### Mathematical Modeling and Computational Calculus

**William Flannery, "Mathematical Modeling and Computational Calculus"**

English | ISBN: 0976413868 | 2013 | 182 pages | PDF | 27 MB

This book will take you from not being able to spell calculus to doing calculus just the way I did it for twenty years as an engineer at high tech firms like Lockheed and Stanford Telecom. You will learn how physical processes are modeled using mathematics and analyzed using computational calculus. Systems studied include satellite orbits, the orbits of the earth and moon, rocket trajectories, the Apollo mission trajectory, the Juno space probe, electrical circuits, oscillators, filters, tennis serves, springs, friction, automobile suspension systems, lift and drag, and airplane dynamics. And not a single theorem in sight.

This book focuses on differential equation models because they are what scientists and engineers use to model processes involving change. Historically, this has presented a big problem for science education because while the models are easy enough to create, solving the differential equations analytically usually requires advanced mathematical techniques and their clever application. But, that was before computers; now, with computers, solutions to differential equations can be computed directly, without the need of theorems or any advanced mathematics, using the formula distance equals velocity times time. It's just that simple. The book will show you how it's done.

Is there a trick here? Of course, here it is: suppose you, as Newton did, want to compute the trajectory of a falling apple, and let's say that the apple's acceleration is constant and equals 10 meters/second/second. So the apple's velocity at the instant it falls is 0 m/s, after 1 second it is 10 m/s, after 2 seconds it is 20 m/s, and after t seconds it is v(t) = 10*t m/s.

You want to know the distance d(t) the apple has fallen after t seconds. This is the problem calculus was developed to solve, that is, given a velocity function v(t), determine the corresponding distance function d(t). To solve it Newton proved theorem after theorem and finally came up with a formula that gives the answer, in this case d(t) = 5*t*t.

But computational calculus bypasses all the theorems and formulas: to calculate how far the apple has fallen after 8 seconds, i.e. d(8), it just subdivides the interval of interest, 8 seconds in this case, into small sub-intervals, say 1 second each, and since the apple's velocity is known at the start of each sub-interval, it uses that velocity to estimate how far the apple falls in the sub-interval using the formula, get ready for it, distance equals velocity times time. The distances for all the sub-intervals are added and that's how far the apple falls in 8 seconds. Capiche?

This is the way it is actually done in the engineering world.

There are two big advantages to the computational method, first, it is very easy to learn, there is only one formula, distance = velocity times time. Second, for most velocity functions v(t) you can't use Newton's method because there is no formula for d(t) that works, none exist. But you can always use computational calculus, no matter how complex the problem, you just compute away and get the answer. Computational calculus has transformed engineering and science.

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