Algorithmic collocation approach for direct solution of fourth-order initial-value problems of ordinary differential equations

This article discusses a multiderivative collocation method for direct solution of the general initial-value problems of ordinary differential equations of the form y((n))(x) = f(x, y, y', y", ..., y(n-1)), y(a) = y(0), y(i)(a) = y(i), i = 1, 2, 3. To ensure the symmetry of the method, collocation of the differential system has been taken at the selected grid points. Furthermore, a predictor for the calculation of the value of y(n+k) and its derivatives that appear in the main method is developed. Taylor series expansion is used to calculate the values of y(n+i), i = 1, 2, 3 and their derivatives which also appear in the main method. The interval of periodicity and the error constant of the method at x = x(n+k) are calculated. Evaluation of the proposed method at x = x(n+k) gives a particular discrete scheme as a special case of the method. Finally, the efficiency of the method is tested on non-stiff initial-value problems.