4.1 Heat equation and diffusion processes

The heat equation and diffusion processes are key applications of Fourier series. These equations describe how heat or substances spread over time, using partial differential equations that can be solved with Fourier techniques.

Initial and boundary conditions are crucial for finding unique solutions to heat and diffusion problems. By applying Fourier methods like separation of variables, we can solve these equations and model real-world phenomena like temperature changes or chemical diffusion.

Heat and Diffusion Equations

Fundamental Equations Describing Heat and Diffusion

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• Heat equation describes the distribution of heat (or variation in temperature) in a given region over time
  • Derived from the principle of conservation of energy and Fourier's law of thermal conduction
  • Expressed as a second-order partial differential equation: $\frac{\partial u}{\partial t} = \alpha \nabla^2 u$, where $u$ is temperature, $t$ is time, and $\alpha$ is thermal diffusivity
• Diffusion equation describes the behavior of the concentration of a substance in a medium over time
  • Derived from the principle of conservation of mass and Fick's law of diffusion
  • Expressed as a second-order partial differential equation: $\frac{\partial \phi}{\partial t} = D \nabla^2 \phi$, where $\phi$ is concentration, $t$ is time, and $D$ is the diffusion coefficient
• Thermal conductivity is a physical property that measures a material's ability to conduct heat
  • Represents the rate at which heat is transferred through a material by conduction (metals like copper and silver have high thermal conductivity)
  • Appears in the heat equation as part of the thermal diffusivity term: $\alpha = \frac{k}{\rho c_p}$, where $k$ is thermal conductivity, $\rho$ is density, and $c_p$ is specific heat capacity

Initial and Boundary Conditions

Specifying Conditions for Unique Solutions

• Initial conditions specify the state of the system at the beginning (t=0)
  • Describe the initial temperature distribution in the heat equation (e.g., $u(x, 0) = f(x)$)
  • Describe the initial concentration distribution in the diffusion equation (e.g., $\phi(x, 0) = g(x)$)
• Boundary conditions specify the behavior of the system at the boundaries of the domain
  • Dirichlet boundary condition: specifies the value of the function at the boundary (e.g., $u(0, t) = u(L, t) = 0$)
  • Neumann boundary condition: specifies the value of the derivative of the function at the boundary (e.g., $\frac{\partial u}{\partial x}(0, t) = \frac{\partial u}{\partial x}(L, t) = 0$)
• Steady-state solution is the solution that does not change with time
  • Obtained by setting the time derivative term to zero in the heat or diffusion equation (e.g., $\nabla^2 u = 0$)
  • Represents the long-term behavior of the system (temperature distribution in a well-insulated room)
• Transient solution is the solution that varies with time
  • Obtained by solving the heat or diffusion equation with the time derivative term included (e.g., $\frac{\partial u}{\partial t} = \alpha \nabla^2 u$)
  • Describes how the system evolves from the initial state to the steady state (cooling of a hot metal rod)

Solution Methods

Techniques for Solving Heat and Diffusion Equations

• Separation of variables is a method for solving partial differential equations by assuming the solution can be written as a product of functions, each depending on only one variable
  • Assume the solution has the form $u(x, t) = X(x)T(t)$
  • Substitute into the heat or diffusion equation and separate the variables to obtain ordinary differential equations for $X(x)$ and $T(t)$
  • Solve the ordinary differential equations and combine the solutions to obtain the general solution (e.g., $u(x, t) = \sum_{n=1}^{\infty} A_n \sin(\frac{n\pi x}{L})e^{-\alpha(\frac{n\pi}{L})^2t}$)
• Fourier series solution is obtained by representing the solution as an infinite series of sine and cosine functions
  • Assume the solution has the form $u(x, t) = \sum_{n=1}^{\infty} A_n(t) \sin(\frac{n\pi x}{L})$ or $u(x, t) = \sum_{n=0}^{\infty} B_n(t) \cos(\frac{n\pi x}{L})$
  • Substitute into the heat or diffusion equation and solve for the coefficients $A_n(t)$ or $B_n(t)$
  • Determine the coefficients using initial conditions and orthogonality properties of sine and cosine functions (e.g., $A_n = \frac{2}{L}\int_0^L f(x) \sin(\frac{n\pi x}{L}) dx$)