1. The problem is to understand the basics of Partial Differential Equations (PDEs). 2. A PDE is an equation involving partial derivatives of a function with respect to multiple variables. 3. For example, the heat equation is a common PDE: $$\frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2}$$ where $u$ depends on time $t$ and space $x$. 4. Important rules: partial derivatives mean we differentiate with respect to one variable while keeping others constant. 5. PDEs describe how physical quantities change over space and time, such as temperature, pressure, or wave displacement. 6. To solve PDEs, we often need initial conditions (values at the start time) and boundary conditions (values at spatial boundaries). 7. Basic types of PDEs include elliptic, parabolic, and hyperbolic, each modeling different phenomena. 8. Understanding PDEs helps in fields like physics, engineering, and finance to model dynamic systems.