In this work, we focus on distributed control of quasi-linear parabolic partial differential equations (PDEs) and address the problem of enforcing a prespecified spatio-temporal behaviour in the closed-loop system using nonlinear feedback control and a sufficiently large number of actuators and sensors. Under the assumption that the desired spatio-temporal behaviour is described by a ‘target pa...

Birth-jump models are designed to describe population models for which growth and spatial spread cannot be decoupled. A birth-jump model is a nonlinear integro-differential equation. We present two different derivations of this equation, one based on a random walk approach and the other based on a two-compartmental reaction-diffusion model. In the case that the redistribution kernels are highly...

Dedicated to Professor Rudolf Gorenflo on the occasion of his 80th anniversary In the paper, maximum principle for the generalized time-fractional diffusion equations including the multi-term diffusion equation and the diffusion equation of distributed order is formulated and discussed. In these equations, the time-fractional derivative is defined in the Caputo sense. In contrast to the Riemann...

In this paper, a numerical method is proposed to approximate the numeric solutions of nonlinear Fisher's reaction– diffusion equation with modified cubic B-spline collocation method. The method is based on collocation of modified cubic B-splines over finite elements, so we have continuity of the dependent variable and its first two derivatives throughout the solution range. We apply modified cu...

This chapter presents a constructive derivation of HDG methods for convection-diffusion-reaction equation using the Rankine-Hugoniot condition. This is possible due to the fact that, in the first order form, convection-diffusion-reaction equation is a hyperbolic system. As such it can be discretized using the standard upwind DG method. The key is to realize that the Rankine-Hugoniot condition n...

The stability and Hopf bifurcation of the positive steady state to a general scalar reaction–diffusion equation with distributed delay and Dirichlet boundary condition are investigated in this paper. The time delay follows a Gamma distribution function. Through analyzing the corresponding eigenvalue problems, we rigorously show that Hopf bifurcations will occur when the shape parameter n ≥ 1, a...

The problem of wave-front propagation for the n-dimensional reaction-diffusion system involving Kolmogorov-Petrovskii-Piskunov kinetics and the diffusion transport with a finite velocity has been considered. By using a scaling procedure we have given an asymptotic derivation of the equation governing the evolution of a reaction front in the long-time large-distance limit. It has been found that...

The known properties of diffusion on fractals are reviewed in order to give a general outlook of these dynamic processes. After that, we propose a description developed in the context of the intrinsic metric of fractals, which leads us to a differential equation able to describe diffusion in real fractals in the asymptotic regime. We show that our approach has a stronger physical justification ...

It is well known that optimal control trajectories can be highly sensitive to perturbations in the model parameters. Computationally efficient numerical algorithms are presented for the worst-case analysis of the effects of parametric uncertainties on boundary control problems for finite-time distributed parameter systems. The approach is based on replacing the full-order model of the system wi...

In nuclear reactor design, the most fundamental task is to follow the neutron distribution as a function of space, direction, energy and time, in order to maintain and control the nuclear fission chain reaction. The fundamental equation governing the average behaviour of neutrons is given by the linear form of Boltzmann transport equation, which is an integro-differential equation. The core sim...