Certainly! The problem involves solving the heat equation in cylindrical coordinates for a cylinder with heat transfer along the radial ($r$) and axial ($z$) directions. We are also considering specific boundary and initial conditions:

Problem Setup:

• Initial temperature: $T(r, z, t=0) = T_0$
• Laser pulse: Applies energy $Q$ on the front face (at $z = 0$).
• Boundary conditions:
  • Front face ($z = 0$): Receives the laser pulse energy $Q$.
  • Rear face ($z = L$): Heat losses, which might be modeled as a convective boundary condition.
  • Cylinder edges (at $r = R$): Also subject to heat losses, likely modeled as convective or insulated depending on the setup.

Governing Equation:

The heat equation in cylindrical coordinates (assuming no angular dependence, $\theta$) is:

$\frac{\partial T}{\partial t} = \alpha \left( \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial T}{\partial r} \right) + \frac{\partial^2 T}{\partial z^2} \right)$

where $\alpha$ is the thermal diffusivity of the material.

Boundary and Initial Conditions:

1. Initial Condition: $T(r, z, 0) = T_0$

2. Boundary Condition at $z = 0$ (front face): This can be modeled as an instantaneous energy addition: $-k \frac{\partial T}{\partial z} \bigg|_{z=0} = \frac{Q}{\pi R^2} \delta(t)$ where $k$ is the thermal conductivity and $\delta(t)$ is the Dirac delta function representing the laser pulse.

3. Boundary Condition at $z = L$ (rear face): If we consider convective losses: $-k \frac{\partial T}{\partial z} \bigg|_{z=L} = h(T(L) - T_\infty)$ where $h$ is the convective heat transfer coefficient, and $T_\infty$ is the ambient temperature.

4. Boundary Condition at $r = R$ (cylinder edge): For convective losses: $-k \frac{\partial T}{\partial r} \bigg|_{r=R} = h(T(R) - T_\infty)$ If the edges are insulated: $\frac{\partial T}{\partial r} \bigg|_{r=R} = 0$

Solution Approach:

The solution will likely involve separation of variables, the application of Fourier-Bessel series for the radial component, and Fourier series for the axial component. The problem is non-trivial and may require numerical methods depending on the complexity of the boundary conditions, particularly if the convective terms are nonlinear.

1. Separation of Variables: Assume $T(r, z, t) = R(r)Z(z)T(t)$, and substitute into the PDE, leading to separate ODEs for $R(r)$, $Z(z)$, and $T(t)$.

2. Radial Component $R(r)$: Solve the Bessel differential equation that arises from the radial part, typically involving Bessel functions $J_n(r)$ and boundary conditions at $r = R$.

3. Axial Component $Z(z)$: Solve using Fourier series with boundary conditions at $z = 0$ and $z = L$.

4. Time Component $T(t)$: Typically involves exponential decay terms, depending on the thermal diffusivity.

5. Superposition and Final Solution: Combine all solutions into a final form, applying the initial and boundary conditions to determine coefficients.

Detailed Calculation:

The detailed calculation involves a lot of steps, including solving Bessel functions and Fourier series, and possibly using numerical methods. Would you like to proceed with these steps, or would you prefer a more specific part of the solution process?