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

Abstract

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.

Keywords:

• Multiderivative method
• Numerical integrator
• Collocation
• Predictor
• Equidistant grid points
• Interval of periodicity
• Error constant