Consider the problem of finding the temperature of a gas contained in a volume V, which is supposed to be a bounded, open and connected set in ℝ3. Denote by u(x, t) the gas temperature at the point x ∈ V at time t. Let Ω be a subdomain of V with smooth boundary ∂Ω. Let ν be the unit normal to ∂Ω; see Fig. 1.1a. In order to write the equation of temperature evolution, we shall consider the law of conservation of thermal energy on the domain Ω, under the assumption that there is no external absorption or application of heat:
KeywordsHeat Equation Heat Kernel Eigenfunction Expansion Path Integral Method Eigenfunction Expansion Method
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