Initial Conditions

PDEs · Axiom Academy

Understanding how to specify the state at t=0 for time-dependent partial differential equations 1. What is an Initial Condition? For a time-dependent PDE, an initial condition specifies the value of the solution (and possibly its time derivatives) at the initial time t = 0. Example: For the heat equation modeling temperature in a rod, u(x,0) = f(x) tells us the initial temperature distribution along the rod. 2. How Many Initial Conditions Do You Need? The number of initial conditions required depends on the order of time derivatives in the PDE. Heat Equation: u t = ku xx (first-order in time) Needs 1 IC: u(x,0) = f(x) Wave Equation: u tt = c²u xx (second-order in time) Needs 2 ICs: u(x,0) = f(x) and u t (x,0) = g(x) The wave equation needs two conditions because it models systems with both position and velocity (like vibrating strings or sound waves). 3. Evolution from Initial Conditions Once you specify initial conditions, the PDE determines how the solution evolves for all future times. Different PDEs propagate initial conditions in dramatically different ways. Diffusion (Heat Equation): Initial sharp features smooth out and spread over time. Heat flows from hot to cold, creating a uniform temperature. Wave Propagation (Wave Equation): Initial disturbances split into waves traveling left and right. The shape is preserved as it propagates. Let's watch the same initial condition evolve under both the heat equation and wave equation:

This is the written version of the interactive lesson above. See the full PDEs course.