Problem 2 - Advanced Engineering Mathematics

Question

As shown in Example 1 in the text, separation of variables leads to \\[ \begin{array}{l} X(x)=c_{1} \cos \alpha x+c_{2} \sin \alpha x \\ Y(y)=c_{3} \cos \beta y+c_{4} \sin \beta y \end{array} \\] and \\[ T(t)+c_{5} e^{-k\left(\alpha^{2}+\beta^{2}\right) t} \\] The boundary conditions \\[ \left.\begin{array}{l} u_{x}(0, y, t)=0, \quad u_{x}(1, y, t)=0 \\ u_{y}(x, 0, t)=0, \quad u_{y}(x, 1, t)=0 \end{array}\right\\} \quad \text { imply } \quad\left\\{\begin{array}{ll} X^{\prime}(0)=0, & X^{\prime}(1)=0 \\ Y^{\prime}(0)=0, & Y^{\prime}(1)=0 \end{array}\right. \\] Applying these conditions to \\[ X^{\prime}(x)=-\alpha c_{1} \sin \alpha x+\alpha c_{2} \cos \alpha x \\] and \\[ Y^{\prime}(y)=-\beta c_{3} \sin \beta y+\beta c_{4} \cos \beta y \\] gives \(c_{2}=c_{4}=0\) and \(\sin \alpha=\sin \beta=0 .\) Then \\[ \alpha=m \pi, m=0,1,2, \ldots \quad \text { and } \quad \beta=n \pi, n=0,1,2, \ldots \\] By the superposition principle \\[ u(x, y, t)=A_{00}+\sum_{m=1}^{\infty} A_{m 0} e^{-k m^{2} \pi^{2} t} \cos m \pi x+\sum_{n=1}^{\infty} A_{0 n} e^{-k n^{2} \pi^{2} t} \cos n \pi y \\] \\[ +\sum_{m=1}^{\infty} \sum_{n=1}^{\infty} A_{m n} e^{-k\left(m^{2}+n^{2}\right) \pi^{2} t} \cos m \pi x \cos n \pi y \\] We now compute the coefficients of the double cosine series: Identifying \(b=c=1\) and \(f(x, y)=x y\) we have \\[ \begin{aligned} A_{00} &=\int_{0}^{1} \int_{0}^{1} x y d x d y=\left.\int_{0}^{1} \frac{1}{2} x^{2} y\right|_{0} ^{1} d y=\frac{1}{2} \int_{0}^{1} y d y=\frac{1}{4} \\ A_{m 0} &=2 \int_{0}^{1} \int_{0}^{1} x y \cos m \pi x d x d y=\left.2 \int_{0}^{1} \frac{1}{m^{2} \pi^{2}}(\cos m \pi x+m \pi x \sin m \pi x)\right|_{0} ^{1} y d y \\ &=2 \int_{0}^{1} \frac{\cos m \pi-1}{m^{2} \pi^{2}} y d y=\frac{\cos m \pi-1}{m^{2} \pi^{2}}=\frac{(-1)^{m}-1}{m^{2} \pi^{2}} \\ A_{0 n} &=2 \int_{0}^{1} \int_{0}^{1} x y \cos n \pi y d x d y=\frac{(-1)^{n}-1}{n^{2} \pi^{2}} \end{aligned} \\] and \\[ A_{m n}=4 \int_{0}^{1} \int_{0}^{1} x y \cos m \pi x \cos n \pi y d x d y=4 \int_{0}^{1} x \cos m \pi x d x \int_{0}^{1} y \cos n \pi y d y \\] \\[ =4\left(\frac{(-1)^{m}-1}{m^{2} \pi^{2}}\right)\left(\frac{(-1)^{n}-1}{n^{2} \pi^{2}}\right) .\\]

Step-by-Step Solution

Answer

The solution is constructed using calculated coefficients \(A_{00}, A_{m0}, A_{0n}, A_{mn}\) in the superposition formula, satisfying the boundary conditions and initial function.

Step 1: Understanding the Problem

In this exercise, we are dealing with a boundary value problem that uses separation of variables to find a solution to a partial differential equation (PDE). The solution involves functions of three variables \(X(x), Y(y), \text{ and } T(t)\) and the boundary conditions that must be satisfied.

Step 2: Analyzing Boundary Conditions

The boundary conditions \(u_x(0, y, t) = 0\), \(u_x(1, y, t) = 0\), \(u_y(x, 0, t) = 0\), and \(u_y(x, 1, t) = 0\) translate to the derivatives \(X'(0) = X'(1) = 0\) and \(Y'(0) = Y'(1) = 0\). These imply that the sine terms in \(X(x)\) and \(Y(y)\) should be zero, leading to \(c_2 = c_4 = 0\) and \(\sin \alpha = \sin \beta = 0\). Thus, \(\alpha = m\pi\) and \(\beta = n\pi\), where \(m, n = 0, 1, 2, \ldots\).

Step 3: Setting Up the Superposition Principle

Using the superposition principle, the general solution is given as a combination of terms involving cosine functions and exponential decay with coefficients \(A_{00}\), \(A_{m0}\), \(A_{0n}\), and \(A_{mn}\). These coefficients represent the contribution of each mode to the solution.

Step 4: Calculating \(A_{00}\)

\(A_{00}\) is calculated as an integral over \([0,1]\times[0,1]\) for the function \(f(x, y) = xy\). Evaluating the integral gives \(A_{00} = \frac{1}{4}\).

Step 5: Calculating \(A_{m0}\)

\(A_{m0}\) involves integrating \(xy\cos(m\pi x)\) over \([0,1]\times[0,1]\). After simplification, we find \(A_{m0} = \frac{(-1)^m - 1}{m^2\pi^2}\).

Step 6: Calculating \(A_{0n}\)

Similarly, \(A_{0n}\) is found by integrating \(xy\cos(n\pi y)\) over \([0,1]\times[0,1]\), resulting in \(A_{0n} = \frac{(-1)^n - 1}{n^2 \pi^2}\).

Step 7: Calculating \(A_{mn}\)

\(A_{mn}\) involves a double integral over \([0,1]\times[0,1]\) for \(xy\cos(m\pi x)\cos(n\pi y)\). This results in \(A_{mn} = 4\left(\frac{(-1)^m - 1}{m^2\pi^2}\right)\left(\frac{(-1)^n - 1}{n^2 \pi^2}\right)\).

Step 8: Reconstructing the Solution

With all coefficients calculated, the complete solution is expressed by substituting these coefficients back into the superposition formula. This provides the general form of the PDE solution under the given boundary conditions and initial function \(f(x,y) = xy\).

Key Concepts

Boundary Value Problem

The concept of a boundary value problem (BVP) is crucial in understanding how partial differential equations (PDEs) work. In a BVP, the solution to a PDE must satisfy certain conditions, known as boundary conditions, at the boundaries of the domain. For instance, in the current exercise, the boundary conditions are given as derivatives of the solution: \( u_x(0, y, t) = 0 \), \( u_x(1, y, t) = 0 \), \( u_y(x, 0, t) = 0 \), and \( u_y(x, 1, t) = 0 \). These conditions dictate how the solution behaves along the edges of the area we're studying.
These boundary conditions tell us that the solution does not change at the boundaries in the x or y directions. Essentially,

• When dealing with PDEs, boundary conditions help restrict and shape the solution to be unique or well-defined.
• They also ensure that the physical behavior of the problem is modeled correctly.

This adds meaningful context to the mathematical process being employed, making the resultant solution not only mathematically correct but physically relevant too.

Separation of Variables

The separation of variables is a method often used to solve partial differential equations. It simplifies the problem by breaking it into smaller, manageable parts. It assumes that the solution can be written as a product of functions, each dependent on a single coordinate. In our exercise, we separate the solution into three functions: \( X(x) \), \( Y(y) \), and \( T(t) \). Each function depends solely on one variable.
The separation process involves:

• Introducing this idea into the PDE and systematically applying it to reformulate the equation.
• After rearranging the terms, one obtains ordinary differential equations (ODEs) for each of the separate functions.

In this exercise, applying separation of variables leads to equations where the coefficients are determined by the boundary conditions. This transforms complex PDEs into simpler ODEs, which are often easier to solve.
This method is particularly useful because it allows us to tackle each equation independently, ultimately leading to a solution that combines these parts.

Superposition Principle

In mathematics and physics, the superposition principle states that the net response at a given place and time is the sum of the responses from each individual source. This principle is vital in linear systems, as they allow for combining solutions to form more complex solutions. In the current boundary value problem, we've used this principle to build the final solution from several simpler solutions. These simpler solutions involve terms such as \( \cos(m\pi x) \) and \( \cos(n\pi y) \).
Applying the superposition principle involves:

• Initiating by finding specific solutions for different values of parameters, \( m \) and \( n \).
• Then summing these individual solutions, multiplied by coefficients, to form a complete solution.

In the context of our exercise, the principle facilitates forming a complete set of solutions expressed as a series - which, when summed together, fulfill the entire boundary conditions and initial values given by the exercise.

Double Cosine Series

A double cosine series is a powerful tool in describing functions in two dimensions within a rectangular domain. By using the specific orthogonal property of cosine functions, these series can represent complex, multi-dimensional behaviors over a set region. In this exercise, a double cosine series is used to express the solution involving terms that are functions of both \( x \) and \( y \), such as \( \cos(m\pi x) \cos(n\pi y) \).
The important aspects include:

• Each term represents a harmonic mode of the solution across the domain.
• The coefficients are determined by integrating the product of the original function and the cosine basis functions over the defined region.

Additionally, evaluating these integrals over the region \([0,1] \times [0,1]\) allows us to find the coefficients necessary for constructing the series. This step crucially determines how much each mode will contribute to the solution, which enables us to model complex phenomena in a structured way.