Differential equations-real life application of numerical methods

After the introduction of Euler’s method, I did some research on numerical method and numerical analysis, and found it very much intriguing. By definition, numerical analysis is the study of numbers, the approximation techniques for solving mathematical problems. Unlike other topics, numerical methods have an extremely wide range for real life application.

In this case, it is very easy to apply the abstract idea in the real life. For example, assume there is a partial curve of a quadratic equation, by using Euler’s method, a group of line segment can be drawn to simulate the original curve, and if the original partial curve ends at point (3,2), the simulation created by Euler’s method can still keep going, and predict where the original curve is going to go. It is known that the curve of a quadratic function is a parabola, which is the same curve where a projectile travels under the influence of gravity, and there goes the real life application. Imagine there is a projectile, maybe a bomb shell about to be launched in certain direction with a certain initial velocity, and all people know is how the direction and speed is going to change, but the exact curve of the track of projectile cannot be reached because there are other variables such as wind speed. At this time, numerical method can be applied and used to predict approximately where the bombshell is going to land and how hard it is going to hit the ground and hence the damage it will create.

Many people may think this application is useful, but not as life changing as those methods and theories like the theory of relativity. On the contrary, however, one of the most important object that once changed the fate of entire human race was a projectile. Sixty-nine years ago, a projectile was released from a B-29 bomber by the name of “Enola Gay” above Hiroshima, caused the death of hundreds of thousands of people, and changed the course of human history, led us into a new age of nuclear energy. Without numerical methods, the army would never know when to drop the bomb, when it would reach the ground, where it would reach, and how much uranium they should put for the bomb to detonate at the exact moment it was supposed to.a-bomb_1456835i

picture from: http://www.telegraph.co.uk/news/picturegalleries/worldnews/5977624/Hiroshima-marks-anniversary-of-atomic-bomb.html?image=3

There are many other use for numerical method other than the prediction of the bomb track. It is used in almost all the known science including biology, economics, astrophysics, even urban planning. It is used to predict the possible population growth of a city, based on the current situation such as the current population, birth rate and death rate of the citizens, the number of immigrants moving into the city (http://www.sosmath.com/matrix/markov/markov.html). It can also be used to simulate the orbit of a satellite around a planet. Most importantly, numerical methods can be programmed into computer algorithms (in fact most of the algorithms are based on numerical methods) to analyze extremely large amount of data, and make important predictions such as the possible future of a stock, the chance of profiting for insurance company  to insure a car, or the preference of customers and voters. We live in an information age, and numerical method is exactly the way analyze information.

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