Problem 12 - Advanced Engineering Mathematics

Question

Suppose \(x_{k}\) is a positive zero of \(J_{0}\). Show that a solution of the boundary-value problem $$ a^{2}\left(\frac{\partial^{2} u}{\partial r^{2}}+\frac{1}{r} \frac{\partial u}{\partial r}\right)=\frac{\partial^{2} u}{\partial t^{2}}, \quad 00 $$ $$ \begin{aligned} &u(1, t)=0, \quad t>0 \\ &u(r, 0)=u_{0} J_{0}\left(x_{k} r\right),\left.\quad \frac{\partial u}{\partial t}\right|_{t=0}=0, \quad 0

Step-by-Step Solution

Answer

The solution is \( u(r, t) = u_0 J_0(x_k r) \cos(a x_k t) \).

Step 1: Recognize the Type of PDE

The given partial differential equation (PDE) is \[ a^{2}\left(\frac{\partial^{2} u}{\partial r^{2}}+\frac{1}{r} \frac{\partial u}{\partial r}\right)=\frac{\partial^{2} u}{\partial t^{2}} \] This is a wave equation in polar coordinates, with radial symmetry.

Step 2: Identifying the Boundary and Initial Conditions

We have the boundary condition \( u(1, t) = 0 \) for \( t > 0 \), meaning that the spatial function must be zero at \( r = 1 \). We also have initial conditions \( u(r, 0) = u_{0} J_{0}(x_{k} r) \) and \( \left.\frac{\partial u}{\partial t}\right|_{t=0} = 0 \) for \( 0 < r < 1 \). These conditions involve Bessel functions due to the radial symmetry of the problem.

Step 3: Assume a Separable Solution

Assume a solution of the form: \[ u(r, t) = R(r)T(t) \] where \( R(r) \) is a spatial function and \( T(t) \) is a temporal function. Substitute this into the PDE to obtain a separation-of-variables equation.

Step 4: Derive Two Ordinary Differential Equations (ODEs)

Substituting \( u(r, t) = R(r)T(t) \) into the PDE and separating variables gives:\[ a^2 \left( R''(r) + \frac{1}{r} R'(r) \right) T(t) = R(r) \frac{d^2 T}{dt^2} \]Dividing both sides by \( a^2 R(r) T(t) \), we get two ODEs:1. \( R''(r) + \frac{1}{r} R'(r) + \frac{x_k^2}{a^2} R(r) = 0 \) 2. \( \frac{d^2 T}{dt^2} + x_k^2 T(t) = 0 \)

Step 5: Solve the Radial Bessel ODE

The radial equation \( R''(r) + \frac{1}{r} R'(r) + \frac{x_k^2}{a^2} R(r) = 0 \) is the standard Bessel's equation of order zero. The solution in terms of Bessel functions is:\[ R(r) = J_0(x_k r) \] since \( J_0(x_k) = 0 \) satisfies the boundary condition \( R(1) = 0 \).

Step 6: Solve the Temporal ODE

The temporal equation \( \frac{d^2 T}{dt^2} + x_k^2 T(t) = 0 \) is a simple harmonic oscillator with solution:\[ T(t) = A \cos(x_k t) + B \sin(x_k t) \]The initial condition \( \left.\frac{\partial u}{\partial t}\right|_{t=0} = 0 \) implies that \( B = 0 \), leading to:\[ T(t) = A \cos(x_k t) \]

Step 7: Combine the Solutions

Combine the spatial and temporal solutions to obtain the form:\[ u(r, t) = J_0(x_k r) A \cos(x_k t) \]Now use the initial condition \( u(r, 0) = u_0 J_0(x_k r) \) to determine that \( A = u_0 \).

Step 8: Final Solution

The final solution to the boundary-value problem is:\[ u(r, t) = u_0 J_0(x_k r) \cos(a x_k t) \] This solution satisfies the given PDE, boundary, and initial conditions as required.

Key Concepts

Wave Equation in Polar Coordinates

The wave equation in polar coordinates is a version of the wave equation adapted for problems with radial symmetry over circular or spherical domains. In this equation, time dynamics interact with changes in space, represented in polar coordinates (radius and angle). Polar coordinates are used when problems exhibit circular symmetry, making them ideal for situations involving waves moving through circular regions.

In the given scenario, the wave equation is expressed as:

• \[a^{2}\left(\frac{\partial^{2} u}{\partial r^{2}}+\frac{1}{r} \frac{\partial u}{\partial r}\right)=\frac{\partial^{2} u}{\partial t^{2}}\]

Here, \(r\) is the radial coordinate, and \(t\) is time. This form highlights how changes in the radial direction relate to changes over time. Since the equation includes radial operations and coefficients, it specifically deals with how waves propagate outward from a central point.

The wave equation is used in various fields, including acoustics, electromagnetics, and fluid dynamics, where spherical or circular wave patterns are observed. Solving these equations helps predict how waves will behave over time and space in circular domains.

Boundary-Value Problem

Boundary-value problems require finding a solution to a differential equation that satisfies specific conditions on the boundaries of the domain. In this exercise, we are solving a boundary-value problem involving a wave equation with certain conditions applied at the boundaries.The boundary and initial conditions given are:

• \(u(1, t) = 0\) for the radial boundary where \(r = 1\).
• \(u(r, 0) = u_{0} J_{0}(x_{k} r)\) for the initial state at time \(t = 0\).
• \(\left.\frac{\partial u}{\partial t}\right|_{t=0} = 0\) indicating no initial velocity.

A boundary-value problem demands that the solution not only satisfies the differential equation but also adheres to these boundary and initial conditions. The use of Bessel functions, like \(J_0\), highlights the problem's symmetry and boundary behavior. These functions are fundamental solutions of Bessel's differential equation, serving as the natural choice for problems involving circular symmetry. Thus, they ensure the solution's compliance with boundary conditions at any circular edge.

Second-Order Ordinary Differential Equations

Second-order ordinary differential equations (ODEs) are differential equations that involve the second derivative of a function. They frequently arise in physics and engineering whenever the relationship between a variable and its changes (both first and second derivatives) is needed, like with motion, waves, or electricity.In this problem, solving the wave equation through separation of variables leads to two ODEs. The spatial component of the wave yields:

• \[ R''(r) + \frac{1}{r} R'(r) + \frac{x_k^2}{a^2} R(r) = 0 \]

This is a form of Bessel's ODE, crucial in systems with circular symmetry. It requires solutions that involve Bessel functions, such as \(J_0(x_k r)\).The temporal part results in:

• \[ \frac{d^2 T}{dt^2} + x_k^2 T(t) = 0 \]

This equation describes simple harmonic motion, commonly resolved with solutions incorporating sine and cosine functions, as demonstrated with \(T(t) = A \cos(x_k t)\).These second-order ODEs are pivotal because they shape the behavior of functions over time and space. Understanding them facilitates the prediction and modeling in areas as diverse as mechanical vibrations, acoustics, and electrical circuits.