Problem 1 - Advanced Engineering Mathematics

Question

Referring to the solution of Example 1 in the text we have $$ R(r)=c_{1} J_{0}\left(\alpha_{n} r\right) \quad \text { and } \quad T(t)=c_{3} \cos a \alpha_{n} t+c_{4} \sin a \alpha_{n} t $$ where the \(\alpha_{n}\) are the positive roots of \(J_{0}(\alpha c)=0\). Now, the initial condition \(u(r, 0)=R(r) T(0)=0\) implies \(\begin{array}{ll}T(0)=0 \text { and so } c_{3}=0 . \text { Thus } \\ \qquad u(r, t)=\sum_{n=1}^{\infty} A_{n} \sin a \alpha_{n} t J_{0}\left(\alpha_{n} r\right) & \text { and } \frac{\partial u}{\partial t}=\sum_{n=1}^{\infty} a \alpha_{n} A_{n} \cos a \alpha_{n} t J_{0}\left(\alpha_{n} r\right)\end{array}\) we find $$ \begin{aligned} a \alpha_{n} A_{n} &=\frac{2}{c^{2} J_{1}^{2}\left(\alpha_{n} c\right)} \int_{0}^{c} r J_{0}\left(\alpha_{n} r\right) d r \\\ &=\frac{2}{c^{2} J_{1}^{2}\left(\alpha_{n} c\right)} \int_{0}^{\alpha_{n} c} \frac{1}{\alpha_{n}^{2}} x J_{0}(x) d x \\ &=\frac{2}{c^{2} J_{1}^{2}\left(\alpha_{n} c\right)} \int_{0}^{\alpha_{n} c} \frac{1}{\alpha_{n}^{2}} \frac{d}{d x}\left[x J_{1}(x)\right] d x \\\ &=\left.\frac{2}{c^{2} \alpha_{n}^{2} J_{1}^{2}\left(\alpha_{n} c\right)} x J_{1}(x)\right|_{0} ^{\alpha_{n} c}=\frac{2}{c \alpha_{n} J_{1}\left(\alpha_{n} c\right)} \end{aligned} $$ Then \\[ A_{n}=\frac{2}{a c \alpha_{n}^{2} J_{1}\left(\alpha_{n} c\right)} \\] and \\[ u(r, t)=\frac{2}{a c} \sum_{n=1}^{\infty} \frac{J_{0}\left(\alpha_{n} r\right)}{\alpha_{n}^{2} J_{1}\left(\alpha_{n} c\right)} \sin a \alpha_{n} t \\]

Step-by-Step Solution

Answer

The solution is \( u(r, t) = \frac{2}{ac} \sum_{n=1}^{\infty} \frac{J_{0}(\alpha_{n} r)}{\alpha_{n}^{2} J_{1}(\alpha_{n} c)} \sin(a \alpha_{n} t) \).

Step 1: Understand Given Functions

We are provided with two functions for a radial and temporal component: \( R(r)=c_{1} J_{0}(\alpha_{n} r) \) and \( T(t)=c_{3} \cos(a \alpha_{n} t) + c_{4} \sin(a \alpha_{n} t) \). These are solutions from Example 1, with \( \alpha_{n} \) being the roots of \( J_{0}(\alpha c)=0 \).

Step 2: Applying Initial Conditions

We are given the condition \( u(r, 0)=R(r) T(0)=0 \). Since the product is zero, \( T(0) \) must equal zero, which implies \( c_{3}=0 \). Thus, \( T(t) = c_{4} \sin(a \alpha_{n} t) \).

Step 3: Revised Solution

With \( c_{3}=0 \), the solution can be expressed as a sum: \( u(r, t) = \sum_{n=1}^{\infty} A_{n} \sin(a \alpha_{n} t) J_{0}(\alpha_{n} r) \). The derivation of \( u \) with respect to time is: \( \frac{\partial u}{\partial t} = \sum_{n=1}^{\infty} a \alpha_{n} A_{n} \cos(a \alpha_{n} t) J_{0}(\alpha_{n} r) \).

Step 4: Define Coefficient Formula

To find \( A_{n} \), we have the formula: \[ a \alpha_{n} A_{n} = \frac{2}{c^{2} J_{1}^{2}(\alpha_{n} c)} \int_{0}^{c} r J_{0}(\alpha_{n} r) \, dr \]. Further with integration parts solved, we deduce: \( A_{n}=\frac{2}{a c \alpha_{n}^{2} J_{1}(\alpha_{n} c)} \).

Step 5: Final Expression for Solution

Substitute \( A_{n} \) into the integral series expression to get: \[ u(r, t) = \frac{2}{a c} \sum_{n=1}^{\infty} \frac{J_{0}(\alpha_{n} r)}{\alpha_{n}^{2} J_{1}(\alpha_{n} c)} \sin(a \alpha_{n} t) \].

Key Concepts

Initial Conditions

The concept of initial conditions is crucial in understanding how to solve differential equations involving Bessel functions. In this problem, the initial condition given is \( u(r, 0) = R(r) T(0) = 0 \). This tells us that when time \( t = 0 \), the function \( u(r, t) \) should also be zero. Because the functions involved are products, if \( T(0) = 0 \), then the entire expression becomes zero, satisfying the condition. Since \( T(t) = c_{3} \cos(a \alpha_{n} t) + c_{4} \sin(a \alpha_{n} t) \), we know \( cos(0) = 1 \) and \( sin(0) = 0 \). Therefore, for \( T(0) = 0 \), it must be that \( c_{3} = 0 \). This simplifies the temporal component to \( T(t) = c_{4} \sin(a \alpha_{n} t) \). Initial conditions like this one help us to tailor the general solutions to specific scenarios by zeroing in on the values of constants like \( c_{3} \) and \( c_{4} \).

Radial and Temporal Components

To fully grasp the behavior of the function \( u(r, t) \), we need to separate it into radial and temporal components. \( R(r) = c_{1} J_{0}(\alpha_{n} r) \) describes a radial component, where \( J_{0}(\alpha_{n} r) \) is a Bessel function of the first kind. Bessel functions, like \( J_{0} \), often appear in problems involving cylindrical or spherical symmetry, such as vibrations in a circular drum. They are solutions to Bessel's differential equation and depend on the roots \( \alpha_{n} \) of the equation \( J_{0}(\alpha c) = 0 \). These roots define the specific modes or "harmonics" of the solution. For the temporal component, \( T(t) = c_{4} \sin(a \alpha_{n} t) \), we observe a sinusoidal form. This implies oscillatory behavior over time, which is typical of wave-like solutions. This separation into radial and temporal factors allows us to apply the boundary and initial conditions effectively, aiding in finding a solution that satisfies the problem's constraints.

Coefficient Formula

Determining the coefficients \( A_{n} \) is vital for constructing the complete solution for \( u(r, t) \). These coefficients can be thought of as 'weights' that indicate how much of each Bessel function mode is included in the solution. The formula used here is:- \[ a \alpha_{n} A_{n} = \frac{2}{c^{2} J_{1}^{2}(\alpha_{n} c)} \int_{0}^{c} r J_{0}(\alpha_{n} r) \, dr \]This formula begins with an integration using \( J_{0} \), leveraging integration by parts to simplify it to:- \[ A_{n} = \frac{2}{a c \alpha_{n}^{2} J_{1}(\alpha_{n} c)} \]The Bessel function \( J_{1} \) appears due to derivatives involved in integration by parts. Bessel functions \( J_{0} \) and \( J_{1} \) are related but differ in their order, which affects their orthogonal properties key to solving these types of problems. Each component, especially \( \alpha_{n} \) and \( J_{1}(\alpha_{n} c) \), influences the amplitude and shape of the overall solution's distribution. Such formulas ensure each mode \( \alpha_{n} \) is appropriately weighted, providing the complete picture of the function's behavior over time and space.