Heat Equation

Definition and Formulation

Basic Equation

Heat equation: ∂u/∂t = α∇²u, where u = temperature function, t = time, α = thermal diffusivity.

Domain and Variables

Domain: spatial region Ω ⊆ ℝⁿ, n = dimension (1D, 2D, 3D). Variables: x ∈ Ω, t ≥ 0.

Thermal Diffusivity

α = k/(ρc): k = thermal conductivity, ρ = density, c = specific heat capacity. Units: m²/s.

Physical Interpretation

Heat Diffusion

Describes heat flow from high to low temperature regions via conduction.

Energy Conservation

Derived from conservation of energy and Fourier’s law of heat conduction.

Isotropic Medium Assumption

Assumes homogeneous, isotropic material properties for uniform diffusivity.

Mathematical Properties

Type of PDE

Parabolic partial differential equation, linear, second order in space, first order in time.

Well-Posedness

Existence and uniqueness of solutions assured under suitable initial and boundary conditions.

Smoothing Effect

Initial irregularities in temperature distribution smooth out over time.

Boundary Conditions

Dirichlet Boundary Condition

Specifies temperature u on the boundary ∂Ω: u|∂Ω = g(t,x).

Neumann Boundary Condition

Specifies heat flux (normal derivative): ∂u/∂n|∂Ω = h(t,x).

Robin Boundary Condition

Linear combination: a u + b ∂u/∂n = c on ∂Ω; models convective heat transfer.

Initial Conditions

Specification of Initial Temperature

u(x,0) = f(x), ∀ x ∈ Ω. Required for time evolution.

Compatibility with Boundary Conditions

Initial and boundary data must be consistent at t=0 on ∂Ω.

Regularity Requirements

Initial data f(x) usually required to be continuous or piecewise smooth.

Analytical Solutions

Fundamental Solution (Heat Kernel)

In ℝⁿ, u(x,t) = (4παt)^{-n/2} exp(-|x|²/(4αt)) solves heat equation with initial δ distribution.

Method of Images

Constructs solutions in domains with simple boundaries by reflecting fundamental solutions.

Similarity Solutions

Reduce PDE via scaling variables to ODEs; useful in infinite or semi-infinite domains.

Separation of Variables Method

Concept

Assume solution u(x,t) = X(x)T(t); substitute into PDE; separate variables.

Spatial Eigenvalue Problem

Leads to Sturm–Liouville problem: X'' + λX = 0 with boundary conditions.

Temporal ODE

T'(t) + αλ T(t) = 0; solution T(t) = Ce^{-αλt}.

Fourier Series Solutions

Expansion in Eigenfunctions

Initial condition f(x) expanded in terms of eigenfunctions X_n(x): f(x) = Σ a_n X_n(x).

Solution Representation

u(x,t) = Σ a_n X_n(x) e^{-αλ_n t}, where λ_n are eigenvalues.

Convergence and Regularity

Converges to classical solution if f smooth; otherwise to weak solution.

Numerical Methods

Finite Difference Method (FDM)

Discretizes spatial and temporal derivatives; explicit and implicit schemes.

Finite Element Method (FEM)

Variational formulation; handles complex geometries; uses basis functions.

Stability and Convergence

Explicit FDM: stable if time step Δt satisfies CFL condition; implicit schemes unconditionally stable.

Explicit FDM scheme (1D):u_i^{n+1} = u_i^n + (α Δt / Δx²)(u_{i+1}^n - 2u_i^n + u_{i-1}^n)Stability: α Δt / Δx² ≤ 0.5 (CFL condition)

Applications

Engineering

Heat transfer in solids, cooling of electronic components, thermal insulation design.

Physics

Modeling diffusion processes, phase change problems, and Brownian motion analogy.

Biology and Medicine

Thermal therapies, temperature regulation in tissues, drug diffusion modeling.

Extensions and Generalizations

Nonlinear Heat Equations

Incorporate temperature-dependent diffusivity, reaction terms, or source terms.

Anisotropic Media

Generalize α to tensor form; models direction-dependent heat conduction.

Fractional Heat Equation

Use fractional derivatives in space/time; models anomalous diffusion.

Generalized heat equation form:∂u/∂t = ∇ · (D(x) ∇u) + Q(x,t,u)Where D(x): diffusivity tensor; Q: source term

References

• Evans, L. C., Partial Differential Equations, Graduate Studies in Mathematics, vol. 19, AMS, 2010, pp. 200-250.
• John, F., Partial Differential Equations, Applied Mathematical Sciences, vol. 1, Springer, 1982, pp. 150-180.
• Crank, J., The Mathematics of Diffusion, 2nd ed., Oxford University Press, 1975, pp. 45-90.
• Strauss, W. A., Partial Differential Equations: An Introduction, 2nd ed., Wiley, 2007, pp. 100-130.
• Smith, G. D., Numerical Solution of Partial Differential Equations: Finite Difference Methods, 3rd ed., Oxford University Press, 1985, pp. 75-110.