The dynamics of biological processes are often modeled as systems of nonlinear ordinary differential equations (ODE). An important feature of nonlinear ODEs is that seemingly minor changes in initial conditions or parameters can lead to radically different behaviors. This is problematic because in general it is never possible to know/measure the precise state of any biological system due to mea...

Combining recent moment and sparse semidefinite programming (SDP) relaxation techniques, we propose an approach to find smooth approximations for solutions of problems involving nonlinear differential equations. Given a system of nonlinear differential equations, we apply a technique based on finite differences and sparse SDP relaxations for polynomial optimization problems (POP) to obtain a di...

Numerical simulations of phase-field models require long time computations and therefore it is necessary to develop efficient and highly accurate numerical methods. In this paper, we propose fast and stable explicit operator splitting methods for both oneand two-dimensional nonlinear diffusion equations for thin film epitaxy with slope selection and Cahn-Hilliard equation. The equations are spl...

The diode clipper circuit with an embedded low-pass filter lies at the heart of both diode clipping “Distortion” and “Overdrive” or “Tube Screamer” effects pedals. An accurate simulation of this circuit requires the solution of a nonlinear ordinary differential equation (ODE). Numerical methods with stiff stability – Backward Euler, Trapezoidal Rule, and second-order Backward Difference Formula...

The phenomenon of coherent energetic pulse propagation in exciton-phonon molecular chains such as α-helix protein is studied using an ODE system model of Davydov-Scott type, both with numerical studies using a new unconditionally stable fourth-order accurate energy-momentum conserving time discretization and with analytical explanation of the main numerical observations. Impulsive initial data ...

We present a class of one-dimensional systems nonlinear parabolic equations for which the phase dynamics at large time can be described by an ODE with Lipschitz vector field in $$\mathbb R^n$$ . In considered case Dirichlet boundary value problem, sufficient conditions finite-dimensional reduction turn out to much wider than known this kind periodic situation.

The problem of unknown input observer design is considered for coupled PDE/ODE systems subject to incremental sector bounded nonlinearities and boundary inputs. Assuming available measurements at the distributed domain, synthesis based on Lyapunov methods recast in terms some matrix inequalities. Numerical simulations support confirm theoretical findings, illustrating robust estimation performa...

Much effect has been devoted over the past decade to inference of gene regulatory networks (GRNs). However, the previous methods infer GRNs containing large amount of false positive edges, which could result in awful influence on biological analysis. In this study, we present a novel hybrid framework to improve the accuracy of GRN inference. In our method, network topologies from linear and non...

Ideas that challenge the status quo either evaporate or dominate. The study of opinion dynamics in the socio-physics literature treats space as uniform and considers individuals in an isolated community, using ordinary differential equation (ODE) models. We extend these ODE models to include multiple communities and their interactions. These extended ODE models can be thought of as being ODEs o...

In this work, we improve a previous minimalistic tree-grass savanna model by taking into account water availability, in addition to fire, since both factors are known to be important for shaping savanna physiognomies along a climatic gradient. As in our previous models, we consider two nonlinear functions of grass and tree biomasses to respectively take into account grass-fire feedbacks, and th...