The beauty behind differential equations
The beauty behind differential equations

December 17, 2019

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It is one of the most difficult courses of the year: Difference and differential equations. Where most us do not really understand what we are actually learning and doing. In this article I will try to explain that differential equations are actually one of the most beautiful concepts in mathematics.

What is a differential equation?

A differential equation is an equation that relates some function with its derivatives. In applications; differential equations give us an understanding how physical phenomena relate to their rates of change. The laws of physics are usually always describable through differential equations. As Steven Strogatz, professor of Applied Mathematics at Cornell University, said: “Since Newton, mankind has come to realize that the laws of physics are always expressed in the language of differential equations.” 
So how do these equations explain certain aspects of our lives? First, we need to know the basics of derivatives, integrals and linear algebra. Without these concepts we cannot understand how differential equations work. As these formulas explain change in certain systems. Moreover, differential equations rather try to explain why for example population sizes shrink or grow rather than explain why a population has a certain size at a certain time. 
To give another example, in physics motion is often described through force. This formula may seem very familiar if you followed the course in physics in high school:

    \begin{equation*} \vv{F} = m\vv{a} \end{equation*}

This is called Newton’s second law of motion. It describes how the total sum of force vectors equals the mass of the object multiplied by the acceleration of that object. What we actually can do with this, is make a differential equation out of it. As acceleration is something that evolves over time we can rewrite that law as:

    \begin{equation*} \vv{F} = m\vv{a}(t) \end{equation*}

Thus our variable acceleration can decrease or increase over time. However, notice that when we integrate acceleration we get the speed of the object at time t. Furthermore, if we integrate the velocity of an object we get the place of that object at time t. So if we first integrate both sides once we get:

    \begin{equation*} \vv{F}t + v_0 = m\vv{v}(t) \end{equation*}

Where v_0 is the initial velocity of the object.. Now integrating both sides again results in:

    \begin{equation*} \frac{1}{2}\vv{F}t^2 + v_0t + s_0 = m\vv{s}(t) \end{equation*}

Where s_0 is the initial position the object starts. So actually knowing only the position of the object over a specific time period is enough to calculate the sum of its force vectors. Therefore we also know its velocity and acceleration through this time period. However, notice that the initial values of position and velocity can be any real number. Thus we have multiple equations that will satisfy these differential equations.

Types of differential equations

There are also different kinds of differential equations. The one we described above is called an ordinary differential equation. These kind of equations usually only depend on the unknown variable time. On the contrary, we also have partial differential equations. These equations have multiple variables as input. A famous example is the heat equation. It describes how heat in a solid body flows over time in three dimensional space:

    \begin{equation*} \frac{\partial T}{\partial t} = \alpha \Big(\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2} \Big) = \alpha \nabla ^2 T \end{equation*}

Where \alpha is a real number and T is the given temperature at a given place in time. To actually solve this equation is really hard. As this is beyond the scope of this article we will try to focus on the one dimensional version of the heat equation.

Heat equation

Imagine holding a rod that is very hot in one end and very cold on the other. Over time the temperature will distribute itself through the rod. So the hot end becomes colder and the other end becomes warmer. To put this in a partial differential equation we need to first establish some intuition of what will happen.
The picture above is an example of a rod with different temperatures through time and space. First of all, we have a change in temperature that is caused by the change in the position we are in. So what place on the rod we are on. Besides that, we also have a change in temperature due to the change in time. To put this mathematically, we can describe this through a partial differential equation that will look like this:

    \begin{equation*} \frac{\partial T}{\partial t} (x, t) = \alpha \Big( \frac{\partial^2 T}{\partial x^2} (x, t) \Big) \end{equation*}

To explain this equation, the change in temperature over time is equal to the change in space. More specifically, the second derivative with respect to x. This is because a second derivative by definition explains the curvature of a graph. Since, we intuitively expect the temperature to change faster where the “curvature” of our graph is “curvy”. To give an illustration:
Here we can see that at the point where the temperature distribution curves the most, the temperature is more likely to change very quickly there. This is why we take the second derivative with respect to our space variable. This is also the case for the three dimensional body. 

Black-Scholes model

Now that we understand the meaning and intuition behind differential equations, we can now see what kind of applications there are. A famous model in the world of finance that broadly explains the prices of options is the Black-Scholes model. The financial insight behind this formula is that one can perfectly eliminate risk when buying or selling options, given that we have a frictionless market. This implies that there is only one correct price to buy or sell options according to the formula:

    \begin{equation*} \frac{\partial V}{\partial t} + \frac{1}{2}\sigma ^2S^2\frac{\partial ^2V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV = 0 \end{equation*}

It is not a straightforward formula and the intuition behind it is beyond the scope of this article. However, the main point here is that even the most complicated systems like financial markets can be brought down to a very elegant formula that can actually explain a large proportion of a certain system. This is the beauty behind differential equations.

This article is written by Sam Ansari


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