Question

Let \(u(x, t)\) have continuous derivatives through the second order in \(x\) and \(t\) for \(t \geq 0\) and \(0 \leq x \leq \pi\) and let \(u(0, t)=u(\pi, t)=0\). Prove that if \(u(x, t)\) satisfies the heat equation (10.91) for \(t>0,0

Short answer

Short Answer: The form of the heat equation with given boundary conditions is \(u(x, t) = \sum_{n=1}^{\infty} C_ne^{-k(n\pi)^2t}\sin(n\pi x)\), where \(C_n\) are constants. It was proven by recalling a result from Problem 6 following Section 10.8 that allows expressing any function satisfying the heat equation in a linear combination of functions of the given form, then rewriting the heat equation in terms of this form, and verifying that the rewritten equation is satisfied for the given domain.

Step-by-step solution

Heat Equation and Boundary Conditions

The heat equation (10.91) is given by: \begin{equation} \frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2}. \end{equation} The boundary conditions are given by: \(u(0, t)=u(\pi, t)=0\).

Write down the form (10.99)

We want to prove that \(u(x, t)\) has the form: \begin{equation} u(x, t) = \sum_{n=1}^{\infty} C_ne^{-k(n\pi)^2t}\sin(n\pi x), \end{equation} where \(C_n\) are constants.

Recall the result from Problem 6 following Section 10.8

The result from Problem 6 following Section 10.8 states that any function \(u(x,t)\) satisfying the heat equation and given boundary conditions can be expressed as a linear combination of functions of the form: \(\sin(n\pi x)e^{-(n\pi)^2kt}\).

Rewrite the heat equation in terms of the form (10.99)

Using the result from Step 3, rewrite the heat equation (10.91) as: \begin{equation} \frac{\partial}{\partial t}\sum_{n=1}^{\infty} C_ne^{-k(n\pi)^2t}\sin(n\pi x) = k \frac{\partial^2}{\partial x^2}\sum_{n=1}^{\infty} C_ne^{-k(n\pi)^2t}\sin(n\pi x). \end{equation}

Verify the rewritten equation

Now, we need to verify that the rewritten equation in Step 4 is satisfied for \(t>0,00,0<x<\pi\), since both sides are equal. Thus, we have proven that \(u(x, t)\) has the form (10.99).

Key concepts

Understanding Boundary Conditions

In mathematics, boundary conditions are constraints that define the behavior of a function on the boundary of its domain. For the heat equation problem, boundary conditions specify the values that a solution must satisfy along the edges of the region in consideration. For example, in the given exercise, it is noted that \(u(0, t) = 0\) and \(u(\pi, t) = 0\).
These conditions imply that the temperature at both ends of the spatial domain \([0, \pi]\) is zero for all times \(t > 0\).
This is characteristic of a rod whose ends are kept at constant temperatures, often due to an external control such as contact with ice.

Boundary conditions are crucial because they have a significant impact on the form of the solution. They determine the coefficients in the solution's expression, typically leading to a solution that matches the initial conditions while respecting the constraints imposed by the boundaries.

• They provide necessary physical restrictions, ensuring a realistic solution that fits the problem's scenario.
• They affect how Fourier series are constructed and what form they will take.

Understanding and applying the right boundary conditions is key to finding the accurate solution of a partial differential equation like the heat equation.

Exploring Fourier Series

A Fourier series is a way to represent a function as a sum of sine and cosine functions. These series are incredibly important in solving partial differential equations, especially in problems involving periodic functions or localized conditions, like our heat equation example.
In the solution, the function \(u(x, t)\) is expressed as a Fourier series involving sine terms:\[u(x, t) = \sum_{n=1}^{\infty} C_n e^{-k(n\pi)^2t}\sin(n\pi x)\]
This specific form is chosen because the boundary conditions dictate a zero value at both \(x = 0\) and \(x = \pi\), meaning the sine terms naturally fit as they are zero at those points.

A Fourier series is useful because:

• It allows the problem to be broken into simpler, orthogonal components that can be solved individually.
• Solutions can be constructed for each frequency component, and added to find the overall solution.

Fourier series thus make it simpler to handle complex boundary conditions and obtain a complete solution. In our case, having identified the solution form using sinusoidal terms ensures compatibility with the physical context defined by boundary conditions.

Diving Into Partial Differential Equations

Partial differential equations (PDEs) involve multiple independent variables and their partial derivatives. They are fundamental in describing numerous physical phenomena such as sound, heat, diffusion, and much more.
The heat equation is a classic example of a PDE and is often used to model the distribution of heat in a given region over time.

In the exercise, the heat equation is expressed as:
\[\frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2}\]
This formulation means that the change in temperature \(u\) over time \(t\) is proportional to the spatial curvature of the temperature profile at any point \(x\), with \(k\) being a constant representing thermal conductivity.

• PDEs like the heat equation require boundary and initial conditions to define a unique solution.
• They can often be solved using methods such as separation of variables, Fourier series, and numerical methods.

Solving PDEs involves understanding how a system evolves under dynamic conditions, making them indispensable tools in both theoretical and applied mathematics. The ability to solve such equations provides insight into how systems behave in response to various forces and constraints.