Exact estimation of the error for numerical solution of linear second order differential equation. (Q2737688)

A linear second order differential equation of the form NEWLINE\[NEWLINE y^{\prime \prime} = - a(x)y, \qquad \tag{1}NEWLINE\]NEWLINE with initial conditions \(y(0)=y_0\), \(y^{\prime}(0)=y^{\prime}_0\), \(a(x) \geq 0\), \(x \in [0, X_0]\) is considered. It is noted that in the case \(a(x) \geq 0\) there is an effect of exponential increasing of the error estimation with increasing \(x\) while the solution has a vibrating character and the error is increasing basically slow.NEWLINENEWLINEA method of finding a correct error estimation of the numerical solution of the differential equation based on a difference scheme is proposed. To get the numerical solution of the problem \(y^{\prime \prime} = f(x,y)\) with \(y(0)=y_0, y^{\prime}(0)=y^{\prime}_0\), with Numerov's method it is necessary to give two initial values \(y_0\) and \(y_1\) so that \(y(0)=y_0\) and \(y_1\) can be found approximately by Taylor's series. After that one can use the formula NEWLINE\[NEWLINE y_m = 2y_{m-1} - y_{m-2} + \frac{h^2}{12}(f(x_m,y_m) +10f(x_{m-1}, y_{m-1} +f(x_{m-2}, y_{m-2})), \quad m \geq 2. NEWLINE\]NEWLINE Two examples of application of the method for an exact estimation of the error of numerical solutions of ordinary differential equations are presented.