AN ALGORITHM FOR SOLVING INITIAL VALUE PROBLEMS OF THIRD ORDER ORDINARY DIFFERENTIAL EQUATIONS

We propose an implicit multi-step method for the solution of initial value problems (IVPs) of third order ordinary differential equations (ODE) which does not require reducing the ODE to first order before solving. The development of the method is based on collocation of the differential system and interpolation of the approximate solution at selected grid points. This generates a system of equations, which are then solved using Gaussian elimination method. Three predictors, each of order 5, are also proposed to calculate some starting values in the main method. Analysis of basic properties is considered to guarantee the accuracy of the method. The results for method of step length k = 5 when compared with that of step length k = 4 show a better level of accuracy. KEYWORD: Zero stable, third order IVPs, predictor method, step length. INTRODUCTION We want to consider the solution of the third order IVPs of the form. ( ) ( ) ( ) ( ) ) 1 . 1 ( , , , , , , , , , 2 1 0 − − − − − = = = = R f y a a y a y a y y y y x f y ll l ll l lll ε η η η without reducing the problem into a system of first order IVPs . Problems of the form (1.1) are important for their application in Science and Engineering, especially in Biological Sciences and Control Theory Awoyemi (1996). Some solutions of IVPs of the form (1.1) exists. Awoyemi (1996) considered a two-step method for IVPs of order two. In his contribution Awoyemi (2003) applied the concept of P-stable method to IVPs of order three. Udo et al (2007) opined that using truncation error a linear multistep method can be derived for a second order IVP. According to Awoyemi (1999, 2001), reducing to first order is inefficient due to computational burden and also uneconomical arising from computer time wastage. Bun and Vasil’ Yer (1992) opined that with the reduction approach, we cannot solve equation explicitly with respect to the derivative of the highest order. In consideration of these setbacks, we consider a method that can solve (1.1) without reduction. Eminent scholars have made efforts to solve higher order IVPs especially the special second order differential equations by a number of different methods. Lambert (1973), Enright (1974), Twizell and Khaliq (1984) independently considered the technique of multiderivative methods of solving second order IVPs. They agreed on the fact that multiderivative methods give high accuracy and possess good stability properties when used to solve first order IVPs. However, Awoyemi (1999, 2001) introduced the concept of multiderivative collocation approach for solving directly higher order IVPs. Awoyemi (2003) developed a P-stable method for step length k = 3 for solving third order IVPs. Awoyemi et al (2006) considered a non-symmetric method for a step length of k = 4, also a multiderivative method for third order IVPs. Thus, in this article, we consider a step length of k = 5 for third order IVPs instead of k = 4 to investigate if such an extension will improve the existing result. 123 M. O. Udo, Department Of Mathematics And Statistics, Cross River University Of Technology, Calabar, Nigeria. D. O. Awoyemi, Department Of Mathematical Sciences, Federal University Of Technology, Akure, Ondo State, Nigeria. 124 M. O. UDO AND D. O. AWOYEMI THE METHOD We define an operator L as in Awoyemi (1999) by ∑ −


INTRODUCTION
We want to consider the solution of the third order IVPs of the form.
( without reducing the problem into a system of first order IVPs .Problems of the form (1.1) are important for their application in Science and Engineering, especially in Biological Sciences and Control Theory Awoyemi (1996).Some solutions of IVPs of the form (1.1) exists.Awoyemi (1996) considered a two-step method for IVPs of order two.In his contribution Awoyemi (2003) applied the concept of P-stable method to IVPs of order three.Udo et al (2007) opined that using truncation error a linear multistep method can be derived for a second order IVP.
According to Awoyemi (1999Awoyemi ( , 2001)), reducing to first order is inefficient due to computational burden and also uneconomical arising from computer time wastage.Bun and Vasil' Yer (1992) opined that with the reduction approach, we cannot solve equation explicitly with respect to the derivative of the highest order.
In consideration of these setbacks, we consider a method that can solve (1.1) without reduction.Eminent scholars have made efforts to solve higher order IVPs especially the special second order differential equations by a number of different methods.Lambert (1973), Enright (1974), Twizell and Khaliq (1984) independently considered the technique of multiderivative methods of solving second order IVPs.They agreed on the fact that multiderivative methods give high accuracy and possess good stability properties when used to solve first order IVPs.However, Awoyemi (1999Awoyemi ( , 2001) ) introduced the concept of multiderivative collocation approach for solving directly higher order IVPs.Awoyemi (2003) developed a P-stable method for step length k = 3 for solving third order IVPs.Awoyemi et al (2006) considered a non-symmetric method for a step length of k = 4, also a multiderivative method for third order IVPs.Thus, in this article, we consider a step length of k = 5 for third order IVPs instead of k = 4 to investigate if such an extension will improve the existing result.

THE METHOD
We define an operator L as in Awoyemi (1999) by where k = 5 will represent the step length of the method.
The first and second characteristic polynomials of (2.13) are respectively.We have that Applying definition 2 of Udo et al (2007) we see that (2.13) is consistent.
(2.18)According to Fatunla (1988) and Awoyemi et al (2006), we ignore the imaginary part and evaluate for 0 as the region of stability of (2.13).
Equation (1.1) suggests that we will need the first and second derivatives of (2.13), which are given as We present below a summary of the properties of method (2.13) and those of its first and second derivatives.The analysis of the properties of methods (2.19) and (2.20) has been confirmed and are omitted here because of space.It follows the same process as that of (2.13).

THE PREDICTOR
In developing the predictor, we employ the same collocation procedure adopted for the main method (2.13) which yields the method of the form (2.10).Again putting Evaluating these coefficients at t = 1 gives our predictor for as ( ) The first and second derivations of (3.1) are ( ) The predictors for y n+4 and y n+3 , their respective first and second derivatives which are similarly derived are here listed as The last problem is known as Blasius equation and has no analytical solution, Awoyemi et al (2006).The existence of solution is guaranteed by a theorem in Lambert (1973).

NUMERICAL SOLUTION TO TEST PROBLEMS
Here we present a manual framework for the solution of problem 1 using (2.13).A full manual solution is not only lengthy but tedious.The following starting values ,are needed for the evaluation of (2.13).The first three are our given initial conditions in (4.1a).The rest are gotten by taking a Taylor series of their respective expansions as in equations (3.1) to (3.9c).Setting n = 0 in (2.13) gives (4.2) where Schemes similar to that of (4.2) can be generated by setting n = 1, 2, 3, .in (2.13).Consequently an iterative process with h = 0.1 can be carried out.
The results of the above solved problems using a computer program are here presented.Tables 5.1, 5.2 and 5.3 displays the exact results (YEX), calculated results (YC) and the errors (ER) arising from their difference.We considered the solution at h equals 0.1, 0.05 and 0.025.

CONCLUSION
An implicit zero stable method for the solution of third order IVPs is developed.The order of the main method and those of its predictors are found to be the same a case highly recommended by Fatunla (1988).The accuracy of this method is encouraging judging from the small error values.Three different test problems were considered for the different sizes of h.It was found that as h decreases, the method recorded improved accuracy (tables 5.4 and 5.5).
In Awoyemi et al (2006), the same set of problems were considered.The performance of the step length k = 5 method over that of step length k = 4 was evidently displayed in table 5.6, as results from both methods are compared with the exact value.However, in terms of computer time the k = 4 method has a slight advantage over that of k = 5 because of the number of variables per iteration.Hence, the choice of which method to adopt depends on accuracy and time.
. Udo, Department Of Mathematics And Statistics, Cross River University Of Technology, Calabar, Nigeria.D. O. Awoyemi, Department Of Mathematical Sciences, Federal University Of Technology, Akure, Ondo State, Nigeria.