02-EulerMethod - Euler Integration Method Just about all...

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Euler Integration Method Just about all systems of differential equations representing actual physical phenomena do not have closed form solution s. And until the development of high speed computers, most remained out of reach. The Navier-Stokes equations, for example, that describe the behavior of fluid flow were developed back in late 1800’s remained unsolvable for a long time. Early computers could solve a simplified version of it and it still took days for a solution to come out. And today, we rely exclusively on the high-speed computers to solve such complex systems of differential equations. Numerical methods are by far the most useful when it comes to solving differential equations. And the fields are still developing and growing as more powerful computers are becoming available. You may find it interesting that, much of the ground work for numerical methods were established in the early 1900’s. But without the computers, the numerical methods remained impractical. The industry-standard method for solving differential equations is the 4 th order Runge-Kutta which was derived in 1900. Discussion of this method is beyond the scope of this course. Euler’s method is another numerical method that is frequently used. It is not as elegant nor as accurate. But it is very simple and in a pinch, it will give a fairly good solution. All of you have met the basic concept of Euler integration during the discussion of the definite integrals. To find the area under a curve, you can subdivide the region into equally-spaced sub-intervals and the area is approximated by adding the rectangular areas. For example, the below graph subdivides a region into 50 individual rectangles. And the area is the sum of these 50 rectangles. 50 1 i i A A Euler’s integration method starts out with this concept but the major difference is that, we will advance ONE rectangle at a time .
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