4.2. Derivation of Multistep Methods by Numerical Integration

Assuming that \(\,y_j\,\), \(\,y_{j+1}\,\) and \(\,y_{j+2}\,\) are known, we aim to determine \(\,y_{j+3}\,\). With this aim we integrate equation (4.1) over the interval \(\,[x_{j+2},x_{j+3}]\,\);

(4.4)\[\begin{split}\begin{aligned} y'~ & = ~ f(x,y) \\ \int_{x_{j+2}}^{x_{j+3}} y' \dx ~&=~ \int_{x_{j+2}}^{x_{j+3}} f(x,y) \dx\\ \biggl[ y(x) \biggr]_{x_{j+2}}^{x_{j+3}} ~&=~ \int_{x_{j+2}}^{x_{j+3}} f(x,y) \dx\\ \therefore\quad y(x_{j+3}) - y(x_{j+2}) ~&=~ \int_{x_{j+2}}^{x_{j+3}} f(x,y) \dx\\ \text{or}\quad y_{j+3} ~&=~ y_{j+2} + \int_{x_{j+2}}^{x_{j+3}} f(x,y) \dx \end{aligned}\end{split}\]

Note

\(y_{j+3}\) and \(y_{j+2}\) are numerical approximations of \(y(x_{j+3})\) and \(y(x_{j+2})\), respectively.

To find the terms on the right hand side, we need to integrate \(\displaystyle \int_{x_{j+2}}^{x_{j+3}} f(x,y)\,\dx\). The straight integration cannot be done since \(y(x)\) is not known. Instead, we integrate by approximating the two-variable function \(\,f(x,y)\,\) as an single-variable interpolating/extrapolating polynomial that is determined by using the previously obtained or given data points (cf. interpolation and their application in numerical integration).