## Mathematics & Science
Learning Center |
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## Numerical Methods for Solving Differential Equations## Euler's Method## Theoretical Introduction
## Deriving the Euler's Method FormulasReminder: We're solving the initial value problem:
As we just saw in the graphical description of the method, the basic idea is to use a known point as a "starter," and then use the tangent line through this known point to jump to a new point. Rather than focus on a particular point in the sequence of points we're going to generate, let's be generic. Let's use the names: - (
*x*,_{n}*y*) for the known point_{n} - (
*x*_{n+1},*y*_{n+1}) for the new point
Our picture, based on previous experience, should look something like this: (Though the proximity of the true solution to the
point ( y)
is, perhaps, a little optimistic.)_{n}Our task here is to find formulas for the coordinates of the new point, the
one on the right. Clearly it lies on the tangent
line, and this tangent line has a known slope,
namely y). Let's mark on our
picture names for the sizes of the _{n}x-jump, and the y-jump as we
move from the known point, (x, _{n}y), to the new point.
Let's also write in the slope of the tangent line that
we just mentioned. Doing so, we get:_{n}The formula relating x_{n+1}
is obvious:
hAlso, we know from basic algebra that
y) = Δ_{n}y / hwhich can be rearranged to solve for Δ Δ y) _{n}But, we're really after a formula for
yAnd, replacing Δ
h f(x, _{n}y)_{n}And that's it! We've derived the formulas required to generate a numerical solution to an initial value problem using Euler's Method. Let's go and look at a summary of the method. |
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ODE Laboratories: A Sabbatical Project by Christopher A. Barker©2017 San Joaquin Delta College, 5151 Pacific Ave., Stockton, CA 95207, USA e-mail: cbarker@deltacollege.edu |