Reconstruction of inaccessible boundary value in a sideways parabolic problem with variable coefficients—Forward collocation with finite integration method

Radial basis functions method for solving the fractional diffusion equations

The non-standard Hamiltonian system, also referred to as a partial Hamiltonian system in the literature, of the form q ˙ i = ∂ H ∂ p i , p ˙ i = − ∂ H ∂ q i + Γ i ( t , q i , p i ) ${\dot q^i} = \frac{{\partial H}}{{\partial {p_i}}},{\text{ }}{\dot p^i} = - \frac{{\partial H}}{{\partial {q_i}}} + {\Gamma ^i}(t,{\text{ }}{q^i},{\text{ }}{p_i})$ appears widely in economics, physics, mechanics, and other fields. The non-standard (partial) Hamiltonian systems arise from physical Hamiltonian structures as well as from artificial Hamiltonian structures. We introduce the term ‘artificial Hamiltonian’ for the Hamiltonian of a model having no physical structure. We provide here explicitly the notion of an artificial Hamiltonian for dynamical systems of ordinary differential equations (ODEs). Also, we show that every system of second-order ODEs can be expressed as a non-standard (partial) Hamiltonian system of first-order ODEs by introducing an artificial Hamiltonian. This notion of an artificial Hamiltonian gives a new way to solve dynamical systems of first-order ODEs and systems of second-order ODEs that can be expressed as a non-standard (partial) Hamiltonian system by using the known techniques applicable to the non-standard Hamiltonian systems. We employ the proposed notion to solve dynamical systems of first-order ODEs arising in epidemics.

10 - Applications of Systems of Ordinary Differential Equations

It is well known that a model of any electrical circuit can be translated into a system of ordinary differential equations. Usually, the simulation is performed in the time-domain using classical discrete approaches convenient for both linear and non-linear systems. An other method consists in the quantization on the variable magnitude which is based on discrete events. The discrete event system specification formalism is adapted to simulate any electrical circuit described with a system of ordinary differential equations using the system state quantification. In this paper, a solution to make more efficient the magnitude quantification for a system of linear ordinary differential equations is given. The proposed technique is used to simulate a simple three phase coupled power systems and it is compared to other solutions given by the explicit integration using formal language and the time-domain discretization using fixed-step fifth order Runge-Kutta integration method.

We consider boundary value problems with periodic boundary conditions for first-order linear systems of impulsive ordinary differential equations with unknown right-hand sides and jumps of solutions at the impulse points entering into the statement of these problems which are assumed to be subjected to some quadratic restrictions. From indirect noisy observations of their solutions on a finite system of intervals, we obtain the optimal, in certain sense, estimates of images of their right-hand sides under linear continuous operators. Under the condition that the unknown correlation functions of noises in observations belong to some special sets, it is established that such estimates and estimation errors are expressed explicitly via solutions of special periodic boundary value problems for linear impulsive systems of ordinary differential equations.

In this paper, we propose a scalable configuration of all-optical solver for systems of ordinary differential equations (SODE) based on the integrated feedback system, which can be easily extended to solve SODEs with various dimensions and forms. We use different optical ordinary differential equation (ODE) solvers to construct different ODEs and then introduce optical feedback waveguides to realize the coupling of them, thus forming the SODE solvers. For a proof of concept, we design and fabricate a two-dimensional (2-D) and a four-dimensional (4-D) SODE solver based on the silicon-on-insulator platform. The corresponding expressions of the SODEs are deduced based on the temporal coupled mode theory. In addition, aiming at the 2-D SODE solver, we also use transfer matrix method to examine the correctness of the SODE and estimate the working bandwidth at the same time. At last, a system experiment is carried out to get the solutions of the SODEs when we use 10-GHz non-return-to-zero (NRZ)-like pulses as input signals, and all the measured output waveforms agree well with the calculated results.

On the comparison of numerical methods for the integration of kinetic equations in atmospheric chemistry and transport models

In this article, a numerical method integrated with statistical data simulation technique is introduced to solve a nonlinear system of ordinary differential equations with multiple random variable coefficients. The utilization of Monte Carlo simulation with central divided difference formula of finite difference (FD) method is repeated n times to simulate values of the variable coefficients as random sampling instead being limited as real values with respect to time. The mean of the n final solutions via this integrated technique, named in short as mean Monte Carlo finite difference (MMCFD) method, represents the final solution of the system. This method is proposed for the first time to calculate the numerical solution obtained for each subpopulation as a vector distribution. The numerical outputs are tabulated, graphed, and compared with previous statistical estimations for 2013, 2015, and 2030, respectively. The solutions of FD and MMCFD are found to be in good agreement with small standard deviation of the means, and small measure of difference. The new MMCFD method is useful to predict intervals of random distributions for the numerical solutions of this epidemiology model with better approximation and agreement between existing statistical estimations and FD numerical solutions.

Journal Article Stabilization of one-dimensional wave equations coupled with an ODE system on general tree-shaped networks Get access Dongyi Liu, Dongyi Liu * Department of Mathematics, University of Tianjin, Weijin Road, Nankai District, Tianjin, 300072, P. R. China *Corresponding author: dyliu@tju.edu.cn Search for other works by this author on: Oxford Academic Google Scholar Liping Zhang, Liping Zhang Department of Mathematics, University of Tianjin, Weijin Road, Nankai District, Tianjin, 300072, P. R. China Search for other works by this author on: Oxford Academic Google Scholar Genqi Xu, Genqi Xu Department of Mathematics, University of Tianjin, Weijin Road, Nankai District, Tianjin, 300072, P. R. China Search for other works by this author on: Oxford Academic Google Scholar Zhongjie Han Zhongjie Han Department of Mathematics, University of Tianjin, Weijin Road, Nankai District, Tianjin, 300072, P. R. China Search for other works by this author on: Oxford Academic Google Scholar IMA Journal of Mathematical Control and Information, Volume 32, Issue 3, September 2015, Pages 557–589, https://doi.org/10.1093/imamci/dnu008 Published: 30 March 2014 Article history Received: 12 July 2013 Revision received: 22 January 2014 Accepted: 08 February 2014 Published: 30 March 2014

The fluid dynamics presented in this paper for a fluidized bed gasifier consists of estimating the volume fraction fields of gas and solid phases, as well as the velocity fields of gas and solid phases and the pressure drop inside the fluidized bed. However, understanding of the flow characteristics has an important role in understanding of the operation of the gasifier. The analyzed system consists of the fundamental equations of mass balance for the gas and solid phases, as well as the equations of momentum balance for gas and solid phases and an equation for the pressure given by the sum of the momentum balance equations of the gas and solid phases. The sets of equations developed form a system of one-dimensional partial differential equations (PDEs). The PDEs system has been transformed in a coupled ordinary differential equation (ODEs) system. The ODEs system has been solved using an implementation of the Runge-Kutta Gill method. The objective of this work is to obtain the profiles of state variables such as volume fraction of gas (eg) and solid phases (es), the velocity profiles of gas (Vg) and solid phase (Vs) and pressure field (Dp). Key words: Fluid dynamics, gasifier, fluidized bed, gas-solid, biomass.

A Hopfield neural network is described by a system of nonlinear ordinary differential equations. We develop a broad range of numerical schemes that are applicable for a wide range of computational problems. We review here our study on an approximate solution of the Fredholm integral equation, and linear and nonlinear singular and hypersingular integral equations, using a continuous method for solving operator equations. This method assumes that the original system is associated with a Cauchy problem for systems of ordinary differential equations on Hopfield neural networks. We present sufficient conditions for the Hopfield networks’ stability defined via coefficients of systems of differential equations.

Non-linear parameter estimation in pesticide degradation

In this paper, an algorithm for researching the stability of linear systems using a computer is proposed. The stability of systems with delay is reduced to finding the limit conditions for the negatives of the real parts of the zeros of the corresponding quasi-polynomials. Verification in practice of such conditions is possible only in the simplest cases. The algorithm proposed in this paper is based on schemes for approximating linear systems with a delay by used sequence special of systems of linear ordinary differential equations. The equivalence of the exponential stability of systems with delay and of the proposed system of ordinary differential equations is established. This allowed us to build an algorithm for studying the location of non-asymptotic roots of quasi-polynomials, which are implemented on a computer.

Consideration was given to the model obeying a system of ordinary differential equations where the subsystems are systems of autonomous ordinary differential equations. If the coupling parameter ɛ = 0, then the model falls apart into decoupled subsystems. For a model consisting of coupled subsystems, considered was the main mode for which the problems of oscillations, bifurcation, and stability were solved, and the results obtained before for the case of two second-order subsystems were generalized.

In the paper was investigated the a priori estimate and the existence of periodic solutions of a fixed period for a system of second-order ordinary differential equations with the main quasi-homogeneous non-linearity. It is proved that an a priori estimate of periodic solutions takes place if the corresponding unperturbed system does not have non-zero bounded solutions. Under the conditions of an a priori estimate, using methods for calculating the mapping degree of vector fields, a criterion for the existence of periodic solutions under any perturbation from a given class is formulated and proven. The results obtained differ from earlier results in that the set of zeros of the main non-linear part is not taken into account.