We consider the direct numerical solution of the general third order initial value problem of the form

where

It is worth noting that this problem (1) can be modeled from the physical problems such as the thin film flow of a liquid in fluid dynamics, electromagnetic waves and gravity driven flow. Therefore, this type of problem is conventionally solved by reducing it to system of first order ordinary differential equations. [1] and some other authors pointed out that this type of problem can be solved directly to circumvent the inherent setbacks posed by the conventional method, [2] - [7] . These scholars proposed different methods of various degrees of accuracies using no other approximate basis functions other than power series.

[8] - [10] independently showed that the direct solution of the general second order initial value problems can be implemented without the need for predictors or starting values from other methods. In their work, they used power series as approximate solution to derive three-step LMM implemented in block modes. [11] investigated and developed a two-point block method in the form of Adams-Moulton type for solving general second order odes directly using variable step size while [5] and [12] proposed a linear multistep method for the direct solution of initial value problems of ordinary differential equations for special third order initial value problem and a hybrid multistep method to solve third order IVPs of ODEs respectively with constant step size. However, [11] developed a two-point four-step block method with variable step-size. In his work, the method was implemented at two points simultaneously in a block using four backward steps. Moreover, these constant and variable step sizes add little or nothing to the accuracy of the results due to the restriction of interpolation points to the order of the problems.

Recently, [13] and [14] figured out that in search for a method that gives better stability condition, the use of approximate solution which combines power series with exponential function is imperative. Therefore, in this work combination of power series and exponential function was used as basic function in determining a symmetric hybrid linear multistep method for the solution of problem (1) directly.

In this work, we considered using a combination of power series and exponential function in the form

as the basic function for the development of the method, where c and i represent the number of collocation and interpolation points respectively.

The differential system of (2) is given as

The basis function (2) is interpolated at all selected points and the differential system (3) is collocated at only the grid points, which gave rise to a system of equation of the form

where

and

Solving (4) for’s, , using Gaussian elimination method and substituting it back into (2) gives a continuous hybrid method of the form

Using the transformation, the continuous coefficients and their first and second derivatives are obtained as,

Evaluating Equations (6), (7) and (8) at the last end grid point where gives the discrete methods

The order p and error constants of Equations (9), (10) and (11) are, , , and respectively.

The starting values of the discrete method (9) obtained from (5) for third order problem of ordinary differential equations are generated in predictor-corrector mode of the same order of accuracy. The predictor methods and its derivatives of the same order with the corrector method are obtained using the same outlines discussed above to give

The order p and error constants of equations (12), (13) and (14) are, , , and respectively.

Other explicit schemes were developed to evaluate other starting values. Taylor series expansion is adopted for, as and their first and second derivatives up to order.

and

To test the effectiveness and the accuracy of the new method, the method is used to solve three test problems below and the results are shown in Tables 1-3.

Problem 1.

Theoretical solution:

Problem 2.

Theoretical solution:

Problem 3.

Theoretical solution:

Problem 4.

Theoretical solution:

A new two-step symmetric hybrid method of order 5 is proposed for the direct solution of third order differential equations. The maim method and the predictors of same order were derived from the same procedure of collocation and interpolation method. The methods are then applied to on some existing problems and the results were displayed on the Tables 1-4. The errors were compared with those of [5] [7] [12] [15] [17] [18] . It was observed from the tables that the new method displayed better accuracy over the existing methods.

The combination of power series and exponential function collocation method was used to produce a two-step continuous-hybrid method. The method obtained was used to solve some mildly-stiff third order ordinary differential equations. The new method compared favorably in terms of accuracy with the existing methods of higher order and step number. Our future research will be focused on more introductions of grid and off-grid points to enhance global error estimations.

Friday Oghenerukevwe Obarhua,Sunday Jacob Kayode, (2016) Symmetric Hybrid Linear Multistep Method for General Third Order Differential Equations. Open Access Library Journal,03,1-8. doi: 10.4236/oalib.1102583