# Partial differential equation (PDE)

## Problems from mathematics and physics

Differential equations describe many different physical systems, ranging form gravitation to fluid mechanics. They are difficult to study, they usually have individual equation, which needs to be analyzed as a separate problem.

A fundamental question for any PDE is the existence and uniqueness of a solution for given boundary conditions. For nonlinear equations these questions are in general hard to answer.

### Heat equation

This problem occurs in the theory of heat flow—heat is transferred by conduction in a rod. The flow of heat occurs only in the $x$ dimension (the surface along the length is insulated). Such problem might sound weird, but it is also the problem of heat flow through the wall.

The equation describing the heat equation problem is

$$k\frac{\partial^2 u}{\partial x^2} = \frac{\partial u}{\partial t},\hskip2em k\gt 0.$$
• $x$ is a point along the rod (a rather simply spatial dimension);
• $t$ is time;
• $k$ is a physical constant related to the material;
• $u(x,t)$ represents temperature at any point at any time.

So here is the equation describing the problem and the problem is defined also by its boundary and initial values or conditions. For example

• temperature at a point is fixed (typically at the beginning/end of the rod);
• distribution of temperature in the body at time $t=0$ is given;
• flow at boundary is not allowed (the end is insulated): $\left .\partial u/\partial x = 0\right|_{x=L}$.

### Laplace's equation

Laplace's equation is useful for solving many physical problems such as electrostatic, gravitational or velocity in fluid mechanics. It can be also interpreted as a steady state temperature distribution. Laplace's equation in two and three dimensions is abbreviated as

\begin{align} &\nabla^2 u = 0,\hskip2em &\text{where} \nonumber \\ &\nabla^2u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \hskip2em &\text{(for 2D)}.\nonumber \end{align}
Steady state solution $u(x,y)$ of temperature distribution at any point $(x,y)$ according to boundary conditions.

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