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

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 the value of \(y\) is not known. Instead, we integrate by approximating the function \(\,f(x,y)\,\) as an interpolating polynomial that is determined by using the previously obtained or given data points (cf. interpolation and their application in numerical integration).