Problem 10 - Advanced Engineering Mathematics

Question

Letting \(u(r, t)=u(r, t)+\psi(r)\) we obtain \(r \psi^{\prime \prime}+\psi^{\prime}=-\beta r .\) The general solution of this nonhomogeneous CauchyEuler equation is found with the aid of variation of parameters: \(\psi=c_{1}+c_{2} \ln r-\beta r^{2} / 4 .\) In order that this solution be bounded as \(r \rightarrow 0\) we define \(c_{2}=0 .\) Using \(\psi(1)=0\) then gives \(c_{1}=\beta / 4\) and so \(\psi(r)=\beta\left(1-r^{2}\right) / 4\) Using \(v=R T\) we find that a solution of \\[ \begin{array}{c} \frac{\partial^{2} v}{\partial r^{2}}+\frac{1}{r} \frac{\partial v}{\partial r}=\frac{\partial v}{\partial t}, \quad 00 \\ v(1, t)=0, \quad t>0 \\ v(r, 0)=-\psi(r), \quad 0< r <1 \end{array} \\] is \\[ v(r, t)=\sum_{n=1}^{\infty} A_{n} e^{-\alpha_{n}^{2} l} J_{0}\left(\alpha_{n} r\right) \\] where \\[ A_{n}=-\frac{\beta}{4} \frac{2}{J_{1}^{2}\left(\alpha_{n}\right)} \int_{0}^{1} r\left(1-r^{2}\right) J_{0}\left(\alpha_{n} r\right) d r \\] and the \(\alpha_{n}\) are defined by \(J_{0}(\alpha)=0 .\) From the result of Problem \(10,\) Exercises 12.6 (see also Problem 2 of this exercise set ) we get \\[ A_{n}=-\frac{\beta J_{2}\left(\alpha_{n}\right)}{\alpha_{n}^{2} J_{1}^{2}\left(\alpha_{n}\right)} \\] Thus from \(u=v+\psi(r)\) it follows that \\[ u(r, t)=\frac{\beta}{4}\left(1-r^{2}\right)-\beta \sum_{n=1}^{\infty} \frac{J_{2}\left(\alpha_{n}\right)}{\alpha_{n}^{2} J_{1}^{2}\left(\alpha_{n}\right)} e^{-\alpha_{n}^{2} t} J_{0}\left(\alpha_{n} r\right) \\]

Step-by-Step Solution

Answer

The solution is \( u(r, t) = \frac{\beta}{4}(1 - r^2) - \beta \sum_{n=1}^{\infty} \frac{J_2(\alpha_n)}{\alpha_n^2 J_1^2(\alpha_n)} e^{-\alpha_n^2 t} J_0(\alpha_n r) \).

Step 1: Review Problem Statement

First, analyze the given problem statement. We are given a nonhomogeneous Cauchy-Euler differential equation in terms of \( \psi(r) \). The problem provides solutions using variation of parameters.

Step 2: Verify Boundary Condition

The problem requires that the solution \( \psi(r) = c_1 + c_2 \ln(r) - \frac{\beta r^2}{4} \) must be bounded as \( r \to 0 \). To ensure this, set \( c_2 = 0 \) as the logarithmic term \( \ln(r) \) would otherwise introduce an unbounded behavior.

Step 3: Calculate Constant Using Given Condition

Using the condition \( \psi(1) = 0 \), solve for \( c_1 \). Substituting into the equation gives: \( \beta(1-1^2)/4 = c_1 \implies c_1 = \beta/4 \). This simplifies \( \psi(r) \) to \( \psi(r) = \frac{\beta(1 - r^2)}{4} \).

Step 4: Set Up Boundary Value Problem for \( v(r, t) \)

We are asked to use the product \( v = R T \) to form a solution for the partial differential equation involving \( v(r, t) \), which includes initial and boundary conditions. \( v(1, t) = 0 \) and \( v(r, 0) = -\psi(r) \).

Step 5: Establish Solution for \( v(r, t) \)

The solution formula for \( v(r, t) \) is given as a sum of an infinite series: \( v(r, t) = \sum_{n=1}^{\infty} A_n e^{-\alpha_n^2 t} J_0(\alpha_n r) \), where the coefficients \( A_n \) are determined using the initial condition.

Step 6: Confirm Coefficient Calculation \( A_n \)

The problem provides revised coefficients \( A_n = -\frac{\beta J_2(\alpha_n)}{\alpha_n^2 J_1^2(\alpha_n)} \) using a result from an exercise. This simplifies finding the coefficients for \( v(r, t) \).

Step 7: Build Final Solution for \( u(r, t) \)

Adding \( \psi(r) \) to \( v(r, t) \) gives the full solution as \( u(r, t) = \frac{\beta}{4}(1 - r^2) - \beta \sum_{n=1}^{\infty} \frac{J_2(\alpha_n)}{\alpha_n^2 J_1^2(\alpha_n)} e^{-\alpha_n^2 t} J_0(\alpha_n r) \), which satisfies the initial and boundary conditions.

Key Concepts

Variation of Parameters

In differential equations, the method of variation of parameters is used to solve nonhomogeneous linear equations. It is a very powerful technique when you can't use simple undetermined coefficients. Instead of trying to guess a particular solution like with undetermined coefficients, variation of parameters allows us to find a particular solution based on the structure of the differential equation itself.

The basic idea is to modify the constants in the general solution of the homogeneous equation to be functions, which will be determined as you proceed. For instance, given the Cauchy-Euler equation: \[ x^2 y'' + x y' + y = g(x)\] variation of parameters modifies the solution of the homogeneous equation, applying the method to get a particular solution fit for the nonhomogeneous form.

This approach is especially helpful as it provides a systematic way to handle the additional complexity introduced by the nonhomogeneous part of the equation. Keep in mind that this process involves calculating integrals, so an understanding of integration techniques is beneficial.

Boundary Value Problem

A boundary value problem (BVP) refers to a differential equation that needs to satisfy certain conditions (boundary conditions) at more than one point. This is a common setup in physics and engineering when the solution must adhere to specific physical constraints or conditions at boundaries.

In the original exercise, a partial differential equation (PDE) problem is presented where the boundary conditions are \(v(1, t) = 0\) and \(v(r, 0) = -\psi(r)\). These conditions must be satisfied by the function \(v(r, t)\) over the given domain. BVPs can be more challenging than initial value problems because they require the solution to meet criteria at two or more distinct points.

Practically, solving a BVP usually involves finding solutions that are smooth and fit the constraints, which is often done using numerical methods when an analytical solution isn't feasible or possible. Understanding these principles is crucial for working through both simple and complex boundary value problems.

Bessel Functions

Bessel functions arise in many physical scenarios where the wave-like behavior appears, such as vibrations, heat conduction, or electromagnetic waves in cylindrical structures. They are solutions to Bessel's differential equation:

\[ x^2 y'' + x y' + (x^2 - n^2) y = 0\]

The functions occur naturally in problems with cylindrical symmetry and are categorized into first kind \(J_n(x)\) and second kind \(Y_n(x)\). In the given problem, \(J_0(\alpha_n r)\) appears, which is a Bessel function of the first kind, zero order. This type is regular at the origin, which is often a requirement for physical problems.

The zeros of Bessel functions are critical in defining eigenvalues, noted as \(\alpha_n\), which play a role in solving PDEs with cylindrical symmetry. In practice, knowing the behavior and property of Bessel functions helps in simplifying and solving differential equations in many applied science and engineering fields.

Series Solution

The series solution is a valuable method for solving differential equations, especially when solutions in terms of basic functions are not readily available. By expressing the solution as an infinite sum, or series, we can break down complex problems into manageable parts.

In the differential equation realm, a series solution frequently involves finding coefficients that ensure the series satisfies the equation and any boundary conditions. For example, in the exercise, a series solution is used to express \(v(r, t)\), aligning with Bessel functions and exponential terms to meet initial conditions.

This strategy involves calculating each term in the series to ensure convergence to the actual solution over the problem's domain. The key aspects include:

• Identifying proper basis functions (like Bessel functions).
• Computing coefficients that align with boundary behaviors.
• Ensuring the series converges to the desired function.

By breaking complex solutions into a sum of simpler parts, series solutions play an essential role in solving both ordinary and partial differential equations.