Eulers Method

1. What?

   • The object of Euler’s method is to obtain approximations to the well-posed initial-value problem
     \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \dfrac{dy}{dt} = f(t,y), \ \ \ \ \ \ a \leq b, \ \ \ \ \ \ y(a) = \alpha \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (5.6)\)

   • A continuous approximation to the solution \(y(t)\) will not be obtained; instead, approximations to \(y\) will be generated at various values, called mesh points, in the interval \([a, b]\).

   • Once the approximate solution is obtained at the points, the approximate solution at other points in the interval can be found by interpolation.

2. Mesh-Points:

   \[t_i = a + ih, \ \ \ \text{for each } i = 0, 1, 2, ... , N\]

   The mesh points are equally distributed throughout the interval \([a, b]\).

3. Step-Size:

   \[h = \dfrac{b − a}{N} = t_{i+1} − t\]

4. Euler’s Method:
   

   Equation \((5.8)\) is called the difference equation associated with Euler’s method.

5. Derivation:
   Show Derivation

6. Algorithm:
   Show Algorithm

7. Geometric Interpetation:
   To interpret Euler’s method geometrically, note that when \(w_i\) is a close approximation to \(y(t_i)\), the assumption that the problem is well-posed implies that \(f(t_i, w_i) \approx y(t_i) = f(t_i, y(t_i))\).

   i.e. each step corresponds to correcting the path by the approximation to the derivative (slope).

Error Bounds for Euler’s Method

1. Comparison Lemmas:
   

   1. Lemma 1:
      
      Show proof

   2. Lemma 2:
      
      Show proof

2. Error Bound:
   
   Show proof

3. Properties of the Error Bound Theorem:
   
   1. The Weakness of Theorem 5.9 lies in the requirement that a bound be known for the second derivative of the solution.
   2. However, if \(\dfrac{\partial f}{\partial t}\) and \(\dfrac{\partial f}{\partial y}\) both exist, the chain rule for partial differentiation implies that
      
      So it is at times possible to obtain an error bound for \(y''(t)\) without explicitly knowing \(y(t)\).
   3. The Principal Importance of the error-bound formula given in Theorem 5.9 is that the bound depends linearly on the step size h.
   4. Consequently, diminishing the step size should give correspondingly greater accuracy to the approximations.

Finite Digit Approximations

1. Euler Method [Finite-Digit Approximations]:
   

   Where \(\delta_i\) denotes the round-off error associated with \(u_i\).

2. Error Bound for fin-dig approx. to \(y_i\) given by Euler’s method:
   

3. Properties of the Error Bound on Finit-digit Approximations:
   
   1. The error bound (5.13) is no longer linear in h.
   2. In fact, since
      \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\) \(\lim_{h\to 0} \ (\dfrac{hM}{2} + \dfrac{\delta}{h}) = \infty,\)
      the error would be expected to become large for sufficiently small values of h.
   3. Calculus can be used to determine a lower bound for the step size h:
      

   4. The Minimal value of \(E(h)\) occurs when,

      \[h = \sqrt{\dfrac{2\delta}{M}} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (5.14)\]

   5. Decreasing h beyond this value tends to increase the total error in the approximation; however, normally, \(\delta\) is so small that the lower bound for h doesn’t affect Euler’s Method.