How to represent the diffusion of ink in water with differential equations?

In the ordinary differential equations of this course, the content of the specific solution of the equation is not the focus, and the real essence lies in the qualitative analysis, including existential uniqueness, stability, and so on. Because most of the equations cannot be solved analytically, but we still have to analyze the properties of the solutions when they cannot be solved concretely, which is the basic spirit of modern ordinary differential equation theory and partial differential equation theory. As for not understanding the lipschitz condition, I can only say that the foundation of the number of points is not solid enough, lipschitz is defined in the number of points continuously, and the uniqueness of the picard iteration proves that it does not go beyond the scope of the number of points.

Knowing how to solve equations but not proving" means that it is likely that they have not mastered abstract mathematical concepts well enough, and can only deal with concrete equations, which is very uncomfortable with this general form. If you want to solve it, I don't say any good way; Some people can get started quickly, and some people can only watch and practice more; It's a good idea to go back and review the points, especially if it's about lipschitz continuity. <>

Excerpt from the <> of "Methods of Mathematical Physics" edited by Liang Kunmiao

This equation describes not only the transport of particles, but also the conduction of heat in a medium.

It is the first-order linear equation, and the natural heat dissipation of boiling water, and the current discharge of a fully charged capacitor resistor, the paratrooper leaves the plane and lands is exactly the same expression, and the rate of change of the concentration of the essence of black juice (first-order differential) is a negative value"Proportional to the current concentration of black juice, the result of defolding is an exponential function with natural e as the base constant, please contact friends who like calculus.

The Langevin equation clarifies the microscopic mechanism of diffusion from the perspective of force, and the physical picture behind it is very clear, and the Langevin equation itself is also a random differential equation. The Focke-Planck equation (which degenerates into Fick's second law when the applied potential field is a constant value) is a parabolic partial differential equation that is equivalent to statistically processing the random forces in a random differential equation (it seems to be called coarse-grained), and finally obtaining the statistical result, the evolution of concentration (or probability density) over time and space. In addition, there are the main equations (the physical picture behind them is a fine equilibrium) and the Chepman-Kolmogolov equations (which are essentially the so-called full probability formulas), which are supposed to be integral equations.

These four equations can be extrapolated from each other (of course, the Langevin equation must be overdamped, i.e. the inertia term can be ignored).

Finding the general solution of ordinary differential calculus has been shown to be infeasible, and only the general solution of a specific equation is found. So personally, I don't think there is a way to find a solution. There is also a method of solving calculus definite integration by finding the original function by transforming the number fraction of higher numbers, which is not realistic.

Now basically, numerical analysis and numerical solutions to differential equations are two practical tools to solve the above two problems.

There are also some physical conditions, such as the relationship between diffusion rate and concentration.

Sick?This formula can only describe the formal effect formed under the action of a certain function, and can only be used in the form that is formed.

fick'If S Law remembers correctly, this can be used to find an approximate solution to the specific diffusion given boundary conditions.

In general, it is an NS equation, which can be simplified depending on the situation: for example, only diffusion is considered and no convection is considered

Methods of Mathematical Physics.

Go back and review Professor Liang Kunmiao's "Methods of Mathematical Physics" by Professor Liang Kunmiao of Nanjing University or "Methods of Mathematical Physics" by Professor Guo Dunren of Peking University.

I have learned everything, but I have forgotten it.

Forget about physics and think of mathematics.

I don't know about ink diffusion, but I think it's the same as sugar diffusion in water, which was done by Einstein's new method of measuring the molecular size.

You can only figure out what the greatest possible look like.

Fick's law should be able to be found.

Diffusion equations, spherical coordinate systems, initial concentration distribution functions with Dirac functions, this ** just meets your needs, and there are examples.

Partial Differential Equation Models for Diffusion Problems - Handout Tutorial - Daoist Baba.

Diffusion and Brownian motion are not opposites. There is no either/or relationship between the two. Rather, during the diffusion of ink in water, the particles of ink do Brownian motion.

It is diffusion, and Brownian motion is not visible to the naked eye.

It's just one way to create a texture. It is a bit similar to transfer printing, taking advantage of the easy diffusion of oil smoke in 1 ink on the water surface.

2 Characteristics of rice paper for adsorption and fixation of ink.

Rendering is a technique in Chinese painting, and a large area is painted flat without revealing the brush marks. That's where tools are needed – like a brush or a wide row pen. The process of making the texture is not so particular, with some ink dripping in the water, stirring it slightly, and covering the surface of the water with a piece of rice paper of a moderate size - the flowing ink marks are instantly fixed on the rice paper.

This natural texture effect is often used to decorate background patterns. Whether it can come in handy is a great deal of chance.

d abc belongs to the macroscopic particles of the mechanical movement of jujube skin round, put a few grains of coarse salt into a cup with water, after a period of time the whole cup of water becomes salty, belongs to the diffusion phenomenon, choose d

The large characters are used for raw Xuan, the small characters are used for cooked Xuan, and when writing regular script, the ink is slightly drier.

You can be sure that you are using raw rice paper, which is characterized by the following: 洇. If you want to use it, the tip of the pen should have less moisture, and the pen should not be too slow. The longer the practice, the more proficient.

Because the higher the temperature, the more intense the molecular movement.

Therefore, the molecules of the ink are more easily dispersed into the water in the hot water.

That is, it dissolves quickly.

First of all, it's not decomposition, it's diffusion or dispersion.

Because in a material with a high temperature, the internal energy is relatively large, and the rate of molecular motion is relatively fast, note that the velocity is not the velocity (vector), which causes the intense Brownian motion, so the ink spreads quickly in hot water.

In the case of black ink, the water temperature is too high, and a small amount of precipitation will appear.

Because the higher the temperature, the more intense the molecular movement. The faster the diffusion rate, so the faster it decomposes.

It must be because of the high temperature and fast molecular movement. Because the diffusion of ink in water is the result of molecular movement.

Whoever came up with it, it's really damn. After three days of painting, you're going to have to spend a lot more.

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A drop of red ink is dropped in a glass of water, and after a period of time, the whole cup of water turns red, this phenomenon is called diffusion phenomenon, which shows that water molecules are constantly doing irregular movements, so choose C