Background on Partial Differential Equations

There are many phenomena in nature, which, even though occurring over finite regions of space and time, can be described in terms of properties that prevail at each point of space and time separately. This description originated with Newton, who with the aid of his differential calculus showed how to grasp a global phenomenon--for example, the elliptic orbit of a planet, by means of a locally applied law, for example $F = ma$. This approach has been extended from the motion of single point particles to the behavior of other forms of matter and energy, including fluids, light, heat, electricity, signals traveling along optical fibers, neurons, and even gravitation. This extension consists of formulating or stating a partial differential equation governing the phenomenon, and then solving that differential equation for the purpose of predicting measurable properties of the phenomenon. PDEs are classified into three categories, hyperbolic (e.g. wave) equations, parabolic (e.g. diffusion) equations, and elliptic (e.g. Laplace) equations, on the basis of their characteristics, or the curves of information propagation. Suppose there exists an unknown function $u$ of $x$. We can denote $u$'s partial derivative with respect to $x$ as follows;

Then,

1. Laplace Equation - An equation for unknown function $u$, of $x$, $y$, and , as follows;
   
   
   Solutions to this equation, known as harmonic functions, serve as the potentials of vector field in physics, such as the gravitational or electrostatic fields. A generalization of Laplace's equation is Poisson's equation:

   $$u_{xx} + u_{yy} + u_{zz} = f$$ (3)

   
   
   where is a given function. The solutions to this equation describe potentials of gravitational and electrostatic fields in the presence of masses or electrical charges, respectively.
2. Diffusion Equation - An equation describing the temperature in a given region over time for unknown function $u$, with respect to $x$, $y$, and , as follows;

   $$u_t = k(u_{xx} + u_{yy} + u_{zz})$$ (4)

   
   
   Solutions will typically ``even out'' over time. The number describes the heat conductivity of the material.
3. Wave Equation - An equation for unknown function $u$, with respect to $x$, $y$, , and $t$, where $t$ is a time variable, as follows;

   $$u_{tt} = c^2(u_{xx} + u_{yy} + u_{zz})$$ (5)

   
   
   Its solutions describe waves such as sound or light waves; $c$ is a number which represents the speed of the wave. In lower dimensions, this equation describes the vibration of a string or drum. Solutions will typically be combinations of oscillating sine waves.