Higher-Order Methods  Alternative Derivations of Euler's  Derivation via Numerical Integration

Derivation via Tangent Line Approximation

Imagine for the moment, that you are a law-enforcement officer walking your beat when you get the word that there's been a hold-up at the bank just around the corner. The single perpetrator had committed the robbery only moments ago. As you arrive at the scene, you quickly ascertain that everyone is safe. Your next duty is to pursue the suspect. You immediately ask an eye-witness outside the bank who saw the perpetrators make their getaway, ``Which way did they go?'' In response the eye-witness points immediately in the direction taken by the robbers.

    Which way do you go? Your choice should be clear. If the suspect left in the specified direction, your pursuit of the suspect should also be in that direction.

    Euler's method can be analogously explained. Recall that the derivative of a function specifies the function's tangent's slope or the immediate direction of the function. With respect to the original differential equation, equation (5.1), the two pieces of information that you have are (1)  , analogous to the initial location of the suspect (the bank) and (2) y'(t), the direction the suspect is headed.

   Figure: 5.2

    In extending these ideas into a computational approximation of the solution to the differential equation, the approach would be described as: starting at the given initial point (  , go along the tangent line until reaching  , your final location being 

    Formulating the above approach mathematically, a requisite component is the equation of the tangent line to the curve. The tangent to the curve, Y(t), which passes through the point  having the slope  is given in point-slope form to be

Then, the above description of our approach is to travel along the tangent line Y(t) until reaching  , whereby we obtain the numerical approximation

Which, when written for the general interval becomes

readily recognizable as Euler's method.

  
 Higher-Order Methods  Alternative Derivations of Euler's  Derivation via Numerical Integration
Wed Oct 28 11:42:13 EST 1998