📢How Initial Conditions Drive Diffusion Gradients

The initial condition serves as the indispensable starting point for solving time-dependent partial differential equations, such as the diffusion equation, as it defines the concentration everywhere in the domain at $t=0$. This initial state establishes the potential energy or concentration gradient that drives the subsequent dynamic evolution toward a steady-state equilibrium. Even in a scenario where the substance is uniformly distributed, meaning the initial concentration is constant throughout the entire relevant volume ($u(\vec{x}, t_0)=u_0$), this uniform initial diffusion state is sufficient, when combined with appropriate boundary conditions, to determine the concentration at any later time.

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How Initial Conditions Drive Diffusion Gradients | Audiosviadean on Notion