Perturbative methods are used to tackle PDEs with variable coefficients, difficult geometries, or nonlinearities. The main idea is to introduce a small parameter \( \varepsilon \) multiplying the “hard” part of the equation. This provides a knob to control the irregularity: when \( \varepsilon = 0 \), the PDE reduces to a simpler problem; as \( \varepsilon \) increases toward \(1\), we recover the original PDE.
For example, recall Laplace’s equation:
If we wish to study a more complicated nonlinear PDE such as
our usual linear techniques (e.g., superposition) no longer apply. Instead, introduce a small parameter:
When \( \varepsilon = 0 \), we return to Laplace’s equation. We then analyze the problem for \( \varepsilon \in (0,1) \) and finally take \( \varepsilon = 1 \) to connect back to the original nonlinear PDE.