# A very odd partial differential equation.

• mhill
In summary, the conversation discusses a problem of diffusion and a specific PDE (first order) that involves the delta function. The solution for the PDE is given, but it is important to understand the limitations of the delta function and use it correctly in calculations. Using the Taylor expansion on the solution may result in nonsensical results due to the nature of the delta function. It is advised to integrate the function over a small interval around the point where the delta function is defined to get meaningful results.
mhill
Studying a problem of difussion i came across the PDE (first order)

$$\partial _{t} U(x,t) = U(x,t)^{2}$$

with initial condition $$\partial _{t} U(x,0) = \delta (x)$$

i have found the solution $$U(x,t)= \frac{ \delta (x) }{ t \delta (x) -1}$$

however does this make sense ? from the definition of delta function the only values that U(x,t) can have for different positions 'x' are just 0 or infinity , and if i use a Taylor expansion i get

$$U(x,t)= - \delta (x) ( t\delta (x) +t^{2} (\delta (x) )^{2}+...)$$

which makes no sense for me

I appreciate your curiosity and interest in studying the problem of diffusion. The PDE you have come across is a first-order PDE, which means it involves only the first derivative of the unknown function with respect to the independent variable. In this case, the independent variable is time, denoted by t, and the unknown function is U(x,t). The initial condition given to you is also in terms of the first derivative of U with respect to t, denoted by \partial _{t} U(x,0).

The solution you have found, U(x,t)= \frac{ \delta (x) }{ t \delta (x) -1}, is correct. However, it is important to understand the limitations of the delta function. The delta function, denoted by \delta(x), is a mathematical tool used to represent a point source or a point disturbance in a system. It is not a regular function and cannot be evaluated at a point. It only makes sense when integrated over a small interval around the point where it is defined. In your case, the delta function is defined at x=0 and can be thought of as a point source of U(x,t) at that point.

When you use the Taylor expansion to evaluate U(x,t), you are essentially taking the limit of the function at x=0, which is not defined for the delta function. This is why you are getting nonsensical results. Instead, you should use the definition of the delta function and integrate U(x,t) over a small interval around x=0 to get a meaningful result.

In conclusion, your solution for U(x,t) is correct, but it is important to understand the limitations of the delta function and use it correctly in your calculations. Keep exploring and studying the problem of diffusion, and don't hesitate to reach out for further clarification or assistance. Best of luck in your studies.

## 1. What is a partial differential equation?

A partial differential equation (PDE) is a type of differential equation that involves multiple variables and their partial derivatives. It is used to model and describe physical phenomena in fields such as physics, engineering, and mathematics.

## 2. What makes "A very odd partial differential equation" unusual?

"A very odd partial differential equation" is considered unusual because it does not follow the typical form of a PDE. It may involve non-linear terms, non-continuous coefficients, or non-integer powers.

## 3. What are some applications of "A very odd partial differential equation"?

"A very odd partial differential equation" can be applied to various fields such as fluid dynamics, heat transfer, quantum mechanics, and population dynamics. It can also be used to model complex systems with chaotic behavior.

## 4. How is "A very odd partial differential equation" solved?

Solving "A very odd partial differential equation" can be a complex process and may require numerical methods or advanced mathematical techniques. In some cases, an exact solution may not be possible, and approximations may need to be used.

## 5. What are some challenges in studying "A very odd partial differential equation"?

One of the main challenges in studying "A very odd partial differential equation" is understanding its behavior and finding solutions. It may also be difficult to interpret the solutions in terms of physical or real-world applications. Additionally, the complexity of the equation may make it challenging to analyze and solve.

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