Abstract
Problem statement: The conventional methods of solving higher order differential equations have been by reducing them to systems of first order equations. This approach is cumbersome and increases computational time. Approach: To address this problem, a numerical algorithm for direct solution of 5th order initial value problems in ordinary differential equations (odes), using power series as basis function, is proposed in this research. Collocation of the differential system is taken at selected grid points to reduce the number of functions to be evaluated per iteration. A number of predictors and their derivatives having the same order of accuracy with the main method are proposed. Results: The approach yields a multiderivative method of order six. Numerical examples solved show increased efficiency of the method with increased number of iterations, converging to the theoretical solutions. Conclusion/Recommendations: The new mutiderivative method is efficient to solve linear and nonlinear fifth order odes without reduction to system of lower order equations.
Highlights
In this research, numerical method of solution of higher order differential equations of the form: y(m) = f (x, y, y′,..., ym−1), y(a) = y0, (1)yi (a) = yi,i = 1(1)m −1, m ≥ 2Attempts have been made by some researchers to solve directly Problem (1) for m = 4 by developing methods of step number k = 4 with varying order of accuracy (Awoyemi, 2005; Kayode, 2008b)
The practice of solving this type of problems has been the reduction to systems of first-order equations and the resulting equations solved by applying any suitable method for first order equations (Awoyemi, 2003)
It is extensively discussed that due to the dimension of the problem after it has been reduced to a system of first order equations, the approach waste a lot of human efforts and computer time
Summary
Attempts have been made by some researchers to solve directly Problem (1) for m = 4 by developing methods of step number k = 4 with varying order of accuracy (Awoyemi, 2005; Kayode, 2008b). None of these could handle Problem (1) directly when m>4 without reducing it to a system of lower order problems. Problem (1) is solved directly by developing a 5 step multiderivative method for m = 5. It is extensively discussed that due to the dimension of the problem after it has been reduced to a system of first order equations, the approach waste a lot of human efforts and computer time
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