Partial Differential Equations
“Are there Differential Equations controlled by more than one variable?”
Differential Equations are ubiquitous in nature. Whether it be in modeling how an object falls or how the economy rises, they always seem to have some form of application. But some of these Differential Equations are very special. Specifically, they are controlled by more than one variable. These Partial Differential Equations govern phenomena such as waves and heat conduction and require their own unique solution methods.
“How can we measure the difference between two sample populations?”
As Engineers and Social/Natural Scientists, we often like to measure the differences between sample populations. However, how can we know how far apart they truly are and whether or not these sample populations are representative of the entire population? Well, what if we were to measure the difference between their median values and then divide that by difference of their standard distributions? This is known as the T-Test and is used to quantify two population’s difference and its measurement accuracy.
“How can we discern real data from noise?”
When working with signal processing, we often want to try to discern real data from random noise. However, doing so requires complex thinking. So how can we use our engineering mindset to solve this? Well, we can simply use an organization of mathematical theorems known as detection theory to solve these problems.
“What is a more accurate way for approximating integrals?”
If we have a particular function, then there are many simple ways to approximate the area under the curve without integrating. However, is there a more accurate way to do this? Well, let’s use our mathematical mindset to find out. If we were to fit a third-degree polynomial over our function and then take its integral, we would arrive with an area very close. This method is known in the English speaking world as Simpson’s Rule, and is represented by the formula Area = (b-a)/6 * ( f(a) + 4f((a+b)/2) + f(b) ), with a being the start point and b being the endpoint.
“Is it possible to simulate traffic to improve it?”
If you have ever ridden a car in a crowded area, then you probably have experienced traffic. It’s slow, annoying, and worst of all unsustainable. Because of this traffic engineers are trying to design better and better ways to assist people with their daily commutes. One of the tools that they use at their disposal is known as a traffic simulation. Traffic simulations use mathematical frameworks to model how travel moves and what can be done about it. With the data generated by these systems, urban planners can optimize future city designs, making our living space safer, quieter, and ultimately less congested!
“How can we use statistics to describe industrial quality?”
Even with the most perfect manufacturing processes, flaws are expected to happen during production. As such, we need a way to quantify how stable a given industrial output is. So how can we use our engineering mindset to solve this? Well, we know that if an item’s measurements become three standard levels of deviations outside of the mean, then it is probably unusable. So what if we were to create a chart that would graph the average of all measurements for each part with along with the mean and the tolerance levels so we can visually see anything out of the ordinary? Well, this is called an X-bar Chart and the control limits can be calculated by the formula UCL/LCL = (mean of all measurements) +/- 3*(standard deviation of all measurements)/sqrt(number of measurements per sample).
“How can we analyze grand scale engineering projects without spending too much money?”
Engineering projects can really range in scale. They can go as large as skyscrapers pushing over a hundred floors, or as small as microdevices that inhabit the mystical world of quantum mechanics. However, if we want to perform any tests at this scale, it would be prohibitively expensive. So how can we use our engineering mindset to get around this problem? Well, we know that equations for physical phenomena tend to be the same regardless of scale. So what if we were to build a model at a more manageable size, perform our tests, retrieve the data, and then use the scale ratio to convert it? Well, this method known as dimensional similitude is used every day by engineers to drastically reduce costs. Dimensional similitude is often employed in wind tunnel tests when full scale aircraft become prohibitively expensive.