A very odd partial differential equation.

[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

 

[jvicens]

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.