🔎Boundary and Initial Conditions for Partial Differential Equations-Types and Uniqueness and Stationa

Boundary and Initial Conditions for Partial Differential Equations-Types and Uniqueness and Stationary States

To find a unique solution for Partial Differential Equations (PDEs) like the diffusion or wave equation, both boundary conditions (BCs), defined on the spatial surface $S$ for all time, and initial conditions (ICs), defined across the volume $V$ at $t=t_0$, must be specified. Common BCs include Dirichlet (fixing the function value $u$ ), Neumann (fixing the normal derivative $n \cdot \nabla u$ or flux), and the generalized Robin condition. The number of ICs needed is determined by the highest-order time derivative. The uniqueness of these solutions is often established through energy methods. Solutions that do not depend on time are called stationary states, which satisfy Poisson's equation ( $\nabla^2 u=-\rho$ ); these require only BCs, with Laplace's equation ( $\nabla^2 u=0$ ) being a special case. Importantly, while solutions to linear problems are generally unique, Poisson's equation with only Neumann conditions results in a solution unique only up to an arbitrary constant, requiring a consistency condition for existence.