Question

In this exercise take it as given that the infinite series \(\sum_{n=1}^{\infty} n^{p} e^{-q n^{2}}\) converges for all \(p\) if \(q>0\), and, where appropriate, use the comparison test for absolute convergence of an infinite series. Let $$u(x, t)=\sum_{n=1}^{\infty} \alpha_{n} e^{-n^{2} \pi^{2} a^{2} t / L^{2}} \sin \frac{n \pi x}{L}$$ where $$\alpha_{n}=\frac{2}{L} \int_{0}^{L} f(x) \sin \frac{n \pi x}{L} d x$$ and \(f\) is piecewise smooth on \([0, L]\). (a) Show that \(u\) is defined for \((x, t)\) such that \(t>0\). (b) For fixed \(t>0\), use Theorem 12.1.2 with \(z=x\) to show that $$u_{x}(x, t)=\frac{\pi}{L} \sum_{n=1}^{\infty} n \alpha_{n} e^{-n^{2} \pi^{2} a^{2} t / L^{2}} \cos \frac{n \pi x}{L}, \quad-\infty0\),$$ u_{x x}(x, t)=-\frac{\pi^{2}}{L^{2}} \sum_{n=1}^{\infty} n^{2} \alpha_{n} e^{-n^{2} \pi^{2} a^{2} t / L^{2}} \sin \frac{n \pi x}{L}, \quad-\inftyt_{0}>0,\) where \(t_{0}\) is an arbitrary positive number. Then argue that since \(t_{0}\) is arbitrary, the conclusion holds for all \(t>0\). (e) Conclude from (c) and (d) that $$u_{t}=a^{2} u_{x x}, \quad-\infty0$$ By repeatedly applying the arguments in (a) and (c), it can be shown that \(u\) can be differentiated term by term any number of times with respect to \(x\) and/or \(t\) if \(t>0\).

Short answer

In this exercise, we were given an infinite series for a function \(u(x,t)\) and asked to prove various properties. We followed a step-by-step approach to find the first and second order partial derivatives of \(u\) with respect to \(x\) and \(t\), and showed that \(u\) is defined for \(t>0\). Finally, we concluded that \(u_t = a^2u_{xx}\) for all \(x\) and \(t>0\).

Step-by-step solution

(a) Show that \(u\) is defined for \(t>0\)

For showing that \(u(x,t)\) is defined for \(t>0\), we need to prove that the infinite series converges for \(t>0\). We can start by considering the exponent of the infinite summation: \(e^{-n^2\pi^2 a^2 t/L^2}\). As \(t>0\), the exponent \((-n^2\pi^2 a^2 t/L^2)\) will always be negative, meaning the exponential will always be less than \(1\). Therefore, we can conclude that the infinite summation will converge for \(t>0\) since we are given the series converges for all \(p\) if \(q>0\). Hence, \(u(x,t)\) is defined for \(t>0\).

(b) Find \(u_x(x,t)\)

We will use Theorem 12.1.2 to differentiate term by term with respect to \(x\). Theorem 12.1.2 states that if an infinite series converges absolutely and uniformly for \(z\) in a domain \(D\), then its derivative can be computed term by term in \(D\). Differentiating the given function, \(u(x,t)\), with respect to \(x\): $$u_x(x, t) = \frac{d}{dx}\left(\sum_{n=1}^{\infty} \alpha_{n} e^{-n^{2}\pi^{2} a^{2} t/L^{2}} \sin \frac{n\pi x}{L}\right) = \sum_{n=1}^{\infty} \frac{d}{dx}(\alpha_{n} e^{-n^{2}\pi^{2} a^{2} t/L^{2}} \sin \frac{n\pi x}{L})$$ As the derivative and summation can be interchanged, we have: $$u_x(x, t) = \sum_{n=1}^{\infty} \alpha_{n} e^{-n^{2}\pi^{2} a^{2} t/L^{2}} \frac{d}{dx} \sin \frac{n\pi x}{L} = \frac{\pi}{L} \sum_{n=1}^{\infty} n \alpha_{n} e^{-n^{2}\pi^{2} a^{2} t/L^{2}} \cos \frac{n\pi x}{L}$$

(c) Find \(u_{xx}(x,t)\)

Now, we apply theorem 12.1.2 with \(z=x\) again to differentiate the result of part (b) with respect to \(x\): $$u_{xx}(x, t) = \frac{d}{dx}\left(\frac{\pi}{L} \sum_{n=1}^{\infty} n \alpha_{n} e^{-n^{2} \pi^{2} a^{2} t / L^{2}} \cos \frac{n \pi x}{L}\right) = \frac{\pi}{L} \sum_{n=1}^{\infty} n \alpha_{n} e^{-n^{2} \pi^{2} a^{2} t / L^{2}} \frac{d}{dx}\cos \frac{n \pi x}{L}$$ As the derivative and summation can be interchanged, we have: $$u_{xx}(x, t) = -\frac{\pi^2}{L^2} \sum_{n=1}^{\infty} n^2 \alpha_{n} e^{-n^{2} \pi^{2} a^{2} t / L^{2}} \sin \frac{n \pi x}{L}$$

(d) Find \(u_t(x,t)\)

For this part, we apply Theorem 12.1.2 with \(z=t\) to differentiate the given function, \(u(x,t)\), with respect to \(t\) and for an arbitrary positive number \(t>t_0>0\): $$u_t(x, t) = \frac{d}{dt}\left(\sum_{n=1}^{\infty} \alpha_{n} e^{-n^{2}\pi^{2} a^{2} t/L^{2}} \sin \frac{n\pi x}{L}\right) = \sum_{n=1}^{\infty} \alpha_{n} \frac{d}{dt}(e^{-n^{2}\pi^{2} a^{2} t/L^{2}}) \sin \frac{n\pi x}{L}$$ As the derivative and summation can be interchanged, we have: $$u_t(x, t) = -\frac{\pi^2 a^2}{L^2} \sum_{n=1}^{\infty} n^2 \alpha_{n} e^{-n^{2} \pi^{2} a^{2} t / L^{2}} \sin \frac{n \pi x}{L}$$ Since the result holds for the arbitrary positive number \(t_0>0\), it must hold for all \(t>0\).

(e) Conclude that \(u_t=a^2u_{xx}\) for all \(t>0\)

We have found \(u_{xx}(x,t)\) and \(u_t(x,t)\) in previous steps. Now we can compare them: $$u_{xx}(x, t) = -\frac{\pi^2}{L^2} \sum_{n=1}^{\infty} n^2 \alpha_{n} e^{-n^{2} \pi^{2} a^{2} t / L^{2}} \sin \frac{n \pi x}{L}$$ $$u_t(x, t) = -\frac{\pi^2 a^2}{L^2} \sum_{n=1}^{\infty} n^2 \alpha_{n} e^{-n^{2} \pi^{2} a^{2} t / L^{2}} \sin \frac{n \pi x}{L}$$ Comparing both equations, we can see that \(u_t = a^2u_{xx}\) for all \(x\) and \(t>0\).

Key concepts

Infinite Series Convergence

Understanding infinite series convergence is crucial when dealing with complex expressions. An infinite series consists of adding an infinite number of terms together. In mathematics, determining whether such a series converges or diverges is significant. Convergence means that as you keep adding more terms, the series approaches a fixed number, known as the limit.
Several tests help assess convergence. For a series to be relevant, it often must converge. In the given exercise, the series \( \sum_{n=1}^{\infty} n^{p} e^{-q n^{2}} \) is considered. It is provided that this series converges for all \( p \) if \( q > 0 \). This ensures that despite the infinite number of terms, the sum doesn’t spiral off to infinity but rather settles near a specific value.
For practical applications, like solving the heat equation, knowing that the involved series converges enables us to perform calculations reliably without worrying about undefined results.

Comparison Test

The comparison test is a useful tool in determining the convergence of series. It works by comparing a series whose convergence is unknown with one whose convergence status is already known. - The comparison test states that if \( 0 \leq a_n \leq b_n \) and \( \sum b_n \) converges, then \( \sum a_n \) also converges.- Conversely, if \( a_n \geq b_n \geq 0 \) and \( \sum b_n \) diverges, then \( \sum a_n \) diverges as well.
In the exercise, this test can be deployed to confirm the absolute convergence of certain series components by comparing them to well-understood convergent series. This approach provides a foundational understanding of the series’ behavior, paving the way for accurate approximations and inferences in problems such as differential equations.

Boundary Conditions

Boundary conditions are essential in solving differential equations, as they define limitations and restrictions within which a solution must remain valid. They are especially relevant in physical problems, like those modeled by the heat equation.
- Boundary conditions can be: - **Dirichlet conditions**, where the solution is fixed to take specific values at the boundary. - **Neumann conditions**, where the derivative of the solution is specified at the boundary.- These conditions ensure uniqueness and stability of the solution, preventing multiple solutions from appearing for the same problem scope.
In the context of the heat equation exercise, boundary conditions determine how the solution behaves at the endpoints of the domain. For instance, the way \( f(x) \) is defined over the interval \([0, L]\) sets up how the initial distribution impacts the series solution \( u(x, t) \). They ensure the solution not only solves the equation, but accurately reflects the physical situation being modeled.

Heat Equation

The heat equation is a fundamental partial differential equation (PDE) in mathematical physics. It describes how heat diffuses through a medium over time.
The heat equation is given by: \[ u_t = a^2 u_{xx} \]where:- \( u \) represents the temperature at point \( x \) and time \( t \),- \( a^2 \) is a constant related to the thermal diffusivity of the material.
This equation links the rate of change of temperature in time (\( u_t \)) to the spatial curvature of temperature (\( u_{xx} \)).
Solving the heat equation requires applying techniques like Fourier series, as demonstrated in the exercise. Identifying terms like \( \alpha_n \) or understanding series convergence is vital. By aligning solutions with physical conditions set by boundary values, this enables the vector \( u(x, t) \) to mirror realistic temperature distributions, adapting seamlessly over dynamics confined by given initial conditions.