Da Wikipedia, l'enciclopedia libera. In analisi matematica il modello reazione-diffusione è l'equazione parabolica la cui omogenea associata è l'equazione della diffusione: il termine di sorgente viene chiamato termine di reazione, in quanto nell'applicazione più frequente dove la funzione incognita è la concentrazione di un composto è dovuto ad una reazione chimica in cui il composto è coinvolto. Modello generale di Reazione-Diffusione[modifica | modifica sorgente] Se nell'equazione di diffusione per la funzione u(x, t) è presente un termine non omogeneo di reazione si ha che l'equazione diviene: Reazione-diffusione Reazione-diffusione
Leggi di Fick Leggi di Fick Da Wikipedia, l'enciclopedia libera. In analisi matematica, l'equazione del calore, anche detta equazione di diffusione, è un'equazione differenziale alle derivate parziali che trova nelle scienze svariate applicazioni: per esempio in fisica modellizza l'andamento della temperatura in una regione dello spazio-tempo sotto opportune condizioni, e in chimica l'andamento della concentrazione chimica di una specie. La buona posizione dei problemi associati all'equazione del calore segue inoltre dall'analisi di buona posizione di un problema parabolico, di cui l'equazione è un classico esempio.
Anisotropic diffusion In image processing and computer vision, anisotropic diffusion, also called Perona–Malik diffusion, is a technique aiming at reducing image noise without removing significant parts of the image content, typically edges, lines or other details that are important for the interpretation of the image.[1][2][3] Anisotropic diffusion resembles the process that creates a scale space, where an image generates a parameterized family of successively more and more blurred images based on a diffusion process. Each of the resulting images in this family are given as a convolution between the image and a 2D isotropic Gaussian filter, where the width of the filter increases with the parameter. This diffusion process is a linear and space-invariant transformation of the original image. Anisotropic diffusion
Next: Diffusion Tensor Interpolation Up: Methods Previous: Hue-balls and Deflection Mapping Subsections Introduction Our goal in this section is to use reaction-diffusion textures as a means of visualizing three-dimensional diffusion tensor data. We start by describing a simple model of reaction-diffusion texture that works in two and three dimensions, and then discuss how to modify its calculation to make the texture reflect measured diffusion tensor data. Reaction-Diffusion Textures Reaction-Diffusion Textures
Gray-Scott reaction-diffusion java applet How can patterns be formed by chemical reactions? A first answer to this question was provided by Alan Turing, who specified mathematical conditions necessary for it to be possible to form spatial patterns in two-component reaction-diffusion systems. The java applet on this page simulates diffusion and reaction between two chemicals U and V. Reaction: U + 2 V -> 3 V The chemical U diffuses faster than V, and is used as fuel to produce chemical V, while chemical V catalyzes its own production. Gray-Scott reaction-diffusion java applet
Most of these images are linked to an applet with the same parameters so you can watch and interact with the pattern evolution. Varying F, k, and diffusion parameters Parameters: F=0.035&k=0.065&diffuseU=0.16&diffuseV=0.08& simwidth=400&simheight=400& wrap=false&varyPhysics=true& diffuseU2=0.06&diffuseV2=0.03&F2=0.06&k2=0.062 This image shows how physical parameters can vary within a single environment. Here the diffusion constants vary on the vertical axis, and the reaction constants vary along the horizontal axis. Java demo: Gray-Scott Reaction-Diffusion Java demo: Gray-Scott Reaction-Diffusion
Introduction Instructions: A click anywhere in the crescent-shaped complex region will take you to a page with images, a movie and a specific description. Each grid square leads to a different page. I have special pages for the uskate-world and certain other exotic patterns. This web page serves several purposes: Reaction-Diffusion by the Gray-Scott Model: Pearson's Parameterization at MROB Reaction-Diffusion by the Gray-Scott Model: Pearson's Parameterization at MROB
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Nervous System – explorations in generative design and natural phenomena » reaction diffusion Nervous System has released Reaction, their first collection of housewares. The collection includes porcelain cups and plates and matching 3D printed lamps. The pieces are intricately embossed with intertwining patterns […] Read Article → As part of a new experimental project we are working on we had to create a reaction-diffusion system that can run on a constantly changing surface. Nervous System – explorations in generative design and natural phenomena » reaction diffusion
Cellular Automata – How the Leopard gets its spots. « Jonathan Pace I’ve been reading up on how you can take a set of individual ‘things’, give each thing a rule to iterate, then sit back and watch them exhibit some interesting behaviour. These systems are classified as ‘Cellular Automata’ (http://en.wikipedia.org/wiki/Cellular_automata) and can be simulated in a number of different ways. One such simulation is an array of pixels in an image. By initially randomly assigning each pixel a value of 1 or 0 (corresponding to white or black) we end up with a ‘noisy’ image like this. Now come the rules. Cellular Automata – How the Leopard gets its spots. « Jonathan Pace
Virtual Laboratory for Simulation and Analysis of Propagating Interfaces more... Sample Numerical Simulation Snapshots Produced with VLSAπ SIMULATION PICS -  Virtual Laboratory for Simulation and Analysis of Propagating Interfaces SIMULATION PICS -  Virtual Laboratory for Simulation and Analysis of Propagating Interfaces
Turing Goes GalacticOnce one starts to look, there seems to be no end to Turing patterns: their forms can be seen in weather systems, the distribution of vegetation across landscapes and even the constellations of galaxies.Image: Galaxy N51, the Whirlpool Galaxy. (European Space Agency)Turing Patterns in CellsTuring patterns can involve not just chemicals, but large, complex systems in which each unit — for example, a cell — is distributed like molecules of pigment.Pictured is a Turing pattern of cells in Dictyostelium, or a slime mold.Image: National Institutes of HealthTuring Patterns in 3-DThat markings on animals are produced by Turing systems of pigments is now generally accepted, but the origin of what appear to be Turing patterns in more complex settings — such as limb and tooth and lung development — is still debated.A basic step towards proving the existence of these three-dimensional Turing patterns is demonstrating a three-dimensional pattern in the lab. Alan Turing’s Patterns in Nature, and Beyond | Wired Science
Gierer-Meinhardt model The Gierer-Meinhardt model Figure 1: Short-range activator and long-range inhibitor in Gierer-Meinhardt model \frac{\partial a}{\partial t} = \rho\frac{a^2}{h} - \mu_a a + D_a \frac{\partial^2 a}{\partial x^2} + \rho_a
Computer simulations Equations used for RD systems14 can be generally described as where u and v are the concentrations of hypothetical factors, f and g are the reaction kinetics and Du and Dv are the hypothetical diffusion coefficients (or their mathematical equivalents) for u and v, respectively15, 37. The reaction rate R was introduced for convenience of parameter adjustment. The reaction kinetics and parameters used for each simulation are as follows: The linear model:14 Blending of animal colour patterns by hybridization : Nature Communications
kondo labo Simulation Programs (to download, just click the labyrinth pattern) reaction-diffusion simulator for windows Vista (much faster): This program calculates the original Turing's equation reaction-diffusion simulator for XP (or older OS): This program calculates the original Turing's equation reaction-diffusion simulator for XP (or older OS) :This program calculates four famous types of RD models; GM model, Schnakenberg model, GS model and oregonator. PapersPigment pattern formation by contact-dependent depolarization.Inaba M, Yamanaka H, Kondo S.