The riccati differential equation and a diffusiontype equation article pdf available in new york journal of mathematics 17 august 2008 with 73 reads how we measure reads. The theory of the principal eigenvalue is developed for an elliptic eigenvalue problem associated with a linear parabolic cooperative system with some zero diffusion coefficients. Pdf the riccati differential equation and a diffusion. Reactiondiffusion rd equations arise naturally in systems consisting of many. Pdf reactiondiffusion waves in biology researchgate. Nonlinear reaction diffusion equation with michaelis.
It is not meaningful to talk about a general theory of reactiondiffusion systems. Standing solitary oscillon rotating pulse, stable localized oscillating pulse, merger of 3. Optimal control of a system of reactiondiffusion equations modeling the wine fermentation process. Fractional differential equations are becoming increasingly used as a powerful modelling approach for understanding the many aspects of nonlocality and spatial heterogeneity. Optimal control of wine fermentation merger, borzi, herzog a control mechanism driven by an external temperature of a water coolingheating system. Pdf the theory of reactiondiffusion waves begins in the 1930s with the works in. The principal ingredients of all these models are equation of the form. Then the basic reproduction number and its computation formulae are established for reactiondiffusion. In this paper, we theoretically and numerically investigate our reactiondiffusion model and discuss a related optimal control problem. However, the numerical approximation of these models is computationally.
An approximate analytical solution to the diffusion equation derived below is obtained and shown to be in good agreement both with solutions obtained using. The key idea is to combine the theories of mittagleffler functions and fractional sobolev embeddings. On nonlinear nonlocal systems of reaction diffusion equations. The above equation is an extension of the fitzhughnagumo equation and in the case of. Pdf using taylor series expansion, multiscaling, and further expansion in powers of a small parameter, we develop general amplitude equations for. Inverse initial problem for fractional reactiondiffusion equation with. It is possible to combine different types of boundary conditions on separate parts of. Pdf amplitude equations for reactiondiffusion systems with cross. To fully specify a reactiondiffusion problem, we need the differential equations, some initial.
Cauchy problem and boundary value problems for the diffusion equation. Basic reproduction numbers for reactiondiffusion epidemic. The reactiondiffusion equations reactiondiffusion rd equations arise naturally in systems consisting of many interacting components, e. We call this new class of pdes the gibbs reaction and diffusion equationsgrade and we demonstrate experiments where grade are used for texture pattern formation, denoising, image enhancement, and. Some of these applications, particularly in chemistry and biology, will be considered along the course. Pdf fourier spectral methods for fractionalinspace. Diffusion equation linear diffusion equation eqworld. In the general context of reactiondiffusion equations the individual quantities in the above equation have following meaning. In this section, we consider typical reactions which may appear as reaction terms for the reaction di. For solutions of the cauchy problem and various boundary value problems, see nonhomogeneous diffusion equation with x, t. Reactiondiffusion equations are important to a wide range of applied areas such as cell processes, drug release, ecology, spread of diseases, industrial catalytic processes, transport of contaminants in the environment, chemistry in interstellar media, to mention a few.