mathematics

History of numerical solution of differential equations using computers

History of numerical solution of differential equations using computers is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand History of numerical solution of differential equations using computers rather than just read about it. In short: Differential equations, in particular Euler equations, rose in prominence during World War II in calculating the accurate trajectory of ballistics, both rocket-propelled and gun or cannon type projectiles. Originally, mathematicians used the simpler calculus of earlier centuries to determine velocity, thrust, elevation, curve, distance, and other parameters.

Key takeaways

History of numerical solution of differential equations using computers belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect History of numerical solution of differential equations using computers to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of History of numerical solution of differential equations using computers from memory before moving on to harder problems.

Reference excerpt

Differential equations, in particular Euler equations, rose in prominence during World War II in calculating the accurate trajectory of ballistics, both rocket-propelled and gun or cannon type projectiles. Originally, mathematicians used the simpler calculus of earlier centuries to determine velocity, thrust, elevation, curve, distance, and other parameters. New weapons, however, such as Germany's giant cannons, the "Paris Gun" (Encyclopedia Astronautica) and "Big Bertha," and the V-2 rocket, meant that projectiles would travel hundreds of miles in distance and dozens of miles in height, in all weathers. As a result, variables such as diminished wind resistance in thin atmospheres and changes in gravitational pull reduced accuracy using the historic methodology. There was the additional problem of planes that could now fly hundreds of miles an hour. Differential equations were applied to stochastic processes. Developing machines that could speed up human calculation of differential equations led in part to the creation of the modern computer through the efforts of Vannevar Bush, John von Neumann and others. According to Mary Croarken in her paper "Computing in Britain During World War II," by 1945, the Cambridge Mathematical Laboratory created by John Lennard-Jones utilized the latest computing devices to perform the equations. These devices included a model "differential analyser," and the Mallock machine, described as "an electrical simultaneous equation solver." According to Croarken, the Ministry was also interested in the new arrival of a differential analyzer accommodating eight integrators. This exotic computing device built by Metropolitan-Vickers in 1939 consisted of wheel and disk mechanisms that could provide descriptions and solutions for differential equations. Output resulted in a plotted graph. At the same time, in the United States, analog computer pioneer Vannevar Bush took on a similar role to that of Lennard-Jones in the military effort after President Franklin Delano Roosevelt entrusted him with the bulk of wartime research into automatic control of fire power using machines and computing devices. According to Sarah Bergbreiter in her paper "Moving from Practice to Theory: Automatic Control after World War II," fire control for the downing of enemy aircraft by anti-aircraft guns was the priority. The analog electro-mechanical computing machines plotted the differential firing data while servos created by H.L. Hazen adapted the data to the guns for precise firing control and accuracy. Other improvements of a similar type by Bell Labs increased firing stability so that output from the differential engines could be fully used to compensate for stochastic behaviors of enemy aircraft and large guns. A new age of intelligent warfare had begun. This work at MIT and Bell Labs would later lead to Norbert Wiener's development of the electronic computer and the science of cybernetics for the same purpose, speeding the differential calculation process exponentially and taking one more giant step toward the creation of the modern digital computer using von Neumann architecture. Dr. von Neumann was one of the original mathematicians employed in the development of differential equations for ballistic warfare.

See also Numerical ordinary differential equations Numerical partial differential equations

References

Croarken, Mary. "Computing in Britain During World War II," IEE History of Technology Summer Meeting 6 July 2002. [1] Bergbreiter, Sarah. "Moving from Practice to Theory: Automatic Control after World War II." Student paper: HIS 285S: History of Science, University of California, Berkeley. [2] MacRae, Norman. John von Neumann: The Scientific Genius Who Pioneered the Modern Computer, Game Theory, Nuclear Deterrence, and Much More. N.Y.: Pantheon Books, 1992.

Worked examples

Example 1 — a first encounter with History of numerical solution of differential equations using computers

Start with the simplest possible case. Write down what History of numerical solution of differential equations using computers claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to History of numerical solution of differential equations using computers before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about History of numerical solution of differential equations using computers ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of History of numerical solution of differential equations using computers

In research: History of numerical solution of differential equations using computers appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses History of numerical solution of differential equations using computers in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: History of numerical solution of differential equations using computers is common in secondary-school and first-year university syllabi. It links to neighbouring topics History of computing, Numerical differential equations, so understanding it makes those chapters shorter.
In everyday life: Look for History of numerical solution of differential equations using computers outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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Frequently asked questions

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Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

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