Duhamel's Principle

PDEs · Axiom Academy

Transform inhomogeneous PDEs into superpositions of homogeneous solutions Consider an inhomogeneous heat equation with a source term: The source term f(x,t) represents external heat being added to the system. Solving this directly can be challenging, but Duhamel's principle offers an elegant approach. 2. Decomposing the Source Term We imagine the source f(x,t) as a continuous accumulation of impulses. At each time τ in [0,t], the source contributes f(x,τ)dτ worth of "instantaneous heat." For each impulse at time τ, we solve a homogeneous problem with initial condition f(x,τ)dτ, and the solution evolves for the remaining time (t-τ). 3. The Auxiliary Homogeneous Problem Define v(x,s;τ) as the solution to the homogeneous heat equation: Here, τ is a parameter indicating when the impulse occurred, and s represents the elapsed time since that impulse. The solution v(x,s;τ) tells us how the initial heat distribution f(x,τ) evolves over time s. 4. Duhamel's Superposition Integral The solution to the original inhomogeneous problem is obtained by integrating the contributions from all impulses: We substitute s = t - τ (elapsed time from impulse to current time) to get: This is Duhamel's Principle : the solution is a weighted superposition of homogeneous solutions, where each weight corresponds to the source strength at that instant. Think of Duhamel's principle as accumulating responses over time : Economics: Total wealth accumulation from continuous deposits with compound interest

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