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Differential Equations, Finite Difference

These methods fall under the general class of one-step methods. The algoritm is rather simple. Suppose we have an initial value for the function y(t) given by \begin{equation} y_0=y(t=t_0). \tag{10} \end{equation} We are interested in solving a differential equation in a region in space [a,b] . We define a step h by splitting the interval in N sub intervals, so that we have \begin{equation} h=\frac{b-a}{N}. \tag{11} \end{equation} With this step and the derivative of y we can construct the next value of the function y at \begin{equation} y_1=y(t_1=t_0+h), \tag{12} \end{equation} and so forth.