# partial differential equation heat equation

I received this homework problem from my professor, but I am unsure of how to get it going. It doesn't correlate exactly with anything in our book.

Consider the heat equation for $$u(x,t)$$

$$u_t = ku_{xx}, -\infty < x < \infty, t > 0 \tag{1}$$

$$u(x,0) = f(x) \tag{2}$$

assume that a network of sensors is uniformly distributed in space, at a spatial increment of $$\Delta x$$, and that the smallest absolute amount of change (variation) in $$u$$ that can be detected by a sensor (sensor's activity) is a constant

$$|\delta u|_{min} = \epsilon > 0 \tag{3}$$

consider the problem detecting the presence of an impulse variation in the initial condition

$$\delta f(x) = a \delta(x - x_{0}) \tag{4}$$

where $$a > 0$$ is a constant coefficient and the location $$x_0$$ of the impulse may be unknown.

1. If $$\Delta x = 1$$, find the maximum value of $$\epsilon$$ s.t. any impulse $$(4)$$ in the initial condition with $$a \geq 1$$ will be detected by at least one sensor.

2. Given $$\epsilon > 0$$ find the maximum value of $$\Delta x$$ to guarantee that any impulse $$(4)$$ in the initial condition with $$a \geq 1$$ will be detected by at least one sensor.

3. Given the network of parameters $$\epsilon > 0$$, $$\Delta x > 0$$, find the minimum value $$A > 0$$ s.t. any impulse $$(4)$$ in the initial condition with $$a \geq A$$ will be detected by at least one sensor.

As the problem is a linear by denoting $$u$$ the solution with $$f$$ as initial condition and $$u + \delta u$$ the solution with $$f + \delta f$$ as initial condition, you have that $$\delta u$$ is solution of $$(\delta u)_t = k(\delta u)_{xx}$$ $$\delta u(x,0)=\delta f(x) a \delta(x-x_0)$$ in this case there is a well known explicit solution, the heat kernel so you obtain $$\delta u(x,t)=\frac{a}{\sqrt{4 \pi t}}e^{-\frac{(x-x_0)^2}{4t}}.$$ When $$x$$ is fixed the study of the function $$t \mapsto \delta u(x,t)$$ show that the maximum is $$a \frac{1}{\sqrt{2 \pi e}} \frac{1}{|x-x_0|}$$ so in the worst case the distance betwwen $$x_0$$ and the closest sensor is $$\frac{ \Delta x}{2}$$ the initial condition will be detected as long as $$a \sqrt{\frac{2}{ \pi e}} \frac{1}{\Delta x} \geq \epsilon.$$