Problem 4 - Advanced Engineering Mathematics

Question

Using \(u=X Y\) and \(-\lambda\) as a separation constant we obtain $$\begin{aligned}&X^{\prime \prime}+\lambda X=0,\\\&X^{\prime}(0)=0,\\\&X^{\prime}(a)=0,\end{aligned}$$ and $$\begin{aligned} &Y^{\prime \prime}-\lambda Y=0,\\\&Y(b)=0.\end{aligned}$$ With \(\lambda=\alpha^{2}>0\) the solutions of the differential equations are $$X=c_{1} \cos \alpha x+c_{2} \sin \alpha x \quad \text { and } \quad Y=c_{3} \cosh \alpha y+c_{4} \sinh \alpha y$$ The boundary and initial conditions imply $$X=c_{1} \cos \frac{n \pi}{a} x \quad \text { and } \quad Y=c_{3} \cosh \frac{n \pi}{a} y-c_{3} \frac{\cosh \frac{n \pi b}{a}}{\sinh \frac{n \pi b}{a}} \sinh \frac{n \pi}{a} y$$ for \(n=1,2,3, \ldots\) since \(\lambda=0\) is an eigenvalue for both differential equations with corresponding eigenfunctions 1 and \(y-b,\) respectively we have $$u=A_{0}(y-b)+\sum_{n=1}^{\infty} A_{n} \cos \frac{n \pi}{a} x\left(\cosh \frac{n \pi}{a} y-\frac{\cosh \frac{n \pi b}{a}}{\sinh \frac{n \pi b}{a}} \sinh \frac{n \pi}{a} y\right).$$ Imposing $$u(x, 0)=x=-A_{0} b+\sum_{n=1}^{\infty} A_{n} \cos \frac{n \pi}{a} x$$ gives $$-A_{0} b=\frac{1}{a} \int_{0}^{a} x d x=\frac{1}{2} a$$ and $$A_{n}=\frac{2}{a} \int_{0}^{a} x \cos \frac{n \pi}{a} x d x=\frac{2 a}{n^{2} \pi^{2}}\left[(-1)^{n}-1\right]$$ so that $$u(x, y)=\frac{a}{2 b}(b-y)+\frac{2 a}{\pi^{2}} \sum_{n=1}^{\infty} \frac{(-1)^{n}-1}{n^{2}} \cos \frac{n \pi}{a} x\left(\cosh \frac{n \pi}{a} y-\frac{\cosh \frac{n \pi b}{a}}{\sinh \frac{n \pi b}{a}} \sinh \frac{n \pi}{a} y\right).$$

Step-by-Step Solution

Answer

The solution to the problem is a series expansion of \( u(x, y) \). Each component satisfies the boundary and initial conditions specified.

Step 1: Understand Differential Equation Problem

We are dealing with a differential equation of the form \( u = X Y \), where the spatial part is separated into two independent ordinary differential equations (ODEs): one for \( X(x) \) and one for \( Y(y) \), each with associated boundary or initial conditions.

Step 2: Solve the X(x) ODE with Boundary Conditions

The differential equation is \( X'' + \lambda X = 0 \) with boundary conditions \( X'(0) = 0 \) and \( X'(a) = 0 \). Solving this, we find the solution \( X = c_1 \cos \alpha x + c_2 \sin \alpha x \). Applying the conditions results in a specific form for \( X \) as \( X = c_1 \cos \frac{n \pi}{a} x \) for integer \( n \).

Step 3: Solve the Y(y) ODE with Initial Condition

The other equation is \( Y'' - \lambda Y = 0 \) with boundary condition \( Y(b) = 0 \). The solution is \( Y = c_3 \cosh \alpha y + c_4 \sinh \alpha y \). This leads to \( Y = c_3 \cosh \frac{n \pi}{a} y - c_3 \frac{\cosh \frac{n \pi b}{a}}{\sinh \frac{n \pi b}{a}} \sinh \frac{n \pi}{a} y \).

Step 4: Determine Eigenvalues and Eigenfunctions

The parameter \( \lambda = \alpha^2 > 0 \) identifies eigenvalues and leads to solutions in terms of these eigenvalues for different \( n = 1, 2, 3, \ldots \). The eigenvalues correspond to the separation constants for each equation and define the periodic nature of \( X(x) \) and \( Y(y) \).

Step 5: Impose Solution Constraints

Applying boundary condition \( u(x, 0) = x = -A_0 b + \sum_{n=1}^{\infty} A_n \cos \frac{n \pi}{a} x \), you find \( -A_0 b = \frac{1}{a} \int_{0}^{a} x \, dx = \frac{1}{2} a \), and \( A_n = \frac{2}{a} \int_{0}^{a} x \cos \frac{n \pi}{a} x \, dx = \frac{2a}{n^2 \pi^2}((-1)^n - 1) \).

Step 6: Formulate the Complete Solution

Substitute values back into the solution to obtain the entire form of \( u(x, y) \): \[ u(x, y) = \frac{a}{2b}(b-y) + \frac{2a}{\pi^2} \sum_{n=1}^{\infty} \frac{(-1)^n - 1}{n^2} \cos \frac{n \pi}{a} x \left( \cosh \frac{n \pi}{a} y - \frac{\cosh \frac{n \pi b}{a}}{\sinh \frac{n \pi b}{a}} \sinh \frac{n \pi}{a} y \right) \].

Key Concepts

Ordinary Differential Equations

An ordinary differential equation (ODE), like the ones we see in this exercise, is an equation that involves functions of only one variable along with their derivatives.
ODEs are crucial in modeling dynamical systems where physical phenomena depend on one variable, such as time or position.
Here, we have two separate ODEs derived from using the method of separation of variables: one for the function \( X(x) \) and another for \( Y(y) \).

• The ODE for \( X(x) \) given as \( X'' + \lambda X = 0 \) is a standard second-order linear homogeneous equation with constant coefficients.
• The ODE for \( Y(y) \) is \( Y'' - \lambda Y = 0 \), also a second-order linear homogeneous equation, but with a slightly different form due to the negative sign in front of \( \lambda \).

The general solutions to these equations utilize functions like the cosine and hyperbolic cosine, which are typical solutions to these forms of ODEs. The chosen solutions must satisfy given boundary conditions to ensure the system accurately models the real-world scenario depicted by the problem.

Boundary Value Problems

Boundary value problems (BVPs) arise when we have differential equations coupled with specific conditions, called boundary conditions, which the solution must adhere to at the boundaries of the domain.
For example, in our exercise:

• The boundary conditions for \( X(x) \) are \( X'(0) = 0 \) and \( X'(a) = 0 \).
• For \( Y(y) \), \( Y(b) = 0 \) is the imposed boundary condition.

In BVPs, the solution to the differential equation must satisfy these conditions, which typically results in a discrete set of solutions.
This is in contrast to initial value problems (IVPs), where solutions are determined from known values at a single point. Boundary conditions are critical because they often determine the specific values of constants in the solutions that give rise to physically meaningful results, as we see here where they determine the form of the eigenfunctions.

Eigenvalues and Eigenfunctions

Eigenvalues and eigenfunctions play a fundamental role when solving differential equations through separation of variables, especially BVPs.
In our exercise, the separation leads us to solve two equations, each giving rise to its set of eigenvalues \( \lambda = \alpha^2 \).

• The term 'eigenvalues' refers to special values of \( \lambda \) that allow non-trivial solutions of the differential equations satisfying all boundary conditions.
• 'Eigenfunctions' are the corresponding solutions, functions of the domain variable that are inherent to each eigenvalue. For \( X \), they take the form \( \cos \frac{n \pi}{a} x \), and for \( Y \), they are combinations of hyperbolic sin and cos functions that satisfy the boundary conditions.

These eigenvalues are fundamental in determining the behavior of the system modeled by the differential equation.
In practice, solving for eigenvalues and eigenfunctions conveys insights such as the natural frequencies of a physical system or the modes of vibration, which are essential for various engineering applications.