The focus of this article is to find an approximate solution on a given interval to third order initial value problems (IVP) of the type
where and . In recent time, direct numerical solution of (1) without reduction to equivalent first-order initial value problems (see:      ) has become subject of research by several authors. This method was extensively discussed in (      ) to mention but a few, they developed Linear Multistep Method (LMM) which mode of implementation is Predictor-Corrector form for the solution of initial value problems of ordinary differential equations of the type (1). As reported by   , the major drawback of this approach of implementation is that the methods are not self-starting and thus required the development of predictors which are usually of lower order, hence reducing the accuracy of the methods.
Recently, in order to remove the difficulties usually encountered by adopting this mode of solution, researchers (   -  ) have proposed direct methods other than Predictor-Corrector methods whose modes of implementation are in block-by-block manner which was first introduced by Milne  as a starting step for predictor-corrector. This monumental success has greatly removed the burden of developing predictors and hence resulted in methods of uniform orders that yielded more accurate results. The block-by-block technique has also made it easier to handle the general type of (1) which has been a major concern in the past years.
The quest for numerical methods with better accuracy has also led to the introduction of hybrid linear multistep methods which have recorded high success since its introduction. These successes motivated us to propose a hybrid method with block extension for the solution of (1).
In the next section, we discuss in detail the derivation of the proposed method with its implementation in block mode, followed by analysis of the proposed method to establish the numerical stability, numerical example to demonstrate the efficiency advantages of the proposed method and subsequently. Conclusion was drawn on the performance of the proposed method when applied to solve the numerical examples.
2. Mathematical Formulation
In order to obtain a numerical formula for the approximate solution of (1), the function
is considered as the basis where is continuous within the interval , c and i denote collocation and interpolation points respectively. Variable ’s are coefficients to be determined. The third derivative of (2) equated to (1) is given as
Evaluating (2) at , (3) at using yield the following interpolation and collocation matrix
where , . Solving the matrix Equation (4) for coefficients ’s and substituting into (2) yields after some simplification the continuous method
with the following coefficients:
Evaluating the continuous scheme (5) at and its first and second derivatives at yield two discrete, one first and second derivatives schemes. These can be represented in a block matrix finite difference form as
where T denotes the transpose,
Vectors and are
3. Analysis of the Proposed Method
This section presents analysis of the proposed method (6) vis-a-vis the order, consistency, zero-stability and convergence.
The linear operator associated with the block method (6) is
where is an arbitrary function which is continuously differentiable on . Following Lambert  and Fatunla  , the term in (6) can be written as a Taylor series expansion about the point x to obtain the expression,
where the constant coefficients are given as follows:
Going by Lambert  , the mutistep collocation method (6) has order p if
Therefore is the error constant and is the principal local truncation error at point . The order of the proposed method (6) and the corresponding error constant are as reported in Table 1.
Definition 1 (consistency).
The proposed method (6) is said to be consistent if the order of method is greater than or equal to one, that is if . In addition to
Table 1. Order of accuracy and error constant of the proposed method.
2) where and are 1st and 2nd characteristics polynomial respectively.
Definition 2 (Zero-stability).
The block method (6) is said to be zero-stable if the roots
satisfies and the roots with , the multiplicity must not exceed one. Applying (9) to the proposed method (6) yields the following
This result shows that the method is zero-stable.
Definition 3 (convergence).
The necessary and sufficient condition for the proposed method (6) to be convergent are that it must be consistent and zero-stable according to Dahlquist see . Hence, by definitions 4 and 5 the method is convergent.
Stability Domain of the Proposed Method
In order to study the stability domain of the proposed method (6), the test equations
are applied to the block method (6) with and represents the roots of the first characteristic polynomial of the block method (6). This is then reformulated as a general linear method as discussed in . The partition matrix is expressed in the form
By solving the stability function
yields the polynomial
(16) and its derivatives are then plotted in the MATLAB environment given the stability region displayed in Figure 1.
Definition 4 (Lambert and Watson  ).
Method (6) is P-stable if its interval of periodicity is . It is clearly
Figure 1. This figure depicts the region of absolute stability of the proposed method generated by plotting Equation (16).
shown in Figure 1 that the block method (6) is P-stable.
4. Numerical Example
The first numerical example to be considered is the oscillatory problem
with the theoretical solution
This example was solved by    . The numerical solution, exact solution and absolute error generated by the proposed method when applied to example I are as presented in Table 2. The last column of the Table shows the errors generated by method in  when applied to example I. It is obvious from the table that the proposed method is better in term of accuracy when compared with the method in .
The second example considered is the special third order problem
with the theoretical solution
Source: . The solution curve is shown in Figure 2.
Another example considered is a general third order problem
with the theoretical solution
Table 2. Results of Example I solved with the proposed method.
Figure 2. Solution curve obtained by our method and the exact solution of example II.
The theoretical solution at is . The errors were obtained at using our method at a fixed step-size . The numerical results are compared with those of . For this example, the maximum error was compared with those reported in  in Table 3 for and it was observed that our method perform better. The ROC, computed solutions and maximum error of the proposed method are reported in Table 4. The Table also shows the performance of our method as compared with method in .
General nonlinear third order equation
with the theoretical solution
is also considered. Source: . Figure 3 is the graph of the solution of this problem.
Application to solve nonlinear Genesio equation
The chaotic Genesio equation reported in  given as
where and are the positive constants that satisfied
Table 3. Results of Example III solved with the proposed method.
Table 4. Results of Example III solved with the proposed method.
Figure 3. Solution curve of example IV obtained by the proposed method.
In this work, hybrid method with block extension for the direct solution of third order ordinary differential equations has been proposed. Numerical examples are considered to demonstrate the efficiency advantage of the method especially the Genesio equation which is chaotic in nature. The analysis, stability and numerical examples revealed that the proposed method is efficient for direct solution of third order ordinary differential equations.
Figure 4. Solution curve of the Genesio equation for and for step size within [0, N]: (a) N = 10 and (b) N = 100.
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