C ha pter 1
Intr od uction
Partial differential equations, or PDEs for short, are an important type of differential equation that
arise as natural mathematical models in many physical problems. They allow us to describe the
behavior of a system in terms of fu nctions that depend on multiple variables, such as time and space.
For example, consider the classical heat equation, which describes the distribution of heat in a
conducting material over time. This equation can be derived from Fourier's law of heat conduction,
which states that the rate of heat transfer through a material is proportional to the temperature
gradient. By applying this law to an infinitesimal volume element in the material, one obtains a
PDE for the temperature distribution.
Similarly, the wave equation can be derived from Newton's second law of motion, which relates
the force acting on a body to its acceleration. By applying this law to a small element of a string
or other vibrating object, one obtains a PDE for the displacement of the element as a function of
time and position.
In bo th of these examples, the PDEs are derived from fundamental physical laws and provide a
mathematical description of the underlying physical phenomenon. By solving these equations, we
can make predictions ab out how the system will behave under different conditions, such as changes
in tempe rature or initial conditions.
Our main focus is to introduce four imp ortant types of PDEs at an entry-level technicality,