Spatial Structure: Patch Models

1 - INTRODUCTION

Stationary kinetic methods are not sufficient for studying chemical processes in which solid heterogeneous catalysts and several phases (gas-liquid-solid) are involved, therefore transient experimental techniques are applied to reveal the molecular processes on the catalyst surfaces. Dynamic methods were used to study the kinetics of a heterogeneously catalyzed reaction i.e. the epoxidation of ethylene with hydrogen peroxide as the oxidizing agent and titanium silicalite (TS-1) as the heterogeneous catalyst. No detailed reaction mechanism and kinetic model has ever been proposed for this process. For the kinetic investigation, step response experiments were carried out in a laboratory-scale trickle bed reactor (TBR).A dynamic isothermal model for the TBR was presented together with a dynamic kinetic model based on a proposed surface reaction mechanism, where the key step is the reaction between adsorbed hydrogen peroxide and dissolved ethylene from the liquid phase. The TBR model consisted of dynamic mass balance equations for the components present in the gas and liquid bulk phases as well as adsorbed species on the catalyst surface. The model formed a coupled system of parabolic partial differential equations (gas and liquid phases) and ordinary differential equations (catalyst surface phase). The coupled system of differential equations, describing the concentration profiles from the step response experiments were solved numerically by the method of lines by discretizing the reactor length coordinate with finite differences and solving the large system of ordinary differential equations by a backward difference algorithm suitable for stiff ordinary differential equations. The parameters for surface kinetics, mass transfer and fluid dynamics were determined by non-linear regression analysis using the maximum likelihood approach. The model was able to predict the transient and stationary behavior of the system.

The system of ordinary differential equations has many uses in contemporary mathematics and engineering. Finding the numerical solution to a system of ordinary differential equations for any arbitrary interval is very appealing to researchers. The numerical solution of a system of fourth-order ordinary differential equations on any finite interval [a,b] is found in this work using a symmetric Bernstein approximation. This technique is based on the operational matrices of Bernstein polynomials for solving the system of fourth-order ODEs. First, using Chebyshev collocation nodes, a generalised approximation of the system of ordinary differential equations is discretized into a system of linear algebraic equations that can be solved using any standard rule, such as Gaussian elimination. We obtain the numerical solution in the form of a polynomial after obtaining the unknowns. The Hyers–Ulam and Hyers–Ulam–Rassias stability analyses are provided to demonstrate that the proposed technique is stable under certain conditions. The results of numerical experiments using the proposed technique are plotted in figures to demonstrate the accuracy of the specified approach. The results show that the suggested Bernstein approximation method for any interval is quick and effective.

Populations may be structured by spatial location. There are two common different ways to include spatial location in a population. One way is by means of metapopulations, that is, populations of populations, with links between them such as a collection of towns and cities connected by a transportation network. The air transport subnetwork includes connecting links between distant communities, and we may study the dynamics of populations of different cities as a function of the flow of people between them and their own local dynamics in this framework. A metapopulation may be divided into patches, with each patch corresponding to a separate location. The corresponding models may be systems of ordinary differential equations, with the population size of each species in each patch as a variable. Thus metapopulation models are often systems of ordinary differential equations of high dimension. Some basic references are Hanski (1999), Hanski and Gilpin (1997), Levin, Powell and Steele (1993), Neuhauser (2001).

Solving multivariate ordinary differential equations (ODE) systems and partial differential equations (PDE) systems is the key to many complex physics and chemistry problems, such as the combustion in process of reacting flow. However, the traditional numerical methods in solving multivariate ODE and PDE systems are limited by computational cost, and sometimes its impossible to obtain the solution due to the high stiffness of ODE or PDE. Coincident with the development of machine learning has been a growing appreciation of applying neural networks in solving physics models. DeepM&M net was proposed to address complicated problems in fluid mechanics based on another neural network: DeepONet, which is used to predict functional nonlinear operators. Inspired by these two nets, a machine learning way of solving certain ODE and PDE systems is proposed with a similar framework to the DeepM&M net, which takes inputs of the initial conditions and outputs the corresponding solutions. The main ideas of this framework are first to explore the relations among solutions of the system by DeepONets and then to train a deep neural network with the assistance of trained DeepONets. The implicit operators between variables in certain ODE systems are verified to have existed and are well predicted by the DeepONet. The feasibility of the proposed framework is implied by the success in building blocks.

We establish a connection between solutions to a broad class of large systems of ordinary differential equations and solutions to retarded differential equations. We prove that solving the Cauchy problem for systems of ordinary differential equations reduces to solving the initial value problem for a retarded differential equation as the number of equations increases unboundedly. In particular, the class of systems under consideration contains a system of differential equations which arises in modeling of multiphase synthesis.

An algorithm for solving the linear Cauchy problem for large systems of ordinary differential equations is presented. The algorithm for systems of first-order differential equations is implemented in the EDELWEISS code with the possibility of parallel computations on supercomputers employing the MPI (Message Passing Interface) standard for the data exchange between parallel processes. The solution is represented by a series of orthogonal polynomials on the interval [0, 1]. The algorithm is characterized by simplicity and the possibility to solve nonlinear problems with a correction of the operator in accordance with the solution obtained in the previous iterative process.

The design an optimal numerical method for solving a system of ordinary differential equations simultaneously is described in this paper. System of differential equations was represented by a system of linear ordinary differential equations of Euler’s parameters called quaternions. The components of angular velocity were obtained by the experimental way. The angular velocity of the centre of gravity was determined from sensors of acceleration located in the plane of the centre of gravity of the machine. The used numerical method for solving was a fourth-order Runge-Kutta method. The stability of solving was based on the orthogonality of a direct cosine matrix. The numerical process was controlled on every step in numerical integration. The algorithm was designed in the C# programming language.

Systems of ordinary differential equations with a small parameter at the derivative and specific features of the construction of their periodic solution are considered. Sufficient conditions of existence and uniqueness of the periodic solution are presented. An iterative procedure of construction of the steady-state solution of a system of differential equations with a small parameter at the derivative is proposed. This procedure is reduced to the solution of a system of nonlinear algebraic equations and does not involve the integration of the system of differential equations. Problems of numerical calculation of the solution are considered based on the procedure proposed. Some sources of its divergence are found, and the sufficient conditions of its convergence are obtained. The results of numerical experiments are presented and compared with theoretical ones.

Chapter 11 - Applications of Systems of Ordinary Differential Equations

In this paper, the parameter identification problem for system of ordinary differential equations is considered. The parameter identification problem for system of ordinary differential equations is investigated by the Dzhumabaev’s parametrization method. At first, conditions for a unique solvability of the parameter identification problem for system of ordinary differential equations are obtained in the term of fundamental matrix of system’s differential part. Further, we establish conditions for a unique solvability of the parameter identification problem for system of ordinary differential equations in the terms of initial data. Algorithm for finding of approximate solution to a unique solvability of the parameter identification problem for system of ordinary differential equations is proposed and the conditions for its convergence are setted. Results this paper can be use for investigating of various problems with parameter and control problems for system of ordinary differential equations. The approach in this paper can be apply to the parameter identification problems for partial differential equations.

Classification of the systems of ordinary differential equations and practical aspects in the numerical integration of large systems

In this chapter, we provide a method for solving systems of linear ordinary differential equations by using techniques associated with the calculation of eigenvalues, eigenvectors and generalized eigenvectors of matrices. We learn in calculus how to solve differential equations and the system of differential equations. Here, we firstly show how to represent a system of differential equations in a matrix formulation. Then, using the Jordan canonical form and, whenever possible, the diagonal canonical form of matrices, we will describe a process aimed at solving systems of linear differential equations in a very efficient way.

In this paper, we study the possibility of the existence of a universal solution of the Cauchy problem for the partial differential equation (PDE) systems in the case if this system is overdetermined so that the new overdetermined system of PDE contains all solutions of the initial PDE system and, in addition, reduces to the ordinary differential equation (ODE) systems, whose solution is then found. To do this, the article discusses the modification of the method of finding particular solutions for any overdetermined systems of differential equations by reduction to overdetermined systems of implicit equations. In the previous papers of the authors, a method was proposed for finding particular solutions for overdetermined PDE systems. In this method, in order to find solutions it is necessary to solve systems of ordinary implicit equations. In this case, it can be shown that the solutions that we need cannot depend on a continuous parameter, i.e. they are no more than countable. In advance, there is a need for such an overriding of the systems of differential equations, so that their general solutions are no more than countable. Such an initial overdetermination is rather difficult to achieve. However, the proposed method also allows to reduce the overdetermined systems of differential equations not only up to systems of implicit equations, but also up to the PDE systems of dimension less than that of the initial systems of PDE. In particular, under certain conditions, reduction to the ODE systems is possible. It is proposed to choose solutions for the overdetermined PDE systems using the parameterized Cauchy problem, which is posed for parameterized ODE systems under certain conditions. The solution of this Cauchy problem is some function of the initial data and their derivatives. In order to find the solution of any corresponding Cauchy problem for the initial system of PDE, it is sufficient to calculate the universal solver for the reduced ODE system once. In this case, the solution will not only exist and be unique, but will also depend continuously on the initial data, since this holds for ODE systems. The purpose of this paper is to study the Cauchy problem with the possibility of its universalization and the parameterized Cauchy problem as a whole for arbitrary PDE systems.

The use of sparse matrix technique in the numerical integration of stiff systems of linear ordinary differential equations