Understanding the Heat Equation

Introduction to the Heat Equation

• Heat equation is a partial differential equation.
• Describes temperature changes over time in a region.
• Used to model heat transfer in materials and systems.

Derivation of the Heat Equation

• Heat flows from hotter to colder regions.
• Rate of heat flow is proportional to temperature gradient.
• Material is homogeneous and isotropic.

Governing Equation

• Heat equation formula is: $\frac{\partial u}{\partial t} = \alpha \nabla^2 u$.
• $u(x, y, z, t)$ represents temperature at position $(x, y, z)$ at time $t$.
• $t$ is time.
• $\alpha = \frac{k}{\rho c}$ is thermal diffusivity.
• $k$ is thermal conductivity.
• $\rho$ is density.
• $c$ is specific heat capacity.
• $\nabla^2$ is the Laplacian operator: $\nabla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2}$.

1D Heat Equation

• Simplified one-dimensional heat equation: $\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}$.

Solving the Heat Equation

• Analytical solutions for simple geometries/conditions involve separation of variables, Fourier series, and Laplace transforms.
• Numerical methods such as Finite Difference Method (FDM), Finite Element Method (FEM), and Finite Volume Method (FVM) are used for complex problems.

Applications of the Heat Equation

• Heat transfer in solids and heat exchangers.
• Thermal engineering.
• Geophysics.
• Financial mathematics (Black-Scholes equation).

Example: 1D Heat Equation with Dirichlet Boundary Conditions

• 1D rod of length $L$ with temperature $u(x, t)$.
• Heat equation: $\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}$, $0 < x < L$, $t > 0$.
• Boundary conditions: $u(0, t) = T_1$, $u(L, t) = T_2$, $t > 0$.
• Initial condition: $u(x, 0) = f(x)$, $0 < x < L$.
• $T_1$ and $T_2$ are constants, and $f(x)$ is a given function.

Solution using Separation of Variables

• Assume solution in the form $u(x, t) = X(x)T(t)$.
• Substituting into the heat equation: $X(x)T'(t) = \alpha X''(x)T(t)$.
• Dividing by $\alpha X(x)T(t)$: $\frac{T'(t)}{\alpha T(t)} = \frac{X''(x)}{X(x)} = -\lambda^2$.
• $\lambda$ is a separation constant.

Solving for $X(x)$

• $X''(x) + \lambda^2 X(x) = 0$.
• General solution: $X(x) = A \cos(\lambda x) + B \sin(\lambda x)$.
• Setting $u(x,t) = v(x,t) + g(x)$ simplifies the application of non-homogeneous boundary conditions $u(0, t) = T_1$ and $u(L, t) = T_2$ by finding a $g(x)$ that satisfies the steady-state conditions $g(0) = T_1$ and $g(L) = T_2$ which is $g(x) = T_1 + \frac{T_2 - T_1}{L}x$
• $v(x,t)$ then satisfies homogeneous boundary conditions $v(0,t) = v(L,t) = 0$.

Solving for $T(t)$

• $T'(t) + \alpha \lambda^2 T(t) = 0$.
• General solution: $T(t) = C e^{-\alpha \lambda^2 t}$.

Superposition Principle

• $u(x, t) = \sum_{n=1}^{\infty} B_n \sin\left(\frac{n \pi x}{L}\right) e^{-\alpha \left(\frac{n \pi}{L}\right)^2 t} + g(x)$.
• Coefficients $B_n$ are determined by the initial condition $u(x, 0) = f(x)$.
• $f(x) - g(x) = \sum_{n=1}^{\infty} B_n \sin\left(\frac{n \pi x}{L}\right)$.
• $B_n$ is found using Fourier series: $B_n = \frac{2}{L} \int_0^L [f(x) - g(x)] \sin\left(\frac{n \pi x}{L}\right) dx$.