Overview of Simple One Step Methods

Explicit

y_{i + 1} = y_{i} + h_{i} f (t_{i}, y_{i})

Implicit

y_{i + 1} = y_{i} + h_{i} f (t_{i + 1}, y_{i + 1})

y_{i + 1} = y_{i} + \frac{h _{i}}{2} [f (t_{i}, y_{i}) + f (t_{i + 1}, y_{i + 1})]

y_{i + 1} = y_{i} + \frac{h _{i}}{2} [f (t_{i}, y_{i}) + f (t_{i + 1}, y_{i} + h_{i} f (t_{i}, y_{i}))]

y_{i + 1} = y_{i} + h_{i} f (t_{i} + \frac{h _{i}}{2}, y_{i} + \frac{h _{i}}{2} f (t_{i}, y_{i}))

q_{1} q_{2} q_{3} q_{4} y_{i + 1} = h f (t_{i}, y_{i}) = h f (t_{i} + \frac{h}{2}, y_{i} + \frac{1}{2} q_{1}) = h f (t_{i} + \frac{h}{2}, y_{i} + \frac{1}{2} q_{2}) = h f (t_{i} + h, y_{i} + q_{3}) = y_{i} + \frac{1}{6} (q_{1} + 2 q_{2} + 2 q_{3} + q_{4})

Ashu's Online Notes