We show that the gradient flow u on L2 generated by the energy functional I[u] := ∫ U Φ(detDu) dx for vector-valued mappings is in some sense “integrable”, meaning that (i) the inverse Jacobian β := (detDu)−1 satisfies a scalar nonlinear diffusion equation, and (ii) we can recover u by solving an ODE determined by β.

Combining recent moment and sparse semidefinite programming (SDP) relaxation techniques, we propose an approach to find smooth approximations for solutions of 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 discrete approximatio...

This paper derives state space error bounds for the solutions of reduced systems constructed using Proper Orthogonal Decomposition (POD) together with the Discrete Empirical Interpolation Method (DEIM) recently developed in [4] for nonlinear dynamical systems. The resulting error estimates are shown to be proportional to the sums of the singular values corresponding to neglected POD basis vecto...

The steady heat transfer through a rectangular longitudinal fin is studied.The thermal conductivity and heat transfer coefficient are assumed to be temperature dependent making the resulting ordinary differential equation (ODE) highly nonlinear. An asymptotic solution is used as a means of understanding the relationship between key parameters. A dynamical analysis is also employed for the same ...

Physiologically realistic simulations of computational islets of beta cells require the long-time solution of several thousands of coupled ordinary differential equations (ODEs), resulting from the combination of several ODEs in each cell and realistic numbers of several hundreds of cells in an islet. For a reliable and accurate solution of complex nonlinear models up to the desired final times...

The promising increase of Electric Vehicles (EVs) in our society poses a challenging problem on the impact assessment their charging/discharging to power distribution grids. This paper addresses framework nonlinear differential equations. Specifically, we address ODE (Ordinary Differential Equation) model for representing spatial profile voltage phasor along feeder, which has been recently intr...

The paper presents a new affine invariant theory on asymptotic mesh independence of Newton’s method for discretized nonlinear operator equations. Compared to earlier attempts, the new approach is both much simpler and more intuitive from the algorithmic point of view. The theory is exemplified at collocation methods for ODE boundary value problems and at finite element methods for elliptic PDE ...

We describe the development of an automated modelling system that uses the genetic programming paradigm. The Genetic Modelling System (GeMS) is capable of generating appropriate algebraic (linear and nonlinear) and ODE systems from experimental data. The utility of GeMS is illustrated through several examples involving laboratory systems and simulated kinetic data. Copyright 2004 IFAC