Problem 9 - Advanced Engineering Mathematics

Question

The temperature in a circular plate of radius \(c\) is determined from the boundary-value problem $$ \begin{aligned} &k\left(\frac{\partial^{2} u}{\partial r^{2}}+\frac{1}{r} \frac{\partial u}{\partial r}\right)=\frac{\partial u}{\partial t}, \quad 00 \\ &u(c, t)=0, \quad t>0 \\ &u(r, 0)=f(r), \quad 0

Step-by-Step Solution

Answer

The solution is a Fourier-Bessel series: \( u(r, t) = \sum_{n=1}^{\infty} A_n J_0\left(\frac{\alpha_n}{c} r\right) e^{-k\left(\frac{\alpha_n}{c}\right)^2 t} \).

Step 1: Understand the Problem

The problem describes a boundary-value problem on a circular plate. We need to solve for the temperature distribution \(u(r, t)\) given the differential equation \(k\left(\frac{\partial^{2} u}{\partial r^{2}}+\frac{1}{r} \frac{\partial u}{\partial r}\right)=\frac{\partial u}{\partial t}\). This partial differential equation describes heat conduction in a circular geometry.

Step 2: Simplify the Differential Equation

Recognize that the equation is a form of the heat equation in polar coordinates. The boundary condition is \(u(c,t) = 0\), implying the edge of the plate is kept at zero temperature. The initial condition is \(u(r,0) = f(r)\).

Step 3: Apply Separation of Variables

Assume a solution of the form \(u(r, t) = R(r)T(t)\). Substitute this into the differential equation: \[k\left(R''(r) + \frac{1}{r}R'(r)\right)T(t) = R(r)T'(t)\]. Divide both sides by \(R(r)T(t)\) and set equal to a separation constant \(-\lambda\) to separate variables.

Step 4: Solve the Temporal Equation

Solving the temporal part yields \(T(t) = e^{-\lambda t}\). This is a standard result when \(T(t)\) is isolated from the spatial variables and set equal to an exponential decay, as imposed by the negative separation constant \(-\lambda\).

Step 5: Solve the Radial Equation

We are left with the equation \(R''(r) + \frac{1}{r}R'(r) + \lambda R(r) = 0\). This is a form of Bessel's equation of order zero. The general solution is \(R(r) = A J_0(\sqrt{\frac{\lambda}{k}} r) + B Y_0(\sqrt{\frac{\lambda}{k}} r)\), where \(J_0\) and \(Y_0\) are Bessel functions of the first and second kind, respectively.

Step 6: Apply Boundary Conditions

Using the boundary condition \(u(c, t) = 0\), set \(R(c) = 0\). This implies that \(A J_0(\sqrt{\frac{\lambda}{k}} c) + B Y_0(\sqrt{\frac{\lambda}{k}} c) = 0\). Since \(Y_0\) becomes singular at zero, typically \(B = 0\) is chosen for continuity at \(r=0\) when \(c\) avoids singularities.

Step 7: Construct the Solution

The solution can be written as a series of Bessel functions due to orthogonality of \(J_0\). By combining these ideas, the solution \(u(r, t)\) becomes a series of the form: \[ u(r, t) = \sum_{n=1}^{\infty} A_n J_0\left(\frac{\alpha_n}{c} r\right) e^{-k\left(\frac{\alpha_n}{c}\right)^2 t} \] where \(\alpha_n\) are the roots of \(J_0\left(\frac{\alpha_n}{c} c \right) = 0\).

Step 8: Use Initial Condition

The initial condition \(u(r,0) = f(r)\) can be used to determine the coefficients \(A_n\) by performing a Fourier-Bessel series expansion of \(f(r)\), which involves integrating \(f(r) J_0(\frac{\alpha_n}{c} r)\) times \(r\) from 0 to \(c\).

Key Concepts

Bessel functions

Bessel functions play a crucial role in solving differential equations that appear in circular or cylindrical geometries. Named after Friedrich Bessel, these functions are solutions to Bessel's differential equation, which is frequently encountered in physics and engineering problems involving wave propagation and static potentials. In our problem, we specifically deal with Bessel functions of order zero, denoted as \( J_0(x) \) and \( Y_0(x) \). These functions help express solutions to problems involving circular symmetry.
- **\( J_0(x) \)** is the Bessel function of the first kind of order zero and is well-behaved at the origin.- **\( Y_0(x) \)** or Neumann function is the Bessel function of the second kind and tends to become singular at the origin. This is why, in many practical applications including the one in our problem, the term involving \( Y_0(x) \) is often neglected or set to zero if solutions remain regular.
For our heat equation problem, Bessel functions allow the radial solution, \( R(r) \), to be expressed in a form that captures the behavior of the temperature distribution in the radial direction of the circle. They provide a series of functions whose properties of orthogonality can be exploited to solve boundary and initial value problems. This orthogonality is key in determining the coefficients of the solution when constructing it from a series of Bessel functions.

Boundary-value problems

A boundary-value problem requires finding a solution to a differential equation that satisfies certain specified conditions, known as boundary conditions, at the boundaries of the domain. In our example, the boundary condition is given by \( u(c, t) = 0 \), signifying that the edge of the circular plate is kept at zero temperature at all times.
- **Boundary conditions** can be of several types: Dirichlet, Neumann, or mixed. In our exercise, the condition \( u(c, t) = 0 \) is a Dirichlet boundary condition, which specifies the value of the function on the boundary.
The importance of boundary conditions lies in their ability to uniquely determine the solution to a differential equation, ensuring it fits the given physical constraint. The initial condition, \( u(r, 0) = f(r) \), sets the starting state of the system and is essential in applying methods like separation of variables.
Understanding boundary-value problems is vital as they appear in various scientific and engineering contexts. They help us determine how physical systems respond to constraints imposed at their boundaries, providing insight into phenomena like heat conduction, wave propagation, and more.

Heat equation

The heat equation is a fundamental partial differential equation in mathematical physics. It describes how heat diffuses through a given region over time. The basic form in our problem is adjusted for polar coordinates, reflecting the circular shape of the plate.
For the circular plate, the heat equation given is:\[ k \left( \frac{\partial^2 u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} \right) = \frac{\partial u}{\partial t} \]\Here, \( k \) is the thermal diffusivity of the material. The equation models the rate of change of temperature \( u \) as influenced by the spatial and temporal changes in the material's heat distribution.
Some essential points about the heat equation include:

• It is parabolic in nature and often entails initial and boundary-value problems for a complete solution.
• In our polar-coordinate context, it incorporates terms that account for the radial symmetry of the circular region.

The solutions to the heat equation, especially in our case, often involve techniques like separation of variables and series expansion using special functions like Bessel functions, which capture the complex diffusion behavior in circular domains.

Separation of variables

Separation of variables is a powerful mathematical method used to solve partial differential equations (PDEs). It works by assuming that the solution can be written as a product of functions, each depending only on a single coordinate. In the exercise, we assume a solution of the form \( u(r, t) = R(r)T(t) \).
This assumption allows us to split the heat equation into two separate ordinary differential equations (ODEs):

• A temporal equation involving \( T(t) \)
• A spatial equation involving \( R(r) \)

This method transforms a complex PDE into simpler, solvable ODEs by introducing a separation constant \(-\lambda\). Each part can be solved independently, and the resulting solutions recombined to form the general solution.
- **Temporal solution:** \( T(t) = e^{-\lambda t} \), representing exponential decay over time.- **Spatial solution:** Solved using Bessel functions to account for circular geometry.
By applying boundary conditions and initial conditions, we can specifically determine the form of each solution. This method is widely used in problems with symmetrical boundaries and is a staple in solving PDEs like the heat, wave, and Laplace equations.