A Multiderivative Collocation Method for 5th Order Ordinary Differential Equations

Problem statement: The conventional methods of solving higher order di fferential 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 pr oposed in this research. Collocation of the differential system is taken at selected grid points to reduce t he number of functions to be evaluated per iteratio n. 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 or der 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.


INTRODUCTION
In this research, numerical method of solution of higher order differential equations of the form: is considered for step number k≥5. This class of Problems (1) has a lot of applications in Science and Engineering, especially in mechanical systems, control theory and celestial mechanics. 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). In the same article, 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.
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). But none of these could handle Problem (1) directly when m>4 without reducing it to a system of lower order problems.
In this article, Problem (1) is solved directly by developing a 5 step multiderivative method for m = 5.

MATERIALS AND METHODS
A power series of a single variable x in the form: is used as trial function to produce the approximate solution as: The 5th derivative of (3) is: Thus from Eq. 1 and 4 we have: k 2 j 5 1v j j 0 j( j 1)( j 2)( j 3)( j 4) x f (x, y, y , y , y , y ) After solving the system of Eq. 7 and 8 for the values of λ j 's and substituting these values into Eq. 3, with the necessary algebraic manipulations, our method is obtained to be: 4 2 5 j n j 2 j 1 n 2 j 1 j 0 j 0 From (8) the coefficients α j (x) and β j (x) are given as follows: The discrete scheme and its first, second, third and fourth derivatives arising from Eq. 9 are as put below when t = 1, which implies that x = x n+4 : n 5 n 4 n 3 n 2 n 1 n 5 n 6 n 3 n 1 y 5y 10y 10y 5y y h [ f 20f 5f ] 24

Predictors:
Except that collocation is not taken at x = x n+5 , the procedure for the development of the main predictor to calculate y n+5 and its required derivatives is the same as for the main method. For y n+j , j = 1(1)4, the predictors are derived by Taylor series method. The discrete schemes and their required derivatives arising from the predictors are as listed below: Where: 1v f  Theoretical solution is y(x) = e x + x 2 .
These results Table 1 are recorded at x = 1 only for n-iterations.
The results Table 1 and 2 were only recorded for Problems (1) and (2) at x = 1 and x = 2 respectively, which showed that as the number of iterations (n) are increasing, the new method is converging to the theoretical solutions.

CONCLUSION
A collocation method with step numbers k = 5 has been proposed to solve Problem (1) when m = 5. Two non-linear test problems were solved by the new method. The results were only recorded for Problems (1) and (2) at x = 1 and x = 2 respectively, which showed that as the number of iterations (n) are increasing, the new method is converging to the theoretical solutions.
In our next article for step number k = 6 we will compare the accuracy of the present method with that one to ascertain whether an odd or even step number method will be better in terms of accuracy for general purpose use.