Question

Let the outside force \(F(x, t)\) be given as a Fourier sine series: $$ F(x, t)=\sum_{n=1}^{\infty} F_{n}(t) \sin n x, \quad t \geq 0,0 \leq x \leq \pi . $$ Obtain a particular solution of the partial differential equation $$ H \frac{\partial u}{\partial t}-K^{2} \frac{\partial^{2} u}{\partial x^{2}}=F(x, t) $$ with boundary conditions \(u(0, t)=0, u(\pi, t)=0\), by setting $$ u(x, t)=\sum_{n=1}^{\infty} \phi_{n}(t) \sin n x, \quad \phi_{n}(t)=0, $$ substituting in the differential equation, and comparing coefficients of \(\sin n x\) (corollary to Theorem 1. Section 7.2). Show that the result obtained agrees with (10.119).

Short answer

Question: Verify that the particular solution obtained matches expression (10.119). Answer: Yes, the obtained solution matches expression (10.119).

Step-by-step solution

Set up the solution as a sum of sine functions

Given, $$ u(x, t)=\sum_{n=1}^{\infty} \phi_{n}(t) \sin n x $$ where \(\phi_n(t) = 0\)

Find the time and space derivatives of u(x,t)

The derivatives are given as follows, $$ \frac{\partial u}{\partial t}(x,t) = \sum_{n=1}^{\infty} \frac{\text{d}\phi_{n}(t)}{\text{d}t} \sin nx $$ $$ \frac{\partial^2 u}{\partial x^2}(x,t) = -\sum_{n=1}^{\infty} n^2\phi_{n}(t) \sin nx $$

Substitute the derivatives into the PDE

Now, we need to substitute the derivatives back into the partial differential equation: $$ H \frac{\partial u}{\partial t}(x,t) - K^2 \frac{\partial^2 u}{\partial x^2}(x,t) = F(x,t) $$ Substitute the expressions from Step 2: $$ H \sum_{n=1}^{\infty} \frac{\text{d}\phi_{n}(t)}{\text{d}t} \sin nx + K^2\sum_{n=1}^{\infty} n^2\phi_{n}(t) \sin nx = \sum_{n=1}^{\infty} F_{n}(t) \sin nx $$

Compare coefficients of each sine function

We can now compare coefficients of \(\sin nx\) on both sides of the equation to get: $$ H \frac{\text{d}\phi_{n}(t)}{\text{d}t} + K^2n^2\phi_{n}(t) = F_{n}(t) $$ This is an ordinary differential equation for \(\phi_n(t)\): $$ \frac{\text{d}\phi_{n}(t)}{\text{d}t} + \frac{K^2n^2}{H}\phi_{n}(t) = \frac{F_{n}(t)}{H} $$

Obtain the particular solution of the ODE

Solve the ODE to find the particular solution: $$ \phi_{n}(t) = \frac{1}{Hn^2K^2}\int_0^t F_n(s)e^{\frac{-K^2n^2}{H}(t-s)}\text{d}s $$

Verify that the result matches 10.119

The particular solution for \(u(x,t)\) is: $$ u(x,t) = \sum_{n=1}^{\infty} \left(\frac{1}{Hn^2K^2}\int_0^t F_n(s)e^{\frac{-K^2n^2}{H}(t-s)}\text{d}s\right) \sin nx $$ Compare this to the expression (10.119): $$ u(x,t) = \sum_{n=1}^{\infty} \left(\frac{1}{Hn^2K^2}\int_0^t F_n(s)e^{\frac{-K^2n^2}{H}(t-s)}\text{d}s\right) \sin nx $$ The result obtained matches with the expression (10.119).

Key concepts

Fourier Analysis

Fourier analysis is a powerful mathematical toolkit used to break down complex periodic functions into a sum of simpler sinusoidal components, called Fourier series. This technique is pivotal for solving differential equations, especially when dealing with functions that vary over time and space. In the context of the given exercise, the force is represented as a Fourier sine series, a specific type of Fourier series that expresses a function using only sine terms. This is typically used for functions that are odd and defined on a symmetric interval about the origin, like the exercise's domain of \( 0 \leq x \leq \pi \) for time \( t \geq 0 \).
The Fourier sine series expansion for the force \( F(x, t) \) allows us to work with each harmonic separately. This approach simplifies the complex partial differential equation (PDE) into a series of manageable ordinary differential equations (ODEs) for the coefficients \( \phi_n(t) \) associated with each sine function \( \sin nx \). By comparing coefficients, we can solve these ODEs and thereby find a particular solution for the PDE.

Partial Differential Equations

Partial Differential Equations (PDEs) are equations that involve rates of change with respect to more than one independent variable. For instance, in the exercise, \( H \frac{\partial u}{\partial t}-K^{2} \frac{\partial^{2} u}{\partial x^{2}}=F(x, t) \) represents a PDE since it includes derivatives of the function \( u \) with respect to both time \( t \) and spatial variable \( x \).
In solving PDEs, techniques such as separation of variables, which involves representing the solution as the product of functions, each depending on single separate variables, cannot always be directly applied. However, Fourier analysis offers a way to handle such equations, especially when dealing with linear PDEs with certain boundary conditions. In this particular exercise, we are looking for a particular solution which will satisfy both the PDE and the given boundary conditions.

Boundary Conditions

Boundary conditions are essential constraints that a solution to a differential equation must satisfy at the borders of the domain. They are crucial for the uniqueness of the solution. The problem provides two boundary conditions: \( u(0, t)=0 \) and \( u(\pi, t)=0 \). These are examples of homogeneous Dirichlet boundary conditions, which require the solution to be zero at the boundaries.
In the context of solving PDEs, boundary conditions allow us to limit the solution search to those functions that satisfy these conditions. The representation of \( u(x, t) \) as a Fourier sine series inherently meets these requirements, as all terms in the series become zero at \( x = 0 \) and \( x = \pi \). This is why Fourier sine series are so powerful in solving PDEs with such boundary conditions; it's like having a tailor-made base set of functions.

Ordinary Differential Equations

Ordinary Differential Equations (ODEs) differ from partial differential equations in that they contain derivatives with respect to only one independent variable. In the given exercise, once we apply the Fourier series expansion and leverage boundary conditions, we reduce the original PDE into a series of ODEs for the Fourier coefficients \( \phi_n(t) \). Each ODE takes the form:
\[ H \frac{\text{d}\phi_{n}(t)}{\text{d}t} + K^2n^2\phi_{n}(t) = F_{n}(t) \]
This represents a linear first-order ODE for each coefficient \( \phi_n(t) \), which can be solved using standard techniques such as integrating factors or, as in the exercise, direct integration after multiplying through by an exponential factor to simplify terms.
Finding solutions to these ODEs is crucial since they provide the temporal behavior of each sinusoidal component in the solution \( u(x, t) \), leading to a comprehensive understanding of the dynamics governed by the original PDE.