dominant dynamics of the PPDE system. Subsequently, a Takagi–Sugeno fuzzy model is utilised to accurately describe the resulting non-linear ODE system. Then, using a novel time-dependent functional and the stochastic analysis technique, an H ∞ sampled-data fuzzy controller with the stochastic missing data via linear matrix inequalities is proposed to stabilise exponentially the non-linear PPDE system in the mean square sense and achieve an H ∞ performance for the derived non-linear ODE system. Finally, an example on a temperature cooling fin is given to verify the proposed design strategy.

SIMULACIÓN NUMERICA DEL PROCESO DE INFECCION TRATAMIENTO DEL VIH1 Y SU CONTROL BAJO LA ACCIÓN DE ANTIRRETROVIRALES

En este trabajo se considera la propuesta, desarrollo y simulación numérica de un sistema que modela el comportamiento del tratamiento del VIH-1, utilizando antirretrovirales. El sistema definido para tres variables dependientes de la variable t, denotadas por X(t)= (x1(t), x2(t), x3(t)) representan a la cantidad de Linfocitos T "Helpers" (CD4), cantidad de Linfocitos T Citotóxicos (CD8) y Carga Viral del proceso de infección del VIH-1 en cualquier instante de tiempo t, respectivamente. El sistema conformado por un conjunto de Ecuaciones Diferenciales Ordinarias No lineales, cuyo dominio existencial del sistema representa el tiempo de evaluación del proceso de infección y eliminación del virus en un paciente con VIH-1. Este conjunto de estados del Sistema Dinámico No lineal, asociado a las condiciones de valor inicial es denominado Problema de Cauchy. Existen pocos estudios relacionados a la solución de este sistema, de los encontrados algunos se reducen al estudio en dos variables y otros sin la obtención de la solución explícita. En el presente trabajo se contribuye con el estudio del sistema para tres variables, y un análisis cualitativo y cuantitativo que comprende: linealización, diagrama de fase, estabilidad cualitativa, existencia y unicidad de solución analítica, donde se prueba una equivalencia de la solución del sistema no lineal a la del sistema linealizado. La equivalencia está basada en el Teorema de Grobman – Hartman y se encuentra la solución explícita mediante el método de Runge Kutta de 4to orden. De este modo se obtienen los resultados del modelo y cuya convergencia, está garantizada por la consistencia y estabilidad del esquema numérico. Palabras clave.- Tratamiento antirretroviral, Modelamiento matemático VIH, Sistema dinámico no lineal, Método de Runge Kutta de cuarto orden. ABSTRACTThis paper considers the proposal, development and numerical simulation of a system that models the behavior of the treatment of HIV-1 using antiretrovirals. The system defined for three dependent variables of the variable t, denoted by X(t)=(x1 (t), x2 (t), x3 (t)) represent the amount of T Lymphocytes "Helpers" (CD4), number of Cytotoxic T Lymphocytes (CD8) Viral load and infection process of HIV-1 at any time t, respectively. The assembly consists of a Nonlinear Ordinary Differential Equations system whose existential domain represents time system evaluation process of infection and viral clearance in a patient with HIV-1. This set of states of Nonlinear Dynamic System, associated with the initial value condition is called Cauchy problem. There are few studies related to the solution of this system, found some of the study are reduced to two variables and others without obtaining the explicit solution. Equivalence of the solution of the system linearization, phase diagram, qualitative stability, existence and uniqueness of analytic solution, where test: In this paper contributes to the study of the system for three variables, and qualitative and quantitative analysis comprising nonlinear the linearized system. The equivalence is based on Theorem Grobman - Hartman and explicit solution by the Runge Kutta 4th order, thus the model results and whose convergence is obtained is, is guaranteed by the consistency and stability of the numerical scheme. Keywords.- Antiretroviral therapy, HIV mathematical modeling, Nonlinear dynamic system, Runge Kutta method of fourth order.

Effects of radiation and thermal conductivity on MHD boundary layer flow with heat transfer along a vertical stretching sheet in a porous medium

A steady two-dimensional free convective flow of a viscous incompressible fluid along a vertical stretching sheet with the effect of magnetic field, radiation and variable thermal conductivity in porous media is analyzed. The nonlinear partial differential equations, governing the flow field under consideration, have been transformed by a similarity transformation into a systemof nonlinear ordinary differential equations and then solved numerically. Resulting non-dimensional velocity and temperature profiles are then presented graphically for different values of the parameters. Finally, the effects of the pertinent parameters, which are of physical and engineering interest, are examined both in graphical and tabular form.

Qualitative analysis of a mathematical model for kinetic mechanisms of complete mineralization of phenol by Fe_2^+, Fe_3^+ and H_2O_2

In this work we study the phenol degradation kinetics through the formulation and analysis of a nonlinear system of ordinary differential equations. The results suggest that the dynamics of the model is consistent with the phenomenon of phenol mineralization.

Stability of the Richardson Extrapolation combined with some implicit Runge–Kutta methods

The implementation of the Richardson Extrapolation in combination with different numerical methods for solving systems of ordinary differential equations (ODEs) is relatively simple, but the important requirement for stability of the computational process may cause serious difficulties. For example, the commonly used by scientists and engineers Trapezoidal Rule has good stability properties, but its combination with the Richardson Extrapolation is unstable. Therefore, it is necessary to study in advance and very carefully the stability of the new numerical methods arising when the scientists and the engineers use this computational device in combination with different algorithms for solving systems of ODEs.We are presenting a systematic investigation of the implementation of Richardson Extrapolation for two implicit Runge–Kutta methods. Three numerical examples, including an atmospheric chemical scheme used successfully in several extensive environmental studies and described mathematically by a very stiff and badly scaled nonlinear system of ODEs, are presented to illustrate the advantages of the presented approach. The numerical results show that not only are the computations stable, but also the achieved accuracy is higher when the Richardson Extrapolation is additionally applied. It will be possible to derive similar stability and accuracy results for other implicit Runge–Kutta methods.

Some Comparison of Solutions by Different Numerical Techniques on Mathematical Biology Problem

We try to compare the solutions by some numerical techniques when we apply the methods on some mathematical biology problems. The Runge-Kutta-Fehlberg (RKF) method is a promising method to give an approximate solution of nonlinear ordinary differential equation systems, such as a model for insect population, one-species Lotka-Volterra model. The technique is described and illustrated by numerical examples. We modify the population models by taking the Holling type III functional response and intraspecific competition term and hence we solve it by this numerical technique and show that RKF method gives good results. We try to compare this method with the Laplace Adomian Decomposition Method (LADM) and with the exact solutions.

Analysis of the brachistochronic motion of a variable mass nonholonomic mechanical system

The paper considers the brachistochronic motion of a variable mass nonholonomic mechanical system [3] in a horizontal plane, between two specified positions. Variable mass particles are interconnected by a lightweight mechanism of the ?pitchfork? type. The law of the time-rate of mass variation of the particles, as well as relative velocities of the expelled particles, as a function of time, are known. Differential equations of motion, where the reactions of nonholonomic constraints and control forces figure, are created based on the general theorems of dynamics of a variable mass mechanical system [5]. The formulated brachistochrone problem, with adequately chosen quantities of state, is solved, in this case, as the simplest task of optimal control by applying Pontryagin?s maximum principle [1]. A corresponding two-point boundary value problem (TPBVP) of the system of ordinary nonlinear differential equations is obtained, which, in a general case, has to be numerically solved [2]. On the basis of thus obtained brachistochronic motion, the active control forces, along with the reactions of nonholonomic constraints, are determined. The analysis of the brachistochronic motion for different values of the initial position of a variable mass particle B is presented. Also, the interval of values of the initial position of a variable mass particle B, for which there are the TPBVP solutions, is determined.

The Local Brewery: A Project for Use in Differential Equations Courses

We describe a modeling project designed for an ordinary differential equations (ODEs) course using first-order and systems of first-order differential equations to model the fermentation process in beer. The project aims to expose the students to the modeling process by creating and solving a mathematical model and effectively communicating their findings in a technical report. The students are required to produce a simple first-order differential equation and find the solution, given varying initial conditions. The students are also required to analyze a more complex, nonlinear ODE system model of the fermentation process. In dealing with the nonlinear system of equations, we provide the students a Mathematica file to reduce the time spent developing the model, but allow for more time to interpret the model. We also share some perspectives on the implementation of the project, provide alternative implementations, and possible extensions to the project.

Numerical solution of initial-boundary system of nonlinear hyperbolic equations

In this article, we present a numerical approximation of the initial-boundary system of nonlinear hyperbolic equations based on spectral Jacobi-Gauss-Radau collocation (J-GR-C) method. A J-GR-C method in combination with the implicit Runge-Kutta scheme are employed to obtain a highly accurate approximation to the mentioned problem. J-GR-C method, based on Jacobi polynomials and Gauss-Radau quadrature integration, reduces solving the system of nonlinear hyperbolic equations to solve a system of nonlinear ordinary differential equations (SNODEs). In the examples given, numerical results by the J-GR-C method are compared with the exact solutions. In fact, by selecting relatively few J-GR-C points, we are able to get very accurate approximations. In this way, the results show that this method has a good accuracy and efficiency for solving coupled partial differential equations.

Finite dimensional guaranteed cost sampled-data fuzzy control for a class of nonlinear distributed parameter systems

In this paper, a finite dimensional guaranteed cost sampled-data fuzzy control (GCSDFC) problem is addressed for a class of nonlinear parabolic partial differential equation (PDE) systems. Initially, applying the Galerkin’s method to the PDE system, a nonlinear ordinary differential equation (ODE) system is derived. The resulting nonlinear ODE system can accurately describe the dominant dynamics of the PDE system, which is subsequently expressed by the Takagi–Sugeno (T–S) fuzzy model. Then, a guaranteed cost sampled-data fuzzy controller is developed to stabilize exponentially the closed-loop slow fuzzy system while providing an upper bound for the quadratic cost function. A novel time-dependent functional is constructed to derive the condition for the existence of the proposed controller which is presented by bilinear matrix inequalities (BMIs). Moreover, a suboptimal GCSDFC problem to minimize the cost bound can be formulated as a BMI optimization problem. A local optimization algorithm that views the BMI as a double linear matrix inequality (LMI) is given to solve this BMI optimization problem, in which a Latin hypercube sampling (LHS) method is proposed to find an initially feasible solution for starting the algorithm. Furthermore, it is shown that the proposed controller can ensure the exponential stability of the closed-loop PDE system. Finally, simulation results on the Fisher equation and the temperature profile of a catalytic rod show that the proposed design strategy is effective.

Unsteady MHD boundary layer flow and heat transfer over the stretching sheets submerged in a moving fluid with Ohmic heating and frictional heating

Abstract This paper is devoted to the analysis of the unsteady magnetohydrodynamic (MHD) boundary layer flow and heat transfer on a permeable stretching sheet embedded in a moving incompressible viscous fluid. The combined effects of Ohmic heating, thermal radiation, frictional heating and internal heat absorption/generation are taken into account. The governing time dependent nonlinear boundary layer equations are converted into a systemof nonlinear ordinary differential equations by similarity transformations. Some analytical results that give the characteristics of the velocity field in the boundary layer are presented and proved. The governing equations are then solved by using the shooting technique along with the fourth order Runge-Kutta method. The analytical properties proved in this paper are consistent with those obtained by the numerical method. Furthermore, the effects of the various parameters on the velocity and temperature fields are presented graphically and discussed in detail.

The <i>k</i> = 1 Finite Element Numerical Solution for the Improved Boussinesq Equation

The improved Boussinesq equation is solved with classical finite element method using the most basic Lagrange element k = 1, which leads us to a second order nonlinear ordinary differential equations system in time; this can be solved by any standard accurate numerical method for example Runge-Kutta-Fehlberg. The technique is validated with a typical example and a fourth order convergence in space is confirmed; the 1- and 2-soliton solutions are used to simulate wave travel, wave splitting and interaction; solution blow up is described graphically. The computer symbolic system MathLab is quite used for numerical simulation in this paper; the known results in the bibliography are confirmed.

Time Domain Inverse Problems in Nonlinear Systems Using Collocation & Radial Basis Functions

In this study, we consider ill-posed time-domain inverse problems for dynamical systems with various boundary conditions and unknown controllers. Dynamical systems characterized by a system of second-order nonlinear ordinary differential equations (ODEs) are recast into a system of nonlinear first order ODEs in mixed variables. Radial Basis Functions (RBFs) are assumed as trial functions for the mixed variables in the time domain. A simple collocation method is de- veloped in the time-domain, with Legendre-Gauss-Lobatto nodes as RBF source points as well as collocation points. The duffing optimal control problem with var- ious prescribed initial and final conditions, as well as the orbital transfer Lambert's problem are solved by the proposed RBF collocation method as examples. It is shown that this method is very simple, efficient and very accurate in obtaining the solutions, with an arbitrary solution as the initial guess. Since methods such as the Shooting Method and the Pseudo-spectral Method can be unstable and require an accurate initial guess, the proposed method is advantageous and has promising ap- plications in optimal control and celestial mechanics. The extension of the present study to other optimal control problems, and other orbital transfer problems with perturbations, will be pursued in our future studies.

Nonlinear Time Spectral Analysis for a Dynamic Contact Model with Buckling for an Elastic Plate

In the present paper a dynamic nonlinear model with contact and buckling for an elasticplate is considered. The model consists of two coupled nonlinear hyperbolic type partial differentialequations. The plate is subjected to compressive and/or tensile moving loads on its edges. The foundationsare nonlinear elastic Winkler and Pasternak models. The initial-boundary value problems forthe model are solved with the use of the time spectral method for spatial discretization and after thediscretization the Newmark- time-stepping iterative scheme for the obtained system of nonlinear ordinarydifferential equations. The model is tested for the Winkler-type and shear Pasternak-type andas well for several values of the physical constants of the foundations.

An integral-balance nonlinear model to simulate changes in soil moisture, groundwater and surface runoff dynamics at the hillslope scale

We present a system of ordinary differential equations (ODEs) capable of reproducing simultaneously the aggregated behavior of changes in water storage in the hillslope surface, the unsaturated and the saturated soil layers and the channel that drains the hillslope. The system of equations can be viewed as a two-state integral-balance model for soil moisture and groundwater dynamics. Development of the model was motivated by the need for landscape representation through hillslopes and channels organized following stream drainage network topology. Such a representation, with the basic discretization unit of a hillslope, allows ODEs-based simulation of the water transport in a basin. This, in turn, admits the use of highly efficient numerical solvers that enable space–time scaling studies. The goal of this paper is to investigate whether a nonlinear ODE system can effectively replicate observations of water storage in the unsaturated and saturated layers of the soil. Our first finding is that a previously proposed ODE hillslope model, based on readily available data, is capable of reproducing streamflow fluctuations but fails to reproduce the interactions between the surface and subsurface components at the hillslope scale. However, the more complex ODE model that we present in this paper achieves this goal. In our model, fluxes in the soil are described using a Taylor expansion of the underlying storage flux relationship. We tested the model using data collected in the Shale Hills watershed, a 7.9-ha forested site in central Pennsylvania, during an artificial drainage experiment in August 1974 where soil moisture in the unsaturated zone, groundwater dynamics and surface runoff were monitored. The ODE model can be used as an alternative to spatially explicit hillslope models, based on systems of partial differential equations, which require more computational power to resolve fluxes at the hillslope scale. Therefore, it is appropriate to be coupled to runoff routing models to investigate the effect of runoff and its uncertainty propagation across scales. However, this improved performance comes at the expense of introducing two additional parameters that have no obvious physical interpretation. We discuss the implications of this for hydrologic studies across scales.

Computational analysis of the model describing HIV infection of CD4+T Cells.

An analysis of the model underpinning the description of the spread of HIV infection of CD4+T cells is examined in detail in this work. Investigations of the disease free and endemic equilibrium are done using the method of Jacobian matrix. An iteration technique, namely, the homotopy decomposition method (HDM), is implemented to give an approximate solution of nonlinear ordinary differential equation systems. The technique is described and illustrated with numerical examples. The approximated solution obtained via HDM is compared with those obtained via other methods to prove the trustworthiness of HDM. Moreover, the lessening and simplicity in calculations furnish HDM with a broader applicability.

Transient flow and heat transfer of pseudoplastic fluids on a stretching sheet

Non-isothermal pseudoplastic flow within the laminar boundary layer on stretching sheet is analyzed. Using similarity transformation, the time dependent governing equations are reduced to a highly-correlated non-linear ordinary differential equation system. The system is then numerically integrated using a novel combination a Runge–Kutta algorithm and shooting technique. Effect of several values of governing parameters such as unsteadiness parameter, pseudoplasticity index and Prandtl number on velocity and temperature profiles, apparent viscosity and heat transfer coefficient are explored in detail.

Melting phenomenon in magneto hydro-dynamics steady flow and heat transfer over a moving surface in the presence of thermal radiation

The Lie group method is applied to present an analysis of the magneto hydro-dynamics (MHD) steady laminar flow and the heat transfer from a warm laminar liquid flow to a melting moving surface in the presence of thermal radiation. By using the Lie group method, we have presented the transformation groups for the problem apart from the scaling group. The application of this method reduces the partial differential equations (PDEs) with their boundary conditions governing the flow and heat transfer to a system of nonlinear ordinary differential equations (ODEs) with appropriate boundary conditions. The resulting nonlinear system of ODEs is solved numerically using the implicit finite difference method (FDM). The local skin-friction coefficients and the local Nusselt numbers for different physical parameters are presented in a table.