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t_{1} | = t_{0} + dt, | y_{1} | = y_{0} + G(t_{0},y_{0})dt | ||||
t_{2} | = t_{1} + dt, | y_{2} | = y_{1} + G(t_{1},y_{1})dt | ||||
t_{3} | = t_{2} + dt, | y_{3} | = y_{2} + G(t_{2},y_{2})dt |
Differentiable functions, no matter how difficult or how strange their global behavior, are on a local scale composed of segments which are nearly straight line segments. Differential calculus is the formal acknowledgment and application of this simple idea.
With differential calculus, we can build nice functions by specifying how they are built up locally out of our base function, the straight line. We would write the local change in value of the function over a time interval of infinitesimal length dt informally as
or more typically and formally, as
This says that the initial point is specified, and the slope at the point is determined (through G) by the time and the function value. This is the usual expression of an Ordinary Differential Equation (ODE). Then we can infinitesimally extend the function with that information (along the tangent straight line) and repeat. Of course, this is just an expression in words of the Euler method for numerically solving the differential expression.
Simple Example 1 The simplest differential equation is df = r dt where r is a constant. Take an initial condition to be f(0) = b. This is just the differential equation expression of a straight line, since it says that the slope is everywhere constant. In fact, it is the straight line with slope r and intercept b, f(t) = rt + b. Looking ahead to the next section, , we could say that f is drifting or rending upward at rate r.
Simple Example 2 The next simplest differential equation is df = tdt. This differential equation says that the slope is increasing, in fact increasing exactly as the time. The solution may be easily guessed (using the Fundamental Theorem of Calculus!) as f = t^{2}/2 + C, where C would be specified by the initial value.
Simple Example 3 The next simplest and first non-trivial differential equation is df = f dt. Here the differential equation says that the slope at a pont is the same as the function value. The solution may be guessed to be f(t) = Ce^{t} where the constant C is determined from the initial condition. Again looking ahead, we could write the differential equation as df/f = dt and interpret it to say the relative rate of increase is proportional to the time observed.
Simple Example 4 The next simplest differential equation is df = sf dt. Here the differential equation says that the slope at a point is proportional to the function value. The solution may be guessed to be f(t) = Ce^{st} where the constant C is determined from the initial condition. Again looking ahead, we could write the differential equation as df/f = sdt and interpret it to say the relative rate of increase is proportional to the time observed.
Sketch the graph of the solution of dy/dt = y + t with y(0) = 1. We don’t know a formula for the solution, so how can we possibly sketch its graph? Actually, the equation y' = t + y tells us that at the point (t,y) on the graph of the solution the slope of the solution has a value given by the sum of t and y. In particular, because the curve must pass the initial condition (0, 1) (since y(0) = 1), it’s slope there must be 1 = 0 + 1. So a small portion of the solution curve through the point (0, 1) looks like a short line segment with slope 1. As a guide to sketching the rest of the curve, let’s draw short line segments a lot of points (t,y), each with slope t + y. The result is called a slope field and is shown in the figure. For instance the line segment at the point (1, 2) has slope 1 + 2 = 3. The slope field allows us to visualize the general shape of the solution curves by indicating the direction in which the curves proceed at each point.
Now we can sketch the solution curve through the point (0, 1) by following the slope field.
The solution that Euler’s method produces numerically is the same as the solution produced graphically using the slope field.
The formula for Euler’s method is based on the definition of the derivative, or equivalently on the tangent line approximation:
t_{1} | = t_{0} + dt, | y_{1} | = y_{0} + f(t_{0},y_{0})dt | ||||
t_{2} | = t_{1} + dt, | y_{2} | = y_{1} + f(t_{1},y_{1})dt | ||||
t_{3} | = t_{2} + dt, | y_{3} | = y_{2} + f(t_{2},y_{2})dt |
and so on, as needed.
Think of the slope field as a set of sign posts directing you across the plane. Pick a starting point (corresponding to the initial value), and calculate the slope at that point using the differential equation. This slope is a signpost telling you the direction to take. Head off a small distance in this direction. Stop and look at the new signpost. Recalculate the slope from the differential equation, using the coordinates of the new point. Change direction to correspond to the new slope, and move another small distance and so on.
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