Question

Suppose \(\lambda\) and \(\mu\) are constants and either \(p_{n}(x)=\cos n \lambda x\) or \(p_{n}(x)=\sin n \lambda x,\) while either \(q_{n}(t)=\cos n \mu t\) or \(q_{n}(t)=\sin n \mu t\) for \(n=1,2,3, \ldots\). Let $$ u(x, t)=\sum_{n=1}^{\infty} k_{n} p_{n}(x) q_{n}(t). $$ where \(\left\\{k_{n}\right\\}_{n=1}^{\infty}\) are constants. (a) Show that if \(\sum_{n=1}^{\infty}\left|k_{n}\right|\) converges then \(u(x, t)\) converges for all \((x, t)\). (b) Use Theorem 12.1.2 to show that if \(\sum_{n=1}^{\infty} n\left|k_{n}\right|\) converges then (A) can be differentiated term by term with respect to \(x\) and \(t\) for all \((x, t) ;\) that is, $$ u_{x}(x, t)=\sum_{n=1}^{\infty} k_{n} p_{n}^{\prime}(x) q_{n}(t) $$ and $$ u_{t}(x, t)=\sum_{n=1}^{\infty} k_{n} p_{n}(x) q_{n}^{\prime}(t) $$ (c) Suppose \(\sum_{n=1}^{\infty} n^{2}\left|k_{n}\right|\) converges. Show that $$ u_{x x}(x, y)=\sum_{n=1}^{\infty} k_{n} p_{n}^{\prime \prime}(x) q_{n}(t). $$ and $$ u_{t t}(x, y)=\sum_{n=1}^{\infty} k_{n} p_{n}(x) q_{n}^{\prime \prime}(t). $$ (d) Suppose \(\sum_{n=1}^{\infty} n^{2}\left|\alpha_{n}\right|\) and \(\sum_{n=1}^{\infty} n\left|\beta_{n}\right|\) both converge. Show that the formal solution $$ u(x, t)=\sum_{n=1}^{\infty}\left(\alpha_{n} \cos \frac{n \pi a t}{L}+\frac{\beta_{n} L}{n \pi a} \sin \frac{n \pi a t}{L}\right) \sin \frac{n \pi x}{L} $$ of Equation 12.2.1 satisfies \(u_{t t}=a^{2} u_{x x}\) for all \((x, t)\). This conclusion also applies to the formal solutions defined in Exercises \(17,34,\) and 49 .

Short answer

#Answer# The formal solution converges and satisfies the equation \(u_{tt} = a^2 u_{xx}\) if both series \(\sum_{n=1}^{\infty}\left|k_{n}\right|\) and \(\sum_{n=1}^{\infty} n^{2}\left|k_{n}\right|\) converge.

Step-by-step solution

Absolute convergence of \(u(x, t)\).

We are given that the sum \(\sum_{n=1}^{\infty}\left|k_{n}\right|\) converges. Since \(\left| p_n(x) q_n(t) \right| \leq 1\), we can apply the comparison test to show that \(u(x, t) = \sum_{n=1}^{\infty} k_{n} p_{n}(x) q_{n}(t)\) converges. **Step 2: Using Theorem 12.1.2**

Term by term differentiation of \(u_x\) and \(u_t\)

Since the sum \(\sum_{n=1}^{\infty} n\left|k_{n}\right|\) converges, we can apply Theorem 12.1.2 to show that the series \(u_x(x, t) = \sum_{n=1}^{\infty} k_{n} p_{n}^{\prime}(x) q_{n}(t)\) and \(u_t(x, t) = \sum_{n=1}^{\infty} k_{n} p_{n}(x) q_{n}^{\prime}(t)\) can be differentiated term by term with respect to \(x\) and \(t\), respectively. **Step 3: Finding conditions for convergence of second derivatives**

Convergence of \(u_{xx}\) and \(u_{tt}\) for given conditions

Suppose \(\sum_{n=1}^{\infty} n^{2}\left|k_{n}\right|\) converges. This condition allows us to show the convergence of the second derivatives: $$ u_{x x}(x, y)=\sum_{n=1}^{\infty} k_{n} p_{n}^{\prime \prime}(x) q_{n}(t) $$ and $$ u_{t t}(x, y)=\sum_{n=1}^{\infty} k_{n} p_{n}(x) q_{n}^{\prime \prime}(t) $$ **Step 4: Show that the formal solution satisfies the given equation**

Formal solution satisfying \(u_{tt} = a^2 u_{xx}\) for given conditions

Suppose \(\sum_{n=1}^{\infty} n^{2}\left|\alpha_{n}\right|\) and \(\sum_{n=1}^{\infty} n\left|\beta_{n}\right|\) both converge. Then we can show that the formal solution $$ u(x, t)=\sum_{n=1}^{\infty}\left(\alpha_{n} \cos \frac{n \pi a t}{L}+\frac{\beta_{n} L}{n \pi a} \sin \frac{n \pi a t}{L}\right) \sin \frac{n\pi x}{L} $$ satisfies \(u_{t t}=a^{2} u_{x x}\) for all \((x, t)\). This conclusion also applies to the formal solutions defined in Exercises \(17, 34,\) and 49$.

Key concepts

Term by Term Differentiation

Term by term differentiation is a powerful tool used in mathematical analysis, especially in the study of series and Fourier series.
It allows us to differentiate each term of an infinite series individually, under certain conditions.
This approach is particularly useful when dealing with series representations of functions.In the given problem, the ability to differentiate the series term by term is derived from the convergence of the series \(\sum_{n=1}^{\infty} n\left|k_{n}\right|\).
This ensures that the resulting series formed by the derivatives \(u_x(x, t) = \sum_{n=1}^{\infty} k_{n} p_{n}^{\prime}(x) q_{n}(t)\) and \(u_t(x, t) = \sum_{n=1}^{\infty} k_{n} p_{n}(x) q_{n}^{\prime}(t)\) are valid and converge.

• Term by term differentiation only holds when specific convergence criteria are met.
• It simplifies finding how a function represented by an infinite series changes with respect to its variables.
• Theorem 12.1.2 supports this by providing necessary conditions for the series and its differentiated form to make sense both mathematically and numerically.

Understanding and applying these conditions correctly ensures that you can transition smoothly from the original series to its derivative counterparts, maintaining both precision and correctness in your analysis.

Convergence Tests

Convergence tests help determine if an infinite series adds up to a finite value.
This is crucial in calculus and analysis, allowing us to establish whether the sums described make sense or yield meaningful results.
Various tests exist to help confirm convergence, and understanding these can unravel much of the complexity seen in series like Fourier series.In the problem context,:

• The initial series \(\sum_{n=1}^{\infty} \left|k_{n}\right|\) convergence guarantees the absolute convergence of the function \(u(x, t)\).
• The convergence of \(\sum_{n=1}^{\infty} n\left|k_{n}\right|\) allows term by term differentiation.
• Finally, \(\sum_{n=1}^{\infty} n^{2}\left|k_{n}\right|\) ensures the convergence of the second derivative series \(u_{xx}(x, t)\) and \(u_{tt}(x, t)\).

These conditions are extensions and applications of classical convergence tests such as the Comparison Test, Ratio Test, Root Test, and others.
By deriving specific series properties, these tests allow the safe manipulation of infinite sums, especially when dealing with their derivatives.
Overall, the correct use of convergence tests provides the backbone for dealing with functions described by infinite series, maintaining accuracy in calculations.

Partial Differential Equations

Partial differential equations (PDEs) are equations that involve multivariable functions and their partial derivatives.
They are fundamental in describing a wide array of physical phenomena such as heat conduction, sound, and fluid dynamics.
In our exercise, Fourier series play a pivotal role in crafting solutions to PDEs.The crux of solving PDEs using series lies in approximating the function through infinite sums.
This approximation allows tackling the equation systematically by applying integration and differentiation techniques to each term.
The given equation \(u_{tt} = a^2 u_{xx}\) exemplifies a wave equation.

• The Fourier series representation effectively breaks down complex equations into simpler components.
• This method reduces a PDE to a more manageable format by solving the problem in terms of spatial and temporal harmonics separately.
• Every term of the series \(u(x, t)=\sum_{n=1}^{\infty}(...) \sin \frac{n\pi x}{L}\) corresponds to a mode of vibration or solution to the wave equation.

The convergence of series like \(\sum_{n=1}^{\infty} n^{2}\left|\alpha_{n}\right|\) and \(\sum_{n=1}^{\infty} n\left|\beta_{n}\right|\) ensures the accuracy and viability of the solutions.
Proper understanding and application of these concepts ensure a robust mathematical framework to address PDEs successfully.