5.2 Heat Equation: Derivation and Solutions

The heat equation is a crucial tool in mathematical physics, describing how temperature changes over time in a material. It's derived from energy conservation and Fourier's law, connecting heat flux to temperature gradients.

Solving the heat equation involves techniques like separation of variables and Fourier series. These methods allow us to find solutions for different geometries and boundary conditions, helping us understand heat transfer in various real-world scenarios.

Derivation and Fundamental Concepts

Derivation of heat equation

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• Conservation of energy principle states change in internal energy of a system equals heat added minus work done by the system
  • In heat conduction, work term is negligible since no significant mechanical work is performed
• Fourier's law relates heat flux to negative temperature gradient
  • $q = -k \frac{\partial T}{\partial x}$, where $q$ is heat flux (W/m²), $k$ is thermal conductivity (W/m·K), and $\frac{\partial T}{\partial x}$ is temperature gradient (K/m)
• Continuity equation describes rate of change of temperature in terms of divergence of heat flux
  • $\rho c \frac{\partial T}{\partial t} = -\nabla \cdot q$, where $\rho$ is density (kg/m³), $c$ is specific heat capacity (J/kg·K), and $\frac{\partial T}{\partial t}$ is rate of change of temperature (K/s)
• Combining Fourier's law and continuity equation yields the heat equation
  • $\frac{\partial T}{\partial t} = \alpha \nabla^2 T$, where $\alpha = \frac{k}{\rho c}$ is thermal diffusivity (m²/s) and $\nabla^2$ is the Laplacian operator

Solution Methods

Separation of variables for heat equation

• Assume solution is a product of two functions: $T(x, t) = X(x) \cdot \tau(t)$
  • $X(x)$ depends only on spatial variable $x$
  • $\tau(t)$ depends only on time variable $t$
• Substituting this form into heat equation leads to two ordinary differential equations (ODEs)
  • $\frac{X''}{X} = \frac{1}{\alpha} \frac{\tau'}{\tau} = -\lambda$, where $\lambda$ is separation constant (eigenvalue)
• Spatial ODE $X'' + \lambda X = 0$ has solutions depending on sign of $\lambda$ and boundary conditions
  • For $\lambda > 0$, solutions are sines and cosines (oscillatory)
  • For $\lambda < 0$, solutions are exponentials (growth or decay)
• Temporal ODE $\tau' + \alpha \lambda \tau = 0$ has exponential decay solution: $\tau(t) = e^{-\alpha \lambda t}$
• General solution is linear combination of products of spatial and temporal solutions
  • $T(x, t) = \sum_{n=1}^{\infty} C_n e^{-\alpha \lambda_n t} X_n(x)$, where $C_n$ are constants determined by initial conditions and $X_n(x)$ are eigenfunctions

Solutions in different geometries

• Infinite domain (unbounded): Solution is Gaussian function that spreads and decays with time
  • $T(x, t) = \frac{1}{\sqrt{4 \pi \alpha t}} e^{-\frac{x^2}{4 \alpha t}}$, representing diffusion of initial heat distribution
• Semi-infinite domain (bounded on one side): Solution combines infinite domain solution and its mirror image
  • Boundary condition at $x = 0$ (fixed temperature or insulated) determines sign of mirror image
  • Represents heat transfer in a half-space (ground, thick wall)
• Finite domain (bounded on both sides): Solution is Fourier series with coefficients determined by boundary conditions
  • Steady-state solution: $T(x) = C_1 x + C_2$, where $C_1$ and $C_2$ are constants (linear temperature profile)
  • Transient solution: Exponential decay of Fourier modes with time (approach to steady-state)

Fourier series for boundary conditions

• Fourier series represents periodic function as infinite sum of sines and cosines
  • $f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left(a_n \cos\left(\frac{n \pi x}{L}\right) + b_n \sin\left(\frac{n \pi x}{L}\right)\right)$, where $L$ is period and $a_n$, $b_n$ are Fourier coefficients
• Fourier coefficients depend on boundary conditions
  • Dirichlet (fixed temperature): $T(0, t) = T_0$, $T(L, t) = T_L$ lead to cosine series
  • Neumann (fixed heat flux): $\frac{\partial T}{\partial x}(0, t) = q_0$, $\frac{\partial T}{\partial x}(L, t) = q_L$ lead to sine series
  • Mixed (combination of Dirichlet and Neumann) lead to mixed series
• Substituting Fourier series into heat equation and solving for time-dependent coefficients yields
  • $T(x, t) = \sum_{n=1}^{\infty} C_n e^{-\alpha \lambda_n t} \sin\left(\frac{n \pi x}{L}\right)$, where $\lambda_n = \left(\frac{n \pi}{L}\right)^2$ are eigenvalues