In my previous article on the Fluctuation-dissipation theorem, I discussed how to find the velocity v(t) of a Brownian particle in 1D travelling through a viscous medium with damping coefficient γ. …

In physics and other areas of applied mathematics, such as engineering and finance, we tend to model real-world phenomena using partial differential equations (PDES) consisting of more than one variable. Some common PDEs studied in undergraduate and graduate courses at university include: All of these PDEs are what we call…

In my previous article on the Fokker-Planck Equation (FPE), I discussed a few examples involving Brownian motion with and without drift and how to obtain the time-dependent probability density function (PFD) p(x, t) of the systems. …

The OU process is a stochastic process that appears quite frequently in finance and physics, and its main feature is that it somewhat resembles a random walk whose variance reaches a stationary state as time goes to infinity. This type of process is found frequently in physics because every system…

The Fokker-Planck Equation (FPE) is a general partial differential equation (PDE) whose solution describes how the probability density of a certain physical system, such as a Brownian particle moving through space; evolves with time. …

Smooth continuous functions over a finite domain can generally be represented as an infinite sum of sinusoidal functions. This is a realization Joseph Fourier made a long time ago, and the so called “Fourier series” representation of functions has had deep consequences in mathematics and science, particularly in the solution…

In my previous article The heat equation, I derived the analytic solution to the 1D heat equation with constant boundary conditions using the technique of separation of variables. We will now use that same method again, but to solve a slightly different equation, known as the wave equation: where v…

A routine problem in quantum mechanics courses is the solution to the time-dependent Schrodinger equation for the particle in a box problem. This is usually taught in undergraduate courses in 1 dimension, and sometimes in 2 and 3 dimensions as well, as it illustrates rather nicely how the energy levels…

Partial differential equations (PDEs) are used extensively in physics and engineering for modeling real-world phenomena that varies in space and time. These are however notoriously hard to solve and most of them have no analytic solution. But when an analytic solution exists, it is extremely useful and satisfying, because numerical…