# How do we show that it is finite?

• MHB
• evinda
In summary, the solution of the initial and boundary value problem for the heat equation in a bounded interval $[0,\ell]$ with homogenous Neumann boundary conditions and $k=1$ holds the following inequality:$$|u(x,t)-\frac{1}{\ell} \int_0^{\ell} \phi(x) dx| \leq C e^{-\left( \frac{\pi}{\ell}\right)^2 t}, 0 \leq x \leq \ell, t \geq 1$$
evinda
Gold Member
MHB
Hello! (Wave)

We consider the initial and boundary value problem for the heat equation in a bounded interval $[0, \ell]$ with homogenous Neumann boundary conditions and $k=1$, and we suppose that the initial value $\phi$ is piecewise continuously differentiable and that $\phi'(0)=\phi'(\ell)=0$. I want to show that for the solution holds the following inequality.

$$|u(x,t)-\frac{1}{\ell} \int_0^{\ell} \phi(x) dx| \leq C e^{-\left( \frac{\pi}{\ell}\right)^2 t}, 0 \leq x \leq \ell, t \geq 1$$

where $C$ is a constant that depends on the quantity $\int_0^{\ell} [\phi(x)]^2 dx$.

I have done the following.

Our initial and boundary value problem is the following:

$\left\{\begin{matrix} u_t=u_{xx}, & 0<x<\ell,t>0,\\ u_x(0,t)=u_x(\ell,t)=0, & t \geq 0,\\ u(x,0)=\phi(x), & 0 \leq x \leq \ell \end{matrix}\right.$

We know that the solution of the problem is

$$u(x,t)=\frac{a_0}{2}+ \sum_{n=1}^{\infty} a_n e^{-\left( \frac{n \pi}{\ell}\right)^2 t} \cos{\left( \frac{n \pi x}{\ell}\right)}$$

with $a_n=\frac{2}{\ell} \int_0^{\ell} \phi(x) \cos{\left( \frac{n \pi x}{\ell}\right)}dx$ and $\phi(x)=\frac{a_0}{2}+\sum_{n=1}^{\infty} a_n \cos{\left( \frac{n \pi x}{\ell}\right)}$.I have shown that $$\left| u(x,t)-\frac{1}{\ell} \int_0^{\ell} \phi(x) dx\right| \leq e^{-\left( \frac{\pi}{\ell}\right)^2 t} \left( \frac{2}{\ell} \int_0^{\ell} \phi^2(x) dx\right)^{\frac{1}{2}} \left( \sum_{n=1}^{\infty} e^{-2 \left( \frac{\pi}{\ell}\right)^2 (n^2-1)}\right)^{\frac{1}{2}}$$How can we show that $\int_0^{\ell} \phi^2(x) dx<+\infty$ given the information that $\phi'(0)=\phi'(\ell)=0$ and not that $\phi(0)=\phi(\ell)=0$ ?

evinda said:
We consider the initial and boundary value problem for the heat equation in a bounded interval $[0, \ell]$ with homogenous Neumann boundary conditions and $k=1$, and we suppose that the initial value $\phi$ is piecewise continuously differentiable and that $\phi'(0)=\phi'(\ell)=0$.

How can we show that $\int_0^{\ell} \phi^2(x) dx<+\infty$ given the information that $\phi'(0)=\phi'(\ell)=0$ and not that $\phi(0)=\phi(\ell)=0$ ?

Hey evinda!

As $\phi$ is continuous on the bounded interval $[0,\ell]$, doesn't that imply that $\phi$ is bounded? (Wondering)

Suppose $|\phi(x)|<M$ on the interval.
Doesn't that imply that $\left|\int_0^\ell \phi^2(x)\,dx\right| < M^2 \ell < +\infty$? (Wondering)

I like Serena said:
Hey evinda!

As $\phi$ is continuous on the bounded interval $[0,\ell]$, doesn't that imply that $\phi$ is bounded? (Wondering)

So don't we need to know anything about $\phi(0)$ and $\phi(\ell)$ , in order to deduce that $\phi$ is bounded? (Thinking)

evinda said:
So don't we need to know anything about $\phi(0)$ and $\phi(\ell)$ , in order to deduce that $\phi$ is bounded?

No. It suffices that they exist, which is implied by the fact that $\phi$ is a function on a closed interval.
And its continuity implies that the function is bounded on the interval itself. (Nerd)

I like Serena said:
No. It suffices that they exist, which is implied by the fact that $\phi$ is a function on a closed interval.
And its continuity implies that the function is bounded on the interval itself. (Nerd)

Ah I see... thank you! (Smile)

## 1. What is meant by "finite"?

"Finite" refers to something that has a limit or end, and therefore is measurable or countable. It is the opposite of infinite, which has no limit or end.

## 2. How do we determine if something is finite?

To determine if something is finite, we need to observe its characteristics and determine if it has a limit or end. This can be done through measurement, counting, or logical reasoning.

## 3. What are some examples of finite things?

Examples of finite things include a typical human lifespan, the number of fingers on a hand, the amount of water in a glass, or the number of planets in our solar system. These things have a clear limit or end.

## 4. Can something be both finite and infinite?

No, something cannot be both finite and infinite. These terms are mutually exclusive - if something has a limit or end, it is finite, and if it has no limit or end, it is infinite.

## 5. Why is it important to show that something is finite?

Showing that something is finite allows us to understand its boundaries and limitations. This can help us make informed decisions and predictions, and can also aid in solving problems and developing new knowledge and technologies.

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