An Order Six Zero-Stable Method for Direct Solution of Fourth Order Ordinary Differential Equations

Abstract: A zero-stable numerical method for direct solution of fourth order differential equations reduces the computational burden and computer time wastage involved in the method of reducing such equations to a system of first order equations. The method adopted is the collocation of the differential system and the interpolation of the approximate solution to the problem using the power series as a basis function. The method is consistent and symmetric with optimal order p = 6. A consistent and symmetric main predictor of order five is also developed for the evaluation of the implicit scheme.


INTRODUCTION
The empirical problems leading to higher order differential equations of the form ... , 2 , 1 , ..., , , (1) are often encountered especially by Scientists and Engineers. The solutions of such equations have engaged the attention of many mathematicians, both the theorists and numerical analysts. Many of such empirical results yielding higher order differential equations are not solvable analytically. Numerical methods adopted for such higher order differential equations are only capable of handling first order equations of type ( This implies that such problems will be reduced to system of first order equations [1], [8], [10] . The approach of reducing such equations to a system of first order equations leads to serious computational burden and wastage in computer time [2], [3] . Many attempts have been made to formulate numerical algorithms capable of solving special problem of type (1) without reducing it to system of first order equations [7], [9], [11], [12] . Multiderivative methods for direct solution of problem (1), taking m = 4, have been considered using canonical polynomial as the basis functions [4], [5] . They produced methods with many function evaluations. Efforts are made to develop a zero-stable order six method of higher step-number with reduced functions evaluation for direct solution of problem (1) for m = 4. The reduced functions evaluation of this method further lessened the computational burden [3], [6] .

METHODS OF SOLUTION
The proposed zero-stable numerical method of order six for direct solution of fourth order differential equations is of the form of a continuous linear multistep Let the approximate solution ) (x y to problem (1) be taken to be a partial sum of a power series ) ( x j ϕ of a single variable The first, second and third order derivatives of (4a) are respectively given as ) ( Equation (5)   Using the transformation [6] ] 1 , 0 ( The first, second and third order derivatives of (9) were computed and given as follows: Fist derivative of (10): For any sample discrete scheme to be determined from the continuous method (9) and its first, second and third derivatives respectively, the values of t could be taken in the interval I = (0, 1]. In this work, the value of t was taken to be 1 to obtain a zero-stable discrete method and its derivatives as Using the local truncation error approach [13] , the order P and error constant C p+2 of method (14) were found to be 6 and 720 respectively.
The discrete method (14) was found to be consistent and zero stable, satisfying the necessary and sufficient conditions for the convergence of Linear Multistep Methods [9], [13] .

IMPLIMENTATION OF THE METHOD
The continuous method (9) was designed to solve general differential equations of type (1) [13] . To implement the sample discrete method (14) and its derivatives, additional x ihy x y ih x y y n n n n n n n i n y y y y are the initial values given in the problems.

SAMPLE TESTS AND RESULTS
The accuracy of the method (14) was tested with two test problems and their results compared with existing method [6] as shown in the tables below. Test Problem 1: Theoretical solution: Test Problem 2:  [6] and New Method (14) for Problem 1 X Exact solution Computed Method (14) Awoyemi and Kayode [6] Error in Method (14) 0  Table 3: Comparison of errors arising from Awoyemi and Kayode [6] and New Method (14) for Problem 2 X Exact solution Computed Method (14) Awoyemi and Kayode [6] Error in Method (14) 0 . The maximum errors arising from the problems when solved with the existing methods [6] for a mesh size h=1/32 were compared with the errors of the new method (14) for the same mesh size as shown in the Tables. CONCLUSION The order six method with three functions evaluation developed through collocation approach is capable of solving linear and non-linear general fourth order ordinary differential equations directly without reduction to system of first order equations. This reduced the computational burden and its inevitable effects on computer time. The method is consistent and zero stable, satisfying the basic requirements for convergence of Linear Multistep methods (LMM). All the predictors and their derivatives are consistent and zero stable. Efforts were made to ensure that the orders of the predictors and their derivatives are close to the order of the method in order to reduce the effects this could have on accuracy. While the order of accuracy of the new method is 6, the orders of the predictors and their derivatives are 5 and 6. The zero stability property and the reduced functions evaluation as well as the comparative high order of the predictors of the new method (14) serve as advantages over the existing method [6] . In spite of its lower order, this comparative advantage probably accounts for better accuracy of the new method as shown in Tables 1 and 2.