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The heat conduction equation is fundamental in understanding how heat dissipates through materials. In our problem, we deal with a circular plate, which means we use polar coordinates to describe the heat distribution. The heat conduction equation given in this scenario 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, \(u(r, t)\) represents the temperature at a radius \(r\) and time \(t\), while \(k\) is the thermal conductivity of the material. Heat conduction is concerned with the transfer of heat in a medium over time and variations in space. The equation indicates that the rate of change of temperature at a point in the material is related to the curvature of the temperature profile at that point. Heat flows from regions of high temperature to low temperature, smoothing out temperature differences over time.

As we explore the heat conduction problem, we arrive at Bessel's differential equation, a common result when dealing with radial components in circular geometries. Recognizing this differential equation:

• \[ R''(r) + \frac{1}{r} R'(r) + \lambda R(r) = 0 \]

This is a form of Bessel's differential equation of the zeroth order. Solutions to this type of equation are Bessel functions, specifically \(J_0\) and \(Y_0\).
However, due to physical reasons, we usually discard \(Y_0\) because it becomes singular at \(r = 0\). For boundary-value problems like ours, which require the function to be zero at the boundary, we typically select Bessel functions of the first kind \(J_0\) to satisfy this.
These functions are critical in solving boundary problems involving circular or cylindrical shapes. The parameters \(\lambda\) are determined by the boundary condition \(J_0(\sqrt{\lambda}c) = 0\), where \(\alpha_n\) are roots of this equation, ensuring the solution remains finite.

Separation of variables is a powerful mathematical tool used to solve partial differential equations like the one we have here. The idea is to break down a complex problem into simpler, one-dimensional problems. We start by assuming a solution form:

• \( u(r, t) = R(r)T(t) \)

By substituting this form into the original heat conduction equation, we can isolate dependencies on \(r\) and \(t\) by dividing through by \(kRT\), resulting in two ordinary differential equations. Each can be solved separately.
The method allows us to focus on solving a problem in one variable while considering the effects of the other as a constant. This approach is particularly effective for problems with clear boundary conditions, such as the zero-temperature condition \(u(c, t)=0\) on our plate. Separation of variables leads us to identify the radial part as Bessel’s differential equation and the temporal part as a simple first-order differential equation.

The Fourier-Bessel series is a mathematical tool similar to Fourier series but adapted for circular or radial systems. It is used to expand functions in terms of Bessel functions due to their orthogonal properties.

• Our solution is represented as a series of terms: \[ u(r, t) = \sum_{n=1}^{\infty} A_n J_0(\alpha_n r) e^{-\alpha_n^2 k t} \]

The coefficients \(A_n\) are crucial in representing the initial temperature distribution \(u(r, 0) = f(r)\). These coefficients are determined using integral formulas and leverage the orthogonality of Bessel functions, which means they remain independent of one another under integration.
Finding \(A_n\) involves integrating the product of the initial condition function and the corresponding Bessel function:

• \[ A_n = \frac{2}{c^2 J_1^2(\alpha_n c)} \int_0^c r f(r) J_0(\alpha_n r) \, dr \]

This is a hallmark of solving such boundary value problems — decomposing complex initial shapes into simpler, manageable components.