Heat Equation Summary

PDEs · Axiom Academy

A comprehensive recap of the heat equation: fundamental properties, solution methods, boundary conditions, and key analytical results. Fundamental equation: u_t = k ^2 u describes how temperature (or concentration) diffuses through a medium over time. Rate of change depends on curvature: The time derivative u_t is proportional to the Laplacian ^2 u , which measures spatial curvature. Always smooths - never sharpens: Heat flows from hot to cold regions, eliminating sharp gradients and irregularities. Physical interpretation: The constant k is the thermal diffusivity - higher k means faster diffusion. Key assumption: Assume product form u(x,t) = X(x)T(t) to separate spatial and temporal dependence. Eigenvalue problem for X: Substitution yields X = kT = - , giving the spatial ODE X'' + X = 0 . Exponentially decaying T: The temporal part satisfies T' + k T = 0 , giving T(t) = e^ -k t . General solution: Sum over all modes: u(x,t) = _ n=1 ^ c_n X_n(x) e^ -k _n t . Dirichlet (fixed value): u(0,t) = T_0 , u(L,t) = T_1 . Temperature held constant at boundaries. Leads to sine series: X_n(x) = ( L ) . Neumann (insulated): x |_ x=0 = 0 . No heat flux through boundary (perfect insulation). Leads to cosine series: X_n(x) = ( L ) . Robin (convection): x + u = 0 . Models Newton's law of cooling. Gives transcendental eigenvalue equation with non-simple eigenvalues.

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